A Desingularization of the Main Component of the Moduli Space of Genus-One Stable Maps into ℙ^{n}

Ravi Vakil^{∗} and Aleksey Zinger^{†}

August 23, 2007

Abstract

We construct a natural smooth compactification of the space of smooth genus-one curves with k distinct points in a projective space. It can be viewed as an analogue of a well-known smooth compactification of the space of smooth genus-zero curves, i.e. the space of stable genus-zero maps \overline{𝔐}_{0,k}(ℙ^{n}, d). In fact, our compactification is obtained from the singular space of stable genus-one maps \overline{𝔐}_{1,k}(ℙ^{n}, d) through a natural sequence of blowups along “bad” subvarieties. While this construction is simple to describe, it requires more work to show that the end result is a smooth space. As a bonus, we obtain desingularizations of certain natural sheaves over the “main” irreducible component \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d) of \overline{𝔐}_{1,k}(ℙ^{n}, d). A number of applications of these desingularizations in enumerative geometry and Gromov-Witten theory are described in the introduction, including the second author’s proof of physicists’ predictions for genus-1 Gromov-Witten invariants of a quintic threefold.

Contents

1 Introduction 2 1.1 Background and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Description of the Desingularization . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Outline of the Main Desingularization Construction . . . . . . . . . . . . . . . . . . 12 1.4 Localization Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Blowups of Moduli Spaces of Curves 22 2.1 Blowups and Subvarieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 Moduli Spaces of Genus-One and Zero Curves . . . . . . . . . . . . . . . . . . . . . . 23 2.3 A Blowup of a Moduli Space of Genus-One Curves . . . . . . . . . . . . . . . . . . . 26 2.4 A Blowup of a Moduli Space of Genus-Zero Curves . . . . . . . . . . . . . . . . . . . 29

3 A Blowup of a Moduli Space of Genus-Zero Maps 32 3.1 Blowups and Immersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2 Moduli Spaces of Genus-Zero Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3 Initial Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.4 Inductive Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4 A Blowup of a Moduli Space of Genus-One Maps 46

∗Partially supported by an NSF grant DMS–0228011 †Partially supported by an NSF Postdoctoral Fellowship

4.1 Idealized Blowups and Immersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.3 Inductive Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5 Proof of Theorem 1.2 63 5.1 Pushforwards of Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.2 Construction of Bundle Homomorphism . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.3 Structure of the Cone V_{1,k}^{d} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.4 Desingularization Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

A Most Frequently Used Symbols and Notation 76

1 Introduction

1.1 Background and Applications

The space of degree-d genus-g curves with k distinct marked points in ℙ^{n} is generally not compact, but admits a number of natural compactifications^{1}. Among the most prominent compactifications is the moduli space of stable genus-g maps, \overline{𝔐}_{g,k}(ℙ^{n}, d), constructed in [Gr] and [FuP]. It has found numerous applications in classical enumerative geometry and is a central object in Gromov-Witten theory. However, most applications in enumerative geometry and some results in GW-theory have been restricted to the genus-zero case. The reason for this is essentially that the genus-zero moduli space has a particularly simple structure: it is smooth and contains the space of smooth genus-zero curves as a dense open subset. On the other hand, the moduli spaces of positive-genus stable maps fail to satisfy either of these two properties. In fact, \overline{𝔐}_{g,k}(ℙ^{n}, d) can be arbitrarily singular according to [V2]. It is thus natural to ask whether these failings can be remedied by modifying \overline{𝔐}_{g,k}(ℙ^{n}, d), preferably in a way that leads to a range of applications. As announced in [VZ] and shown in this paper, the answer is yes if g =1.

We denote by 𝔐_{1,k}(ℙ^{n}, d) the subset of \overline{𝔐}_{1,k}(ℙ^{n}, d) consisting of the stable maps that have smooth domains. This space is smooth and contains the space of genus-one curves with k distinct marked points in ℙ^{n} as a dense open subset, provided d ≥ 3. However, 𝔐_{1,k}(ℙ^{n}, d) is not compact. Let \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d) be the closure of 𝔐_{1,k}(ℙ^{n}, d) in the compact space \overline{𝔐}_{1,k}(ℙ^{n}, d). While \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d) is not smooth, it turns out that a natural sequence of blowups along loci disjoint from 𝔐_{1,k}(ℙ^{n}, d) leads to a desingularization of \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d), which will be denoted by 𝔐̃_{1,k}^{0}(ℙ^{n}, d).

The situation is as good as one could possibly hope. A general strategy when attempting to desingularize some space is to blow up the “most degenerate” locus, then the proper transform of the “next most degenerate locus”, and so on. This strategy works here, but with a novel twist: we apply it to the entire space of stable maps \overline{𝔐}_{1,k}(ℙ^{n}, d). The most degenerate locus is in fact an entire irreducible component, and blowing it up removes it^{2}. Hence one by one we erase the “bad” components of \overline{𝔐}_{1,k}(ℙ^{n}, d). Each blowup of course changes the “good” component \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d), and miraculously at the end of the process the resulting space 𝔐̃_{1,k}^{0}(ℙ^{n}, d) is nonsingular. We note that this cannot possibly be true for an arbitrary g, as 𝔐_{g,k}(ℙ^{n}, d) behaves quite badly according to [V2]. The sequential blowup construction itself is beautifully simple. It is completely described in the part of Subsection 1.2 ending with the main theorem of the paper, Theorem 1.1. However, showing that 𝔐̃_{1,k}^{0}(ℙ^{n}, d) is in fact smooth requires a considerable amount of preparation (which takes up Subsections 2.1-4.2) and is finally completed in Subsection 4.3.

1We call a space \overline{𝔐} a compactification of 𝔐 if \overline{𝔐} is compact and contains 𝔐. In particular, 𝔐 need not be dense in \overline{𝔐}.

2Blowing up an irreducible component of a stack will result in the component being removed (or “blown out of existence”), and the remainder of the stack is blown up along its intersection with the component in question.

Since the smooth space 𝔐̃_{1,k}^{0}(ℙ^{n}, d) is obtained from \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d) by blowing up along loci disjoint from 𝔐_{1,k}(ℙ^{n}, d), 𝔐̃_{1,k}^{0}(ℙ^{n}, d) is a smooth compactification of 𝔐_{1,k}(ℙ^{n}, d). One would hope that there is a modular interpretation of this new compactification, and that one could then for example use this interpretation to construct the space 𝔐̃_{1,k}^{0}(ℙ^{n}, d) directly and show that it is smooth. Unfortunately, we have not managed to do this.

The desingularization 𝔐̃_{1,k}^{0}(ℙ^{n}, d) of \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d) possesses a number of “good” properties and has a variety of applications to enumerative algebraic geometry and Gromov-Witten theory. It has already been observed in [Fo] that the cohomology of 𝔐̃_{1,k}^{0}(ℙ^{n}, d) behaves in a certain respect like the cohomology of the moduli space of genus-one curves, \overline{M}_{1,k}. The space 𝔐̃_{1,k}^{0}(ℙ^{n}, d) can be used to count genus-one curves in ℙ^{n}, mimicking the genus-zero results of [KM] and [RT] (though perhaps not their simple recursive formulas). Proceeding analogously to the genus-zero case (e.g. as in [P], [V1], and [Z3]), Theorem 1.1 can then be used to count genus-one curves with tangency conditions and singularities. In all cases, such counts can be expressed as integrals of natural cohomology classes on \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d) or 𝔐̃_{1,k}^{0}(ℙ^{n}, d). Integrals on the latter space can be computed using the localization theorem of [ABo], as 𝔐̃_{1,k}^{0}(ℙ^{n}, d) is smooth and inherits a torus action from ℙ^{n} and \overline{𝔐}_{1,k}(ℙ^{n}, d).

We next discuss two types of applications of Theorem 1.1 in Gromov-Witten theory, as well as a bonus result of this paper, Theorem 1.2. It is shown in [Z4] and [Z6] that the space \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d) has a natural generalization to arbitrary almost Kähler manifolds and gives rise to new symplectic reduced genus-one GW-invariants. These reduced invariants are yet to be constructed in algebraic geometry. However, the spaces 𝔐̃_{1,k}^{0}(ℙ^{n}, d) do possess a number of “good” properties and give rise to algebraic invariants of algebraic manifolds; see the first and last sections of [VZ]. It is not clear whether these are the same as the reduced genus-one invariants, but it may be possible to verify this by using Theorem 1.2.

Theorem 1.1 also has applications to computing Gromov-Witten invariants of complete intersections, once it is combined with Theorem 1.2. Let a be a nonnegative integer. For a general s∈H^{0}(ℙ^{n}, O_{ℙ^{n}}(a)),

Y ≡ s^{−1}(0) ⊂ ℙ^{n}

is a smooth hypersurface. We denote its degree-d GW-invariant by GW_{g,k}^{Y}(d; ·), i.e.

GW_{g,k}^{Y}(d; ψ) ≡ ⟨ψ, [\overline{𝔐}_{g,k}(Y, d)]^{vir}⟩  for all  ψ ∈H^{∗}(\overline{𝔐}_{g,k}(Y, d); ℚ).

Suppose 𝔘 is the universal curve over \overline{𝔐}_{g,k}(ℙ^{n}, d), with structure map π and evaluation map ev:

\xymatrix{𝔘 \ar[d]^{π} \ar[r]^{ev} & ℙ^{n} \\ \overline{𝔐}_{g,k}(ℙ^{n}, d).}

It can be shown that

GW_{0,k}^{Y}(d; ψ) = ⟨ψ · e(π_{∗}ev^{∗}O_{ℙ^{n}}(a)), [\overline{𝔐}_{0,k}(ℙ^{n}, d)]⟩  (1.1)

for all ψ ∈ H^{∗}(\overline{𝔐}_{0,k}(ℙ^{n}, d); ℚ); see [Bea] for example. The moduli space \overline{𝔐}_{0,k}(ℙ^{n}, d) is a smooth orbifold and

π_{∗}ev^{∗}O_{ℙ^{n}}(a) ⟶ \overline{𝔐}_{0,k}(ℙ^{n}, d)

is a locally free sheaf, i.e. a vector bundle. The right-hand side of (1.1) can be computed via the classical localization theorem of [ABo]. The complexity of this computation increases quickly with the degree d, but it has been completed in full generality in a number of different ways; see [Ber], [Ga], [Gi], [Le], and [LLY].

If n=4, so Y is a threefold, then

GW_{1,k}^{Y}(d; ψ) = \frac{d(a−5)+2}{24}GW_{0,k}^{Y}(d; ψ) + ⟨ψ · e(π_{∗}ev^{∗}O_{ℙ^{n}}(a)), [\overline{𝔐}_{1,k}^{0}(ℙ^{4}, d)]⟩  (1.2)

for all ψ ∈ H^{∗}(\overline{𝔐}_{1,k}(ℙ^{4}, d); ℚ); see [LZ, (1.5) and (1.12)]. This decomposition generalizes to arbitrary complete intersections Y and perhaps even to higher-genus invariants. The sheaf

π_{∗}ev^{∗}O_{ℙ^{n}}(a) ⟶ \overline{𝔐}_{1,k}^{0}(ℙ^{4}, d)  (1.3)

is not locally free. Nevertheless, its euler class is well-defined: the euler class of every desingularization of this sheaf is the same, in the sense of [Z5, Subsect. 1.2]. This euler class can be geometrically interpreted as the zero set of a sufficiently good section of the cone

V_{1,k}^{d} ⟶ \overline{𝔐}_{1,k}^{0}(ℙ^{4}, d),

naturally associated to the sheaf (1.3)^{3}; see the second part of the next subsection and Lemma 5.1.

One would hope to compute the last expression in (1.2) by localization. However, since the variety \overline{𝔐}_{1,k}^{0}(ℙ^{4}, d) and the cone V_{1,k}^{d} are singular, the localization theorem of [ABo] is not immediately applicable in the given situation. Let

π̃: 𝔐̃_{1,k}^{0}(ℙ^{4}, d) ⟶ \overline{𝔐}_{1,k}^{0}(ℙ^{4}, d)

be the projection map. As a straightforward extension of the main desingularization construction of this paper, we show that the cone

π̃^{∗}V_{1,k}^{d} ⟶ 𝔐̃_{1,k}^{0}(ℙ^{4}, d)

3V_{1,k}^{d} is a variety such that the fibers of the projection map to \overline{𝔐}_{1,k}^{0}(ℙ^{4}, d) are vector spaces, but not necessarily of the same dimension.

contains a vector bundle

Ṽ_{1,k}^{d} ⟶ 𝔐̃_{1,k}^{0}(ℙ^{4}, d)

of rank da = rk V_{1,k}^{d}|_{𝔐_{1,k}^{0}(ℙ^{4},d)}; see Theorem 1.2. It then follows that

\begin{aligned}⟨ψ · e(π_{∗}ev^{∗}O_{ℙ^{n}}(a)), [\overline{𝔐}_{1,k}^{0}(ℙ^{4}, d)]⟩ & ≡ ⟨ψ · e(V_{1,k}^{d}), [\overline{𝔐}_{1,k}^{0}(ℙ^{4}, d)]⟩ \\ & = ⟨π̃^{∗}ψ · e(Ṽ_{1,k}^{d}), [𝔐̃_{1,k}^{0}(ℙ^{4}, d)]⟩. \end{aligned}  (1.4)

The last expression above is computable by localization. In fact, it is computed explicitly in [Z8], confirming the prediction of [BCOV] for genus-one GW-invariants of a quintic threefold.

Remark: Another approach to computing positive-genus GW-invariants has been proposed in [MaP]. In contrast to the approach of [LZ], it applies to arbitrary-genus invariants, but can at present be used to compute invariants of only low-dimensional and/or low-degree complete intersections.

The main desingularization construction of this paper is the subject of Section 4, but its key aspects are presented in the next subsection. The construction itself and its connections with Sections 2 and 3 are outlined in Subsection 1.3. We suggest that the reader return to Subsections 1.2 and 1.3 before going through the technical details of the blowup constructions in Sections 2-4. In the next subsection, we also describe a natural sheaf over 𝔐̃_{1,k}^{0}(ℙ^{n}, d) which is closely related to the sheaf

π_{∗}ev^{∗}O_{ℙ^{n}}(a) over \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d). It is shown to be locally free in Section 5. Finally, all the data necessary for applying the localization theorem of [ABo] to 𝔐̃_{1,k}^{0}(ℙ^{n}, d) and e(Ṽ_{1,k}^{d}) is given in Subsection 1.4. In Appendix A, we list the most commonly used notation.

Throughout this article we work with Deligne-Mumford stacks. They can also be thought of as analytic orbivarieties. As we work with reduced scheme structures throughout the paper, we will call such objects simply varieties. Also, all immersions will be assumed to be from smooth varieties.^{4}

The authors would like to thank Jun Li for many enlightening discussions and the referee for a timely response with many detailed suggestions.

1.2 Description of the Desingularization

The moduli space \overline{𝔐}_{1,k}(ℙ^{n}, d) has irreducible components of various dimensions. One of these components is \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d), the closure of the stratum 𝔐_{1,k}(ℙ^{n}, d) of stable maps with smooth domains. We now describe natural subvarieties of \overline{𝔐}_{1,k}(ℙ^{n}, d)^{5} which contain the remaining components of \overline{𝔐}_{1,k}(ℙ^{n}, d). They will be indexed by the set

A_{1}(d, k) ≡ {σ =(m; J_{P} , J_{B}): m∈ℤ^{+}, m≤d; [k]=J_{P} ⊔J_{B}},
where [k] = {1, …, k}.

For each σ ∈ A_{1}(d, k), let 𝔐_{1,σ}(ℙ^{n}, d) be the subset of \overline{𝔐}_{1,k}(ℙ^{n}, d) consisting of the stable maps [C, u] such that C is a smooth genus-one curve E with m smooth rational components attached

4The notion of “immersion” is often called “unramified” in algebraic geometry.

5In fact, these will be substacks of the stack \overline{𝔐}_{1,k}(ℙ^{n}, d). They can also be thought of as analytic sub-orbivarieties of the analytic orbivariety \overline{𝔐}_{1,k}(ℙ^{n}, d). As we work with reduced scheme structures throughout the paper, we will call such objects simply varieties.

2

1

d_{1}

d_{2}

d_{3}

2

(1, 0)

(0, d_{1})
(0, d_{2})

1 (0, d_{3})

σ =(3; {2}, {1})
d_{1}+d_{2}+d_{3} =d
d_{1}, d_{2}, d_{3} >0

Figure 1: The domain of an element of 𝔐_{1,σ}(ℙ^{n}, d)

directly to E, u|_{E} is constant, the restriction of u to each rational component is non-constant, and the marked points on E are indexed by the set J_{P}. Here P stands for “principal component”, B stands for “bubble component”, and A stands for “admissible set”. Figure 1 shows the domain of an element of 𝔐_{1,σ}(ℙ^{n}, d), where σ = (3; {2}, {1}), from the points of view of symplectic topology and of algebraic geometry. In the first diagram, each shaded disc represents a sphere; the integer next to each rational component C_{i} indicates the degree of u|_{C_{i}}. In the second diagram, the components of C are represented by curves, and the pair of integers next to each component C_{i} shows the genus of C_{i} and the degree of u|_{C_{i}}. In both diagrams, the marked points are labeled in bold face. Let \overline{𝔐}_{1,σ}(ℙ^{n}, d) be the closure of 𝔐_{1,σ}(ℙ^{n}, d) in \overline{𝔐}_{1,k}(ℙ^{n}, d). The space \overline{𝔐}_{1,σ}(ℙ^{n}, d) has a number of irreducible components. These components are indexed by the splittings of the degree d into m positive integers and of the set J_{B} into m subsets. However, we do not need to distinguish these components.

It is straightforward to check that

\overline{𝔐}_{1,k}(ℙ^{n}, d) = \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d) ∪ ⋃_{σ∈A_{1}(d,k)} \overline{𝔐}_{1,σ}(ℙ^{n}, d).

Dimensional considerations imply that if σ = (m; J_{P} , J_{B}) ∈ A_{1}(d, k) and m ≤ n, then \overline{𝔐}_{1,σ}(ℙ^{n}, d) is a union of components of \overline{𝔐}_{1,k}(ℙ^{n}, d). The converse holds as well: \overline{𝔐}_{1,σ}(ℙ^{n}, d) is contained in \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d) if m > n by [Z4, Theorem 2.3]. However, we will use the entire collection A_{1}(d, k) of subvarieties of \overline{𝔐}_{1,k}(ℙ^{n}, d) to construct 𝔐̃_{1,k}^{0}(ℙ^{n}, d). The independence of the indexing set A_{1}(d, k) of n leads to a number of good properties being satisfied by our blowup construction; see (2) of Theorem 1.1 and the second part of this subsection. It may also be possible to use this construction to define reduced genus-one GW-invariants in algebraic geometry; this is achieved in symplectic topology in [Z6].

We define a partial ordering ≺ on the set A_{1}(d, k) by

σ′ ≡(m′; J′_{P} , J′_{B}) ≺ σ ≡(m; J_{P} , J_{B})  if σ′ ≠σ, m′ ≤m, and J′_{P} ⊆J_{P} .  (1.5)

This relation is illustrated in Figure 2, where an element σ of A_{1}(d, k) is represented by an element of the corresponding space 𝔐_{1,σ}(ℙ^{n}, d). We indicate that the degree of the stable map on every bubble component is positive by shading the disks in the figure. We show only the marked points lying on the principal component. The exact distribution of the remaining marked points between the components is irrelevant.

≺  ^{2}

Figure 2: Examples of partial ordering (1.5)

Choose an ordering < on A_{1}(d, k) extending the partial ordering ≺. The desingularization

π̃: 𝔐̃_{1,k}^{0}(ℙ^{n}, d) ⟶ \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d)

is constructed by blowing up \overline{𝔐}_{1,k}(ℙ^{n}, d) along the subvarieties \overline{𝔐}_{1,σ}(ℙ^{n}, d) and their proper transforms in the order specified by <. In other words, we first blow up \overline{𝔐}_{1,k}(ℙ^{n}, d) along \overline{𝔐}_{1,σ_{min}}(ℙ^{n}, d), where

σ_{min} ≡ (1; ∅, [k])

is the smallest element of A_{1}(d, k). We then blow up the resulting space along the proper transform of \overline{𝔐}_{1,σ_{2}}(ℙ^{n}, d), where σ_{2} is the smallest element of A_{1}(d, k)−\{σ_{min}\}. We continue this procedure until we blow up along the proper transform of \overline{𝔐}_{1,σ_{max}}(ℙ^{n}, d), where

σ_{max} = (d; [k], ∅)

is the largest element of A_{1}(d, k). The variety resulting from this last blowup is the proper transform 𝔐̃_{1,k}^{0}(ℙ^{n}, d) of \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d), as all other irreducible components of \overline{𝔐}_{1,k}(ℙ^{n}, d) have been “blown out of existence”.

The first interesting case of this construction, i.e. for \overline{𝔐}_{1,0}^{0}(ℙ^{2}, 3), is described in detail in [VZ]. The space 𝔐̃_{1,0}^{0}(ℙ^{2}; 3) is a smooth compactification of the space of smooth plane cubics. It has a richer structure than the naive compactification, ℙ^{9}, does.

Theorem 1.1 Suppose n, d∈ℤ^{+}, k ∈ℤ̄^{+}, < is an ordering on the set A_{1}(d, k) extending the partial ordering ≺, and

π̃: 𝔐̃_{1,k}^{0}(ℙ^{n}, d) ⟶ \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d)

is the blowup of \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d) obtained by blowing up \overline{𝔐}_{1,k}(ℙ^{n}, d) along the subvarieties \overline{𝔐}_{1,σ}(ℙ^{n}, d) and their proper transforms in the order specified by <.

(1) The variety 𝔐̃_{1,k}^{0}(ℙ^{n}, d) is smooth and is independent of the choice of ordering < extending ≺.

(2) For all m≤n, the embedding \overline{𝔐}_{1,k}^{0}(ℙ^{m}, d)⟶\overline{𝔐}_{1,k}^{0}(ℙ^{n}, d) lifts to an embedding

𝔐̃_{1,k}^{0}(ℙ^{m}, d) ⟶ 𝔐̃_{1,k}^{0}(ℙ^{n}, d)

and the image of the latter embedding is the preimage of \overline{𝔐}_{1,k}^{0}(ℙ^{m}, d) under π̃.

(3) The blowup locus at every step of the blowup construction is a smooth subvariety in the corresponding blowup of \overline{𝔐}_{1,k}(ℙ^{n}, d).

(4) All fibers of π̃ are connected.

Remark: While in Section 4 we analyze the blowup construction starting with the reduced scheme structure on \overline{𝔐}_{1,k}(ℙ^{n}, d), Theorem 1.1 applies to the standard scheme structure on \overline{𝔐}_{1,k}(ℙ^{n}, d) as well. It is known that the space

𝔐_{1,k}^{eff}(ℙ^{n}, d) ≡ \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d) − ⋃_{σ∈A_{1}(d,k)} \overline{𝔐}_{1,σ}(ℙ^{n}, d),

consisting of stable maps with no contracted genus-one component, is a smooth stack (as such maps are unobstructed, see for example [V1, Prop. 5.5(c)]). Thus, its scheme-theoretic closure, \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d), is reduced. During the blowup process all other components of \overline{𝔐}_{1,k}(ℙ^{n}, d) are “blown out of existence”, as is any non-reduced scheme structure.

In Theorem 1.1 and throughout the rest of the paper we denote by ℤ̄^{+} the set of nonnegative integers. We analyze the sequential blowup construction of Theorem 1.1 in Section 4 using the inductive assumptions (I1)-(I15) of Subsection 4.3. One of these assumptions, (I3), implies the second part of the first statement of Theorem 1.1, as different choices of an ordering < extending the partial ordering ≺ correspond to different orders of blowups along subvarieties that are disjoint. For example, suppose

σ_{∅} = (2; ∅, {1, 2}), σ_{1} = (2; {1}, {2}), and σ_{2} = (2; {2}, {1}).

While \overline{𝔐}_{1,σ_{1}}(ℙ^{n}, d) and \overline{𝔐}_{1,σ_{2}}(ℙ^{n}, d) do intersect in \overline{𝔐}_{1,2}(ℙ^{n}, d), their proper transforms are disjoint after the blowup along \overline{𝔐}_{1,σ_{∅}}(ℙ^{n}, d). The second statement of Theorem 1.1 follows immediately from the description of the blowup construction in this and the next subsections, as each step of the construction commutes with the embeddings of the moduli spaces induced by the embedding ℙ^{m} ⟶ℙ^{n}.

The main claim of this paper is that 𝔐̃_{1,k}^{0}(ℙ^{n}, d) is a smooth variety. The structure of 𝔐_{1,k}^{eff}(ℙ^{n}, d) is well understood; see [V1, Prop. 5.5(c)] for example. In particular, 𝔐_{1,k}^{eff}(ℙ^{n}, d) is smooth. Below we describe the structure of the complement ∂𝔐̃_{1,k}^{0}(ℙ^{n}, d) of 𝔐_{1,k}^{eff}(ℙ^{n}, d) in 𝔐̃_{1,k}^{0}(ℙ^{n}, d).

If J is a finite set and g is a nonnegative integer, we denote by \overline{M}_{g,J} the moduli space of stable genus-g curves with |J| marked points, which are indexed by the set J. Similarly, we denote by \overline{𝔐}_{g,J}(ℙ^{n}, d) the moduli space of stable maps from genus-g curves with marked points indexed by J to ℙ^{n}. If j ∈J, let

ev_{j}: \overline{𝔐}_{g,J}(ℙ^{n}, d) ⟶ ℙ^{n}

be the evaluation map at the marked point labeled by j.

If σ =(m; J_{P} , J_{B}) is an element of A_{1}(d, k), we define

\overline{M}_{σ;P} = \overline{M}_{1,[m]⊔J_{P}}  and
\overline{𝔐}_{σ;B}(ℙ^{n}, d) = \{(b_{1}, …, b_{m}) ∈ ∏_{i=1}^{m} \overline{𝔐}_{0,\{0\}⊔J_{i}}(ℙ^{n}, d_{i}): d_{i} >0, ∑_{i=1}^{m} d_{i} =d; ⨆_{i=1}^{m} J_{i} =J_{B};
ev_{0}(b_{i_{1}})=ev_{0}(b_{i_{2}}) ∀ i_{1}, i_{2} ∈[m]}.

d_{1} d_{2}  d_{3} d_{4}  (1, 0) (0, 0) (0, 0)  (0, d_{1}) (0, d_{2}) (0, d_{3}) (0, d_{4})  \begin{matrix}σ =(3; ∅, ∅) \\ d_{1}+d_{2}+d_{3}+d_{4} =d \\ d_{1}, d_{2}, d_{3}, d_{4} >0 \end{matrix}

Figure 3: A point in \overline{𝔐}_{1,σ}(ℙ^{n}, d)⊂\overline{𝔐}_{1,0}(ℙ^{n}, d) with two preimages under ῑ_{σ}

There is a natural node-identifying surjective immersion

ι_{σ}: \overline{M}_{σ;P} ×\overline{𝔐}_{σ;B}(ℙ^{n}, d) ⟶ \overline{𝔐}_{1,σ}(ℙ^{n}, d) ⊂ \overline{𝔐}_{1,k}(ℙ^{n}, d).

As before, P denotes “principal component”, and B denotes “bubble components”. This immersion descends to the quotient:

ῑ_{σ}: (\overline{M}_{σ;P} ×\overline{𝔐}_{σ;B}(ℙ^{n}, d))/G_{σ} ⟶ \overline{𝔐}_{1,σ}(ℙ^{n}, d),

where G_{σ} ≡S_{m} is the symmetric group on m elements. If m≥3, ῑ_{σ} is not an isomorphism as some subvarieties of the left side are identified. An example of a point on the right which is the image of two points on the left is given in Figure 3. In addition to the conventions used in Figure 1, in the first, symplectic-topology, diagram of Figure 3 we leave the components of the domain on which the map is constant unshaded. The subvarieties identified by the map ῑ_{σ} get “unidentified” after taking the proper transform of \overline{𝔐}_{1,σ}(ℙ^{n}, d) in the blowup of \overline{𝔐}_{1,k}(ℙ^{n}, d) at the step corresponding to

σ̃ = max {σ′ ∈A_{1}(d, k): σ′ ≺σ}.

This is insured by the inductive assumption (I13) in Subsection 4.3 and implies the third statement of Theorem 1.1. For example, if m = 3 and k = 0 as in Figure 3, the “identified” subvarieties are “unidentified” after the blowup of the proper transform of \overline{𝔐}_{1,(2;∅,∅)}(ℙ^{n}, d).

Remark: Throughout the paper, we use 𝔐 (fraktur font) to denote moduli spaces of stable maps, of genus-zero or one, into ℙ^{n}. We use M (calligraphic font) to denote moduli spaces of stable curves.

For each i∈[m], let

π_{i}: \overline{𝔐}_{σ;B}(ℙ^{n}, d) ⟶ ⨆_{d_{i}>0,J_{i}⊂J_{B}} \overline{𝔐}_{0,\{0\}⊔J_{i}}(ℙ^{n}, d_{i})

be the natural projection onto the ith component. We put

F_{σ;B} = ⨁_{i=1}^{m} π_{i}^{∗}L_{0},

where L_{0} ⟶ \overline{𝔐}_{0,\{0\}⊔J_{i}}(ℙ^{n}, d_{i}) is the universal tangent line bundle for the marked point 0. In Subsections 2.3 and 3.4, we construct blowups

π̃_{σ;P} ≡π_{1,([m],J_{P})} : M̃_{σ;P} ≡M̃_{1,([m],J_{P})} ⟶ \overline{M}_{σ;P} ≡\overline{M}_{1,[m]⊔J_{P}}  and
π̃_{σ;B} ≡π_{0,([m],J_{B})} : 𝔐̃_{σ;B}(ℙ^{n}, d)≡𝔐̃_{0,([m],J_{B})}(ℙ^{n}, d) ⟶ ℙF_{σ;B} ≡ℙF_{([m],J_{B})}.

M̃_{σ;P} × Z̃_{σ;B}(ℙ^{n}, d)  𝔐̃_{1,k}^{0}(ℙ^{n}, d)

\overline{M}_{σ;P} × \overline{𝔐}_{σ;B}(ℙ^{n}, d) \overline{𝔐}_{1,k}(ℙ^{n}, d)

ι̃_{σ}

ι_{σ}

π̃_{σ;P} π̃_{σ;B} π̃

Figure 4: Changes in the boundary structure of \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d) under the desingularization

We also construct a section

D̃_{σ;B} ≡D̃_{([m],J_{B})} ∈ Γ(𝔐̃_{σ;B}(ℙ^{n}, d); 𝔼_{σ;B}^{∗} ⊗π̃_{σ;B}^{∗}π_{ℙF_{σ;B}}^{∗}ev_{0}^{∗}T ℙ^{n}),  (1.6)

where

ev_{0} : \overline{𝔐}_{σ;B}(ℙ^{n}, d) ⟶ ℙ^{n}  and  π_{ℙF_{σ;B}}: ℙF_{σ;B} ⟶ \overline{𝔐}_{σ;B}(ℙ^{n}, d)

are the natural evaluation map and the bundle projection map, respectively, and

𝔼_{σ;B} ≡𝔼̃ ⟶ 𝔐̃_{σ;B}(ℙ^{n}, d)≡𝔐̃_{0,([m],J_{B})}(ℙ^{n}, d)

is a line bundle. This line bundle is the sum of the tautological line bundle

γ_{σ;B} ⟶ ℙF_{σ;B}

and all exceptional divisors. The section D̃_{σ;B} is transverse to the zero section. Thus, its zero set,

Z̃_{σ;B}(ℙ^{n}, d) ≡ D̃_{σ;B}^{−1}(0) ⊂ 𝔐̃_{σ;B}(ℙ^{n}, d),  (1.7)

is a smooth subvariety. The boundary ∂𝔐̃_{1,k}^{0}(ℙ^{n}, d) of 𝔐̃_{1,k}^{0}(ℙ^{n}, d) is a union of smooth divisors:

∂𝔐̃_{1,k}^{0}(ℙ^{n}, d) = ⋃_{σ∈A_{1}(d,k)} Z̃_{σ}(ℙ^{n}, d)/G_{σ}, where Z̃_{σ}(ℙ^{n}, d) = M̃_{σ;P} ×Z̃_{σ;B}(ℙ^{n}, d);

see the inductive assumptions (I7) and (I8) in Subsection 4.3 and Figure 4. By the inductive assumption (I6) and (I7), the normal bundle of Z̃_{σ}(ℙ^{n}, d) in 𝔐̃_{1,k}(ℙ^{n}, d) is the quotient of the line bundle

𝕃_{σ;P} ⊗ π̃_{σ;B}^{∗}γ_{σ;B} ⟶ M̃_{σ;P} ×Z̃_{σ;B}(ℙ^{n}, d)

by the G_{σ}-action, where

𝕃_{σ;P} ≡𝕃 ⟶ M̃_{σ;P} ≡M̃_{1,([m],J_{P})}

is the universal tangent line bundle constructed in Subsection 2.3. Thus we conclude that 𝔐̃_{1,k}^{0}(ℙ^{n}, d) is smooth, as the open subset 𝔐_{1,k}^{eff}(ℙ^{n}, d) is smooth, and its complement is a union of smooth divisors whose normal sheaves are line bundles (i.e. with their reduced induced scheme structure, they are Cartier divisors).

Remark 1: In the Gromov-Witten theory, the symbol 𝔼 is commonly used to denote the Hodge vector bundle of holomorphic differentials. It is the zero vector bundle in the genus-zero case.

The line bundles over moduli spaces of genus-zero curves and maps we denote by 𝔼, with various decorations, play roles analogous to that of the Hodge line bundle over moduli spaces of genus-one curves. The most overt parallel is described at the end of Subsection 2.2. There are deeper, more subtle, connections as well; compare the structural descriptions of Lemmas 3.8 and 4.10, for example.

Remark 2: Throughout this paper, the symbols D and 𝔇, with various decorations, denote vector bundle sections related to derivatives of holomorphic maps into ℙ^{n} and of holomorphic bundle sections. In most cases, such bundle sections are viewed as vector bundle homomorphisms.

The final claim of Theorem 1.1 follows from the fact that \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d) is unibranch (locally irreducible). If π: Y ⟶ X is a surjective birational map of irreducible varieties, and π^{−1}(x) is not connected for some x∈X, then X is not unibranch at x.

We next describe a desingularization of the sheaf π_{∗}ev^{∗}O_{ℙ^{n}}(a) and of the corresponding cone V_{1,k}^{d}

over \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d). Let 𝔘̃=π̃^{∗}𝔘 be the pullback of 𝔘 by π̃:

\xymatrix{𝔘̃ \ar[d]^{π} \ar[r]^{π̃} & 𝔘 \ar[d]^{π} \ar[r]^{ev} & ℙ^{n} \\ 𝔐̃_{1,k}^{0}(ℙ^{n}, d) \ar[r]^{π̃} & \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d).}

For each σ ∈A_{1}(d, k), let

V_{σ;B} ⟶ \overline{𝔐}_{σ;B}(ℙ^{n}, d)

be the cone induced by the sheaf O_{ℙ^{n}}(a), similarly to V_{g,k}^{d}; see Subsection 5.2 for details. It is a vector bundle of rank da+1. We note that

π̃^{∗}V_{1,k}^{d}|_{Z̃_{σ}(ℙ^{n},d)} = π_{B}^{∗}{π̃_{σ;B}^{∗}π_{ℙF_{σ;B}}^{∗}V_{σ;B}|_{Z̃_{σ;B}(ℙ^{n},d)}}/G_{σ},

where

π_{B}: M̃_{σ;P} ×Z̃_{σ;B}(ℙ^{n}, d) ⟶ Z̃_{σ;B}(ℙ^{n}, d)

is the projection map. Let L=γ^{∗⊗a}, where γ ⟶ℙ^{n} is the tautological line bundle.

Theorem 1.2 Suppose d, n, a∈ℤ^{+} and k ∈ℤ̄^{+}.

(1) The sheaf π_{∗}π̃^{∗}ev^{∗}O_{ℙ^{n}}(a) over 𝔐̃_{1,k}^{0}(ℙ^{n}, d) is locally free and of the expected rank, i.e. da.

(2) If Ṽ_{1,k}^{d} ⊂π̃^{∗}V_{1,k}^{d} is the corresponding vector bundle and σ ∈A_{1}(d, k), then there exists a surjective bundle homomorphism

𝔇̃_{σ;B}: π̃_{σ;B}^{∗}π_{ℙF_{σ;B}}^{∗}V_{σ;B}|_{Z̃_{σ;B}(ℙ^{n},d)} ⟶ 𝔼_{σ;B}^{∗} ⊗π̃_{σ;B}^{∗}π_{ℙF_{σ;B}}^{∗}ev_{0}^{∗}L

over Z̃_{σ;B}(ℙ^{n}, d) such that

Ṽ_{1,k}^{d}|_{Z̃_{σ}(ℙ^{n},d)} = (π_{B}^{∗} ker 𝔇̃_{σ;B})/G_{σ}.

(3) π̃_{∗}π_{∗}π̃^{∗}ev^{∗}O_{ℙ^{n}}(a) = π_{∗}ev^{∗}O_{ℙ^{n}}(a) over \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d).

We prove the first two statements of this theorem by working with the cone

p: \overline{𝔐}_{1,k}(L, d) ⟶ \overline{𝔐}_{1,k}(ℙ^{n}, d).

The sheaves π_{∗}ev^{∗}O_{ℙ^{n}}(a) and π_{∗}π̃^{∗}ev^{∗}O_{ℙ^{n}}(a) are the sheaves of (holomorphic) sections of

V_{1,k}^{d} ≡ \overline{𝔐}_{1,k}(L, d)|_{\overline{𝔐}_{1,k}^{0}(ℙ^{n},d)} ⟶ \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d)

and π̃^{∗}V_{1,k}^{d}, respectively; see Lemma 5.1. In Subsection 5.4, we lift the blowup construction of Subsection 3.4 to \overline{𝔐}_{1,k}(L, d). In particular, we blow up \overline{𝔐}_{1,k}(L, d) along the subvarieties

\overline{𝔐}_{1,σ}(L, d) = p^{−1}(\overline{𝔐}_{1,σ}(ℙ^{n}, d)), σ ∈ A_{1}(d, k),

and their proper transforms. The end result of this construction, which we denote by 𝔐̃_{1,k}^{0}(L, d),

is smooth for essentially the same reasons that \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d) is. The only additional input we need is Lemma 5.7, which is a restatement of the key result concerning the structure of the cone V_{1,k}^{d} obtained in [Z5]. The bundle

p̃: 𝔐̃_{1,k}^{0}(L, d) ⟶ 𝔐̃_{1,k}^{0}(ℙ^{n}, d)

of vector spaces of the same rank contains 𝔐̃_{1,k}^{0}(ℙ^{n}, d) as the zero section. Thus, p̃ is a vector bundle. There is a natural inclusion

𝔐̃_{1,k}^{0}(L, d) ⟶ π̃^{∗}\overline{𝔐}_{1,k}(L, d).

All sections of π̃^{∗}\overline{𝔐}_{1,k}(L, d) must in fact be sections of 𝔐̃_{1,k}^{0}(L, d) and thus the sheaf π_{∗}π̃^{∗}ev^{∗}O_{ℙ^{n}}(a) is indeed locally free. The bundle map

𝔇̃_{σ;B} ≡ 𝔇̃_{([m],J_{B})}

of the second statement of Theorem 1.2 is described in Subsection 5.2. It is the “vertical” part of the natural extension of the bundle map D̃_{σ;B} from stable maps into ℙ^{n} to stable maps into L. Finally, the last statement of Theorem 1.2 is a consequence of the last statement of Theorem 1.1; see Lemma 5.2. At this point, this observation does not appear to have any applications though.

Remark: By applying the methods of Section 5 and of [Z5], it should be possible to show that the standard scheme structure on \overline{𝔐}_{1,k}(ℙ^{n}, d) is in fact reduced.

1.3 Outline of the Main Desingularization Construction

The main blowup construction of this paper is contained in Subsections 4.2 and 4.3. It is a sequence of idealized blowups along smooth subvarieties. In other words, the blowup locus \overline{𝔐}_{1,σ}^{σ−1} at each step comes with an idealized normal bundle N^{ide}. It is a vector bundle (of the smallest possible rank) containing the normal cone N for \overline{𝔐}_{1,σ}^{σ−1}. After taking the usual blowup of the ambient space along \overline{𝔐}_{1,σ}^{σ−1}, we attach the idealized exceptional divisor

E^{ide} ≡ ℙN^{ide}

along the usual exceptional divisor

E ≡ ℙN ⊂ E^{ide}.

The blowup construction summarized in Theorem 1.1 is contained in the idealized blowup construction of Section 4. The latter turns out to be more convenient for describing the proper transforms of \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d), including at the final stage, i.e. 𝔐̃_{1,k}^{0}(ℙ^{n}, d).

The ambient space \overline{𝔐}_{1,k}^{σ} at each step σ ∈ \{0\} ⊔ A_{1}(d, k) of the blowup construction contains a subvariety \overline{𝔐}_{1,σ^{∗}}^{σ} for each σ^{∗} ∈ A_{1}(d, k). We take \overline{𝔐}_{1,σ}^{σ} to be the idealized exceptional divisor for the idealized blowup just constructed, i.e. along \overline{𝔐}_{1,σ}^{σ−1}. If σ^{∗} < σ or σ^{∗} > σ, \overline{𝔐}_{1,σ^{∗}}^{σ} is the proper transform of \overline{𝔐}_{1,σ^{∗}}^{σ^{∗}} or \overline{𝔐}_{1,σ^{∗}}(ℙ^{n}, d), respectively.

Every immersion ι_{σ^{∗}} of Subsection 1.2 comes with an idealized normal bundle N_{ι_{σ^{∗}}}^{ide}. It is a vector bundle of the smallest possible rank containing the normal cone to the immersion ι_{σ} (see Definition 4.1). It is given by

N_{ι_{σ^{∗}}}^{ide} = ⨁_{i∈[m^{∗}]}π_{P}^{∗} L_{i}⊗π_{B}^{∗}π_{i}^{∗}L_{0} if σ^{∗} = (m^{∗}; J_{P}^{∗} , J_{B}^{∗}),

where

π_{P} , π_{B}: \overline{M}_{σ^{∗};P} ×\overline{𝔐}_{σ^{∗};B}(ℙ^{n}, d) ⟶ \overline{M}_{σ^{∗};P} , \overline{𝔐}_{σ^{∗};B}(ℙ^{n}, d)

are the component projection maps. In the case of Figure 1, N_{ι_{σ^{∗}}}^{ide} is a rank-three vector bundle encoding the potential smoothings of the three nodes. At each step σ of the blowup construction, ι_{σ^{∗}} induces an immersion ι_{σ,σ^{∗}} onto \overline{𝔐}_{1,σ^{∗}}^{σ}. Like the domain of ι_{σ^{∗}}, the domain of ι_{σ,σ^{∗}} splits as a Cartesian product. If σ^{∗} > σ, the second component of the domain does not change from the previous step, while the first is modified by blowing up along a collection of disjoint subvarieties, as specified by the inductive assumption (I9) in Subsection 4.3. The idealized normal bundle N_{ι_{σ,σ^{∗}}}^{ide} is obtained from N_{ι_{σ−1,σ^{∗}}}^{ide} by twisting the first factor in each summand by a subset of the exceptional divisors, as specified by the inductive assumption (I11). These blowup and twisting procedures correspond to several interchangeable steps in the blowup construction of Subsection 2.3. For σ^{∗} =σ, the first component in the domain of ι_{σ−1,σ} has already been blown up all the way to M̃_{σ;P} and the first component of every summand of N_{ι_{σ−1,σ}}^{ide} has already twisted to the universal tangent line bundle 𝕃, i.e.

N_{ι_{σ−1,σ}}^{ide} = ⨁_{i∈[m]} π_{P}^{∗} 𝕃⊗π_{B}^{∗}π_{i}^{∗}L_{0} = π_{P}^{∗} 𝕃 ⊗ π_{B}^{∗}F_{σ;B} ⟶ M̃_{σ;P} ×\overline{𝔐}_{σ;B}(ℙ^{n}, d),

if σ =(m; J_{P} , J_{B}). In particular, the domain of ι_{σ,σ},

ℙN_{ι_{σ−1,σ}}^{ide} = M̃_{σ;P} ×ℙF_{σ;B},

still splits as a Cartesian product! The idealized normal bundle for ι_{σ,σ} is the tautological line for ℙN_{ι_{σ−1,σ}}^{ide}:

N_{ι_{σ,σ}}^{ide} = γ_{N_{ι_{σ−1,σ}}^{ide}} = π_{P}^{∗} 𝕃⊗π_{B}^{∗}γ_{F_{σ;B}} ≡ π_{P}^{∗} 𝕃⊗π_{B}^{∗}γ_{σ;B}.

On the other hand, if σ^{∗} < σ, the domain of ι_{σ,σ^{∗}} is obtained from the domain of ι_{σ−1,σ^{∗}} by blowing up the second component along a collection of disjoint subvarieties, as specified by the inductive assumption (I4) in Subsection 4.3. This corresponds to several interchangeable steps of the blowup construction in Subsection 3.4. By the time we are done with the last step of the blowup construction in Subsection 4.3, ℙF_{σ;B} has been blown up all the way to 𝔐̃_{σ;B}(ℙ^{n}, d). In the σ^{∗} <σ case,

N_{ι_{σ,σ^{∗}}}^{ide} = N_{ι_{σ−1,σ^{∗}}}^{ide} ,

since \overline{𝔐}_{1,σ^{∗}}^{σ−1} is transverse to \overline{𝔐}_{1,σ}^{σ−1}.

We study the proper transform \overline{𝔐}_{1,(0)}^{σ} of \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d) in \overline{𝔐}_{1,k}^{σ} by looking at the structure of

Z̄_{σ^{∗}}^{σ} = ι_{σ,σ^{∗}}^{−1}(\overline{𝔐}_{1,(0)}^{σ}).

Given a finite set J, there are natural bundle sections

s_{j} ∈ Γ(\overline{M}_{1,J}; L_{j}^{∗}⊗𝔼^{∗}), j ∈J,  and  D_{0} ∈ Γ(\overline{𝔐}_{0,\{0\}⊔J}(ℙ^{n}, d); L_{0}^{∗}⊗ev_{0}^{∗}T ℙ^{n});

see Subsections 2.2 and 3.2, respectively. By Lemma 4.10, the intersection of

Z̄_{σ^{∗}}^{0} ≡ ι_{σ^{∗}}^{−1}(\overline{𝔐}_{1,k}^{0}(ℙ^{n}, d))

with the main stratum M_{σ^{∗};P} ×𝔐_{σ^{∗};B}(ℙ^{n}, d) of \overline{M}_{σ^{∗};P} ×\overline{𝔐}_{σ^{∗};B}(ℙ^{n}, d) is

Z_{σ^{∗}}^{0} = {b∈M_{σ^{∗};P} ×𝔐_{σ^{∗};B}(ℙ^{n}, d): ker D_{σ^{∗}}|_{b} ≠{0}},

where

D_{σ^{∗}} ∈ Γ(\overline{M}_{σ^{∗};P} ×\overline{𝔐}_{σ^{∗};B}(ℙ^{n}, d); Hom(N_{ι_{σ^{∗}}}^{ide}, π_{P}^{∗} 𝔼^{∗}⊗π_{B}^{∗}ev_{0}^{∗}T ℙ^{n})),
D_{σ^{∗}} |_{π_{P}^{∗} L_{i}⊗π_{B}^{∗}π_{i}^{∗}L_{0}} = π_{P}^{∗} s_{i}⊗π_{B}^{∗}π_{i}^{∗}D_{0}, ∀ i∈[m^{∗}].

In addition, if N Z̄_{σ^{∗}}^{σ} ⊂N_{ι_{σ,σ^{∗}}}^{ide} is the normal cone for the immersion ι_{σ,σ^{∗}}|Z̄_{σ^{∗}}^{σ} into \overline{𝔐}_{1,(0)}^{σ}, then

N Z̄_{σ^{∗}}^{0}|_{Z_{σ^{∗}}^{0}} = ker D_{σ^{∗}} |_{Z_{σ^{∗}}^{0}}

and N Z̄_{σ^{∗}}^{0} is the closure of N Z̄_{σ^{∗}}^{0}|_{Z_{σ^{∗}}^{0}} in N_{ι_{σ^{∗}}}^{ide}. By Lemma 4.5, N Z̄_{σ^{∗}}^{σ} is still the closure of N Z̄_{σ^{∗}}^{0}|_{Z_{σ^{∗}}^{0}} ,

but now in N_{ι_{σ,σ^{∗}}}^{ide} , for all σ <σ^{∗}. In Subsection 2.3, we construct a non-vanishing section

s̃_{i} ∈ Γ(\overline{M}_{1,J}; 𝕃^{∗}⊗𝔼^{∗}) ≈ Γ(\overline{M}_{1,J}; ℂ)

obtained by twisting s_{i} by some exceptional divisors. Since s̃_{i} agrees with s_{i} on M_{σ^{∗};P}, we can replace s_{i} with s̃_{i} in the descriptions of D_{σ^{∗}}, Z_{σ^{∗}}^{0}, and N Z̄_{σ^{∗}}^{0}|_{Z_{σ^{∗}}^{0}} above. In particular, N Z̄_{σ^{∗}}^{σ^{∗}−1} is the closure of

N Z̄_{σ^{∗}}^{0}|_{Z_{σ^{∗}}^{0}} = π_{P}^{∗} 𝕃⊗π_{B}^{∗} ker D_{σ^{∗};B}|_{Z_{σ^{∗}}^{0}} ⊂ π_{P}^{∗} 𝕃⊗π_{B}^{∗}F_{σ^{∗};B}, where
D_{σ^{∗};B} ∈ Γ(\overline{𝔐}_{σ^{∗};B}(ℙ^{n}, d); Hom(F_{σ^{∗};B}, ev_{0}^{∗}T ℙ^{n})), D_{σ^{∗};B}|_{π_{i}^{∗}L_{0}} = π_{i}^{∗}D_{0}, ∀ i∈[m^{∗}].

The bundle homomorphism D_{σ^{∗};B} induces a section

D̃_{0} ∈ Γ(ℙF_{σ^{∗};B}; γ_{σ;B}^{∗} ⊗π_{ℙF_{σ^{∗};B}}^{∗}ev_{0}^{∗}T ℙ^{n}).

By the previous paragraph and Lemma 4.5, Z̄_{σ^{∗}}^{σ^{∗}} is the closure of

M̃_{σ^{∗};P} × D̃_{0}^{−1}(0)∩ℙF_{σ^{∗};B}|_{𝔐_{σ^{∗};B}(ℙ^{n},d)} ⊂ M̃_{σ^{∗};P} × ℙF_{σ^{∗};B} and
N Z̄_{σ^{∗}}^{σ^{∗}} = π_{P}^{∗} 𝕃⊗π_{B}^{∗}γ_{σ;B}|Z̄_{σ^{∗}}^{σ^{∗}} .

Since 𝔐̃_{1,k}^{0}(ℙ^{n}, d)≡𝔐̃_{1,k}^{σ_{max}}(ℙ^{n}, d) is the proper transform of \overline{𝔐}_{1,(0)}^{σ^{∗}} in \overline{𝔐}_{1,k}^{σ_{max}},

Z̃_{σ^{∗}} ≡ ι_{σ_{max},σ^{∗}}^{−1}(𝔐̃_{1,k}^{0}(ℙ^{n}, d))

is still the closure of

M̃_{σ^{∗};P} × D̃_{0}^{−1}(0)∩ℙF_{σ^{∗};B}|_{𝔐_{σ^{∗};B}(ℙ^{n},d)} ⊂ M̃_{σ^{∗};P} × 𝔐̃_{σ^{∗};B}(ℙ^{n}, d).

On the other hand, in the process of constructing the blowup 𝔐̃_{σ;B}(ℙ^{n}, d) of ℙF_{σ^{∗};B} in Subsection 3.4, we also define a bundle section

D̃_{σ^{∗};B} ∈ Γ(𝔐̃_{σ^{∗};B}(ℙ^{n}, d); 𝔼_{σ;B}^{∗} ⊗π̃_{σ;B}^{∗}π_{ℙF_{σ^{∗};B}}^{∗}ev_{0}^{∗}T ℙ^{n})

by twisting D̃_{0} by the exceptional divisors. In particular,

D̃_{σ^{∗};B}^{−1}(0)∩ℙF_{σ^{∗};B}|_{𝔐_{σ^{∗};B}(ℙ^{n},d)} = D̃_{0}^{−1}(0)∩ℙF_{σ^{∗};B}|_{𝔐_{σ^{∗};B}(ℙ^{n},d)}.

Since D̃_{σ^{∗};B} is transverse to the zero set, we conclude that

Z̃_{σ^{∗}} = M̃_{σ^{∗};P} × D̃_{σ^{∗};B}^{−1}(0),

as stated in Subsection 1.2.

Finally, the role played by the blowup construction of Subsection 2.4 in the blowup construction of Section 3 is similar to the role played by the construction of Subsection 2.3 in the construction of Section 4. In the case of Section 3, we blow up a moduli space of genus-zero stable maps, ℙF_{(ℵ,J)}, along certain subvarieties 𝔐̃_{0,ϱ}^{0} and their proper transforms. These subvarieties are images of natural node-identifying immersions ι_{0,ϱ}. The domain of ι_{0,ϱ} splits as the Cartesian product of a moduli space of genus-zero curves and a moduli space of genus-zero maps, defined in Subsections 2.4 and 3.3, respectively. As we modify 𝔐̃_{0,ϱ}^{0} by taking its proper transforms in the blowups of ℙF_{(ℵ,J)} constructed in Subsection 3.4, the first factor in the domain of the corresponding immersion changes by blowups along collections of smooth disjoint subvarieties, as specified by the inductive assumption (I6). This change corresponds to several interchangeable steps in the blowup construction of Subsection 2.4. By the time we are ready to blow up the proper transform of 𝔐̃_{0,ϱ}^{0}, the first component of the domain of the corresponding immersion has been blown up all the way to M̃_{0,ρ_{P}}, the end result in the blowup construction of Subsection 2.4.

In the blowup construction of Subsection 3.4, we twist a natural bundle section

D̃_{0} ∈ Γ(ℙF_{(ℵ,J)}; γ_{(ℵ,J)}^{∗}⊗π_{ℙF_{(ℵ,J)}}^{∗}ev_{0}^{∗}T ℙ^{n})

by the exceptional divisors to a bundle section

D̃_{(ℵ,J)} ∈ Γ(𝔐̃_{(ℵ,J)}(ℙ^{n}, d); 𝔼̃^{∗}⊗π_{0,(ℵ,J)}^{∗}π_{ℙF_{(ℵ,J)}}^{∗}ev_{0}^{∗}T ℙ^{n}).

The two sections enter in an essential way in the main blowup construction of this paper. It is also essential that D̃_{(ℵ,J)} is transverse to the zero set. The section D̃_{0} is transverse to the zero set outside of the subvarieties 𝔐̃_{0,ϱ}^{0} and vanishes identically along 𝔐̃_{0,ϱ}^{0}. Its derivative in the normal direction to ι_{0,ϱ} is described by Lemma 3.11, using Lemma 3.8. The bundle sections s_{i} over a moduli space of genus-zero curves defined in Subsection 2.2 and modified in Subsection 2.4 enter into the expression of Lemma 3.8. In fact, this expression is identical to the expression for D_{σ^{∗}} above, i.e. in the genus-one case. We use Lemma 3.11 to show that with each newly twisted version of D̃_{0} is transverse to the zero set outside of the proper transforms of the remaining subvarieties \overline{𝔐}_{0,ϱ}^{0}, i.e. the ones that have not been blown up yet; see the inductive assumption (I4) in Subsection 3.4. In particular, at the end of the blowup construction of Subsection 3.4, we end up with a twisted version of D̃_{0}, which we call D̃_{(ℵ,J)}, which is transverse to the zero set.

1.4 Localization Data

Suppose the group G = (S^{1})^{n+1} or G = (ℂ^{∗})^{n+1} acts in a natural way on the projective space ℙ^{n}. In particular, the fixed locus consists of n+1 points, which we denote by p_{0}, …, p_{n}, and the only curves preserved by G are the lines passing through pairs of fixed points. The G-action on ℙ^{n} lifts to an action on \overline{𝔐}_{1,k}(ℙ^{n}, d) and on 𝔐̃_{1,k}^{0}(ℙ^{n}, d). The fixed loci of these two actions that are contained in 𝔐_{1,k}^{eff}(ℙ^{n}, d) and their normal bundles are the same and are described in [MirSym, Sects. 27.3 and 27.4]. We note that the four-term exact sequence [MirSym, (27.6)] applies to such loci.

In this subsection, we describe the fixed loci of the G-action on 𝔐̃_{1,k}^{0}(ℙ^{n}, d) that are contained in ∂𝔐̃_{1,k}^{0}(ℙ^{n}, d) and their normal bundles. To simplify the discussion, we ignore all automorphism groups until the very end of this subsection.

The boundary fixed loci Z̃Γ̃ will be indexed by refined decorated rooted trees Γ̃. Figure 5 shows such a tree Γ̃ and the corresponding decorated graph Γ=π(Γ̃). In [MirSym, Section 27.3] the fixed loci Z_{Γ} of the G-action on \overline{𝔐}_{g,k}(ℙ^{n}, d) are indexed by decorated graphs Γ. If Γ is a decorated graph such that Z_{Γ} is a G-fixed locus contained in ∂\overline{𝔐}_{1,k}(ℙ^{n}, d), we will have

Z_{Γ} ∩ \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d) = π̃( ⨆_{π(Γ̃)=Γ}Z̃Γ̃),

where Γ̃ denotes a refined decorated rooted tree.

We now formally describe what we mean by a refined decorated rooted tree and its corresponding decorated graph. A graph consists of a set Ver of vertices and a collection Edg of edges, i.e. of two-element subsets of Ver. In Figure 5, the vertices are represented by dots, while each edge

0 2 12 1 3 2 1 3  2 1 1  2 2  2 3 1  2 1 3 2 1 1 1 1 3  2 12 1 3 2 1 3  2 1 1  2 2  2 3 1 2 1 3 2 1 1 1 1 3 (1, 0)

Figure 5: A refined decorated rooted tree and its decorated graph

{v_{1}, v_{2}} is shown as the line segment between v_{1} and v_{2}. A graph is a tree if it contains no loops, i.e. the set Edg contains no subset of the form

{{v_{1}, v_{2}}, {v_{2}, v_{3}}, …, {v_{N}, v_{1}}}, v_{1}, …, v_{N} ∈Ver, N ≥1.

A tree is rooted if Ver contains a distinguished element v_{0}. It is represented by the large dot in the first diagram of Figure 5. A rooted tree is refined if Ver−{v_{0}} contains two, possibly empty, distinguished subsets Ver_{+} and Ver_{0} such that

Ver_{+}∩Ver_{0} = ∅ and {v_{0}, v}∈Edg ∀ v ∈Ver_{+}∪Ver_{0}.

We put

Edg_{+} = {{v_{0}, v}: v ∈Ver_{+}} and Edg_{0} = {{v_{0}, v}: v ∈Ver_{0}}.

The elements of Edg_{+} and Edg_{0} are shown in the first diagram of Figure 5 as the thick solid lines and the thin dashed lines, respectively. Finally, a refined decorated rooted tree is a tuple

Γ̃ = (Ver, Edg; v_{0}; Ver_{+}, Ver_{0}; μ, 𝔡, η),  (1.8)

where (Ver, Edg; v_{0}; Ver_{+}, Ver_{0}) is refined rooted tree and

μ: Ver−Ver_{0} ⟶ {0, …, n}, 𝔡: Edg−Edg_{0} ⟶ ℤ^{+}, and η: {1, …, k} ⟶ Ver

are maps such that (i) μ(v_{1})=μ(v_{2}) and 𝔡({v_{0}, v_{1}})=𝔡({v_{0}, v_{2}}) for all v_{1}, v_{2} ∈Ver_{+}; (ii) if v_{1} ∈Ver_{+}, v_{2} ∈Ver−Ver_{0}−Ver_{+}, and {v_{0}, v_{2}}∈Edg, then

μ(v_{1}) ≠ μ(v_{2})  or  𝔡({v_{0}, v_{1}})≠𝔡({v_{0}, v_{2}});

(iii) if {v_{1}, v_{2}}∈Edg and v_{2} ∉Ver_{0}∪{v_{0}}, then

μ(v_{2})≠μ(v_{1}) if v_{1} ∉Ver_{0}  and  μ(v_{2})≠μ(v_{0}) if v_{1} ∈Ver_{0};

(iv) if v_{1} ∈Ver_{0}, then {v_{1}, v_{2}}∈Edg for some v_{2} ∈Ver−{v_{0}} and

val(v_{1}) ≡ |{v_{2} ∈Ver: {v_{1}, v_{2}}∈Edg}| + |{l ∈[k]: η(l)=v_{1}| ≥ 3;

(v) ∑_{e∈Edg_{+}}𝔡(e) ≥ 2. In Figure 5, the value of the map μ on each vertex, not in Ver_{0}, is indicated by the number next to the vertex. Similarly, the value of the map 𝔡 on each edge, not in Edg_{0}, is indicated by the number next to the edge. The elements of the set [k] = [3] are shown in bold face. Each of them is linked by a line segment to its image under η. The first condition above implies that all of the thick edges have the same labels, and so do their vertices, other than the root v_{0}. By the second condition, the set of thick edges is a maximal set of edges leaving v_{0} which satisfies the first condition. By the third condition, no two consecutive vertex labels are the same. By the fourth condition, there are at least two solid lines, at least one of which is an edge, leaving from every vertex which is connected to the root by a dashed line. The final condition implies that either the set Edg_{+} contains at least two elements or its only element is marked by at least 2.

A decorated graph is a tuple

Γ = (Ver, Edg; g, μ, 𝔡, η),

where (Ver, Edg) is a graph and

g: Ver ⟶ ℤ̄^{+}, μ: Ver ⟶ {0, …, n}, 𝔡: Edg ⟶ ℤ^{+}, and η: {1, …, k} ⟶ Ver

are maps such that

μ(v_{1}) ≠ μ(v_{2}) if {v_{1}, v_{2}}∈Edg.

The domain [k] of the map η can be replaced by any finite set. A decorated graph can be represented graphically as in the second diagram of Figure 5. In this case, every vertex v should be labeled by the pair (g(v), μ(v)). However, we drop the first entry if it is zero. If Γ̃ is a refined decorated rooted tree as in (1.8), the corresponding decorated graph Γ is obtained by identifying all elements of Ver_{0} with v_{0}, dropping Edg_{0} from Edg, and setting

g(v) = \begin{cases}1, & if v =v_{0}; \\ 0, & otherwise. \end{cases}

In terms of the first diagram in Figure 5, this procedure corresponds to contracting the dashed edges and adding 1 to the label for v_{0}.

The fixed locus Z_{Γ} of \overline{𝔐}_{1,k}(ℙ^{n}, d) consists of the stable maps u from a genus-one nodal curve Σ_{u} with k marked points into ℙ^{n} that satisfy the following conditions. The components of Σ_{u} on which the map u is not constant are rational and correspond to the edges of Γ. Furthermore, if e={v_{1}, v_{2}} is an edge, the restriction of u to the component Σ_{u,e} corresponding to e is a degree-𝔡(e) cover of the line

ℙ_{p_{μ(v_{1})},p_{μ(v_{2})}}^{1} ⊂ ℙ^{n}

passing through the fixed points p_{μ(v_{1})} and p_{μ(v_{2})}. The map u|_{Σ_{u,e}} is ramified only over p_{μ(v_{1})} and p_{μ(v_{2})}. In particular, u|_{Σ_{u,e}} is unique up to isomorphism. The remaining, contracted, components of Σ_{u} correspond to the vertices v ∈Ver such that

val(v) + g(v) ≥ 3.

For such a vertex v, g(v) specifies the genus of the component corresponding to v. The map u takes this component to the fixed point μ(v). Thus,

Z_{Γ} ≈ \overline{M}_{Γ} ≡ ∏_{v∈Ver}\overline{M}_{g(v),val(v)};

0 2 12 1 3 2 1 3  2 1 1  2 2  2 3 1 e_{1}  2 1 3 2 e_{2}  1 1 1 1 3  0  3 1 0 Γ̃_{e_{1}} = 0  3 2  1 1 0 Γ̃_{e_{2}} =

Figure 6: A refined decorated rooted tree and some of its components graphs

see [MirSym, Section 27.3]. For the purposes of this definition, \overline{M}_{0,1} and \overline{M}_{0,2} denote one-point spaces. For example, in the case of the second diagram in Figure 5,

Z_{Γ} ≈ \overline{M}_{Γ} ≡ \overline{M}_{1,10}×\overline{M}_{0,3}×\overline{M}_{0,2}^{2}×\overline{M}_{0,1}^{8} ≈ \overline{M}_{1,10}.

In this case, Z_{Γ} is a locus in \overline{𝔐}_{1,3}(ℙ^{n}, 22), with n≥3.

If Γ̃ is a refined decorated rooted tree as in (1.8), we put

Edg(v_{0}) = {{v_{0}, v_{1}}∈Edg: v_{1} ∈Ver}  and  J_{v_{0}} = {l ∈[k]: μ(l)=v_{0}}.

Similarly, for each v ∈Ver_{0}, we set

Edg(v) = {{v, v_{1}}∈Edg: v_{1} ∈Ver−{v_{0}}}  and  J_{v} = {l ∈[k]: μ(l)=v}.

If e = {v, v_{1}} is an element of Edg(v) for some v ∈ Ver_{0} or of Edg(v_{0}) − Edg_{0} with v = v_{0}, let (Ver_{e}, Edg_{e}) be the branch of the tree (Ver, Edg) beginning at v with the edge e. We put

J_{e} = {l ∈[k]: μ(l)∈Ver_{e}−{v}}  and  d_{e} = ∑_{e′∈Edg_{e}}𝔡(e′).

Let Γ̃_{e} be the decorated graph defined by

Γ̃_{e} = (Ver_{e}, Edg_{e}; g_{e} ≡0, μ_{e}, 𝔡_{e} ≡𝔡|_{Edg_{e}}, η_{e}), where

μ_{e}(v′) = \begin{cases}μ(v′), & if v′ ≠v; \\ μ(v_{0}), & if v′ =v; \end{cases} η_{e}: \{0\}⊔J_{e} ⟶ Ver_{e}, η_{e}(l) = \begin{cases}η(l), & if l ∈J_{e}; \\ v, & if l =0; \end{cases}

see Figure 6 for two examples.

If e is an element of Edg(v_{0})−Edg_{0} or of Edg(v) for some v ∈Ver_{0}, let

ZΓ̃_{e} ⊂ \overline{𝔐}_{0,\{0\}⊔J_{e}}(ℙ^{n}, d_{e})

be the fixed locus corresponding to the decorated graph Γ̃_{e}. We put

σ(Γ̃) = (|Edg(v_{0})|; J_{v_{0}}, [k]−J_{v_{0}}) ∈ A_{1}(d, k),  M̃_{Γ̃;P} = M̃_{σ(Γ̃);P};

Z̄_{Γ̃;B} = ∏_{e∈Edg(v_{0})−Edg_{0}} ZΓ̃_{e} × ∏_{v∈Ver_{0}}(\overline{M}_{0,\{0\}⊔Edg(v)⊔J_{v}} × ∏_{e∈Edg(v)} ZΓ̃_{e}) ⊂ \overline{𝔐}_{σ(Γ̃);B}(ℙ^{n}, d);

F_{Γ̃;B} = ⨁_{e∈Edg_{+}}L_{e;0} ⊂ F_{σ(Γ̃);B} ⟶ Z̄_{Γ̃;B},

where L_{e;0} ⟶ ZΓ̃_{e} is the tangent line bundle for the marked point 0. If e = {v_{0}, v_{1}} is an element of Edg_{+}, let

μ_{+}(Γ̃) = μ(v_{1}), 𝔡_{+}(Γ̃) = 𝔡(e), and dim_{+}(Γ̃) = \begin{cases}|Edg_{+}|−2, & if 𝔡_{+}(Γ̃)=1; \\ |Edg_{+}|−1, & if 𝔡_{+}(Γ̃)≥2. \end{cases}

By the assumption (i) above, the numbers μ_{+}(Γ̃) and 𝔡_{+}(Γ̃) are independent of the choice of e ∈ Edg_{+}. Furthermore, if e, e′ ∈ Edg_{+}, then the line bundles L_{e;0} and L_{e′;0} are G-equivariantly isomorphic. Thus,

F_{Γ̃;B} ≈ ℂ^{|Edg_{+}|} ⊗ L_{e;0} if e∈Edg_{+}.

The group G acts trivially on ℂ^{|Edg_{+}|}. Let

F′_{Γ̃;B} = \begin{cases}\{(w_{e})_{e∈Edg_{+}} ∈ℂ^{Edg^{+}} : ∑_{e∈Edg_{+}}w_{e} =0\}, & if 𝔡_{+}(Γ̃)=1; \\ ℂ^{Edg^{+}}, & if 𝔡_{+}(Γ̃)≥2; \end{cases}

Z̃_{Γ̃;B} = ℙ(F′_{Γ̃;B} ⊗L_{e;0}) ≈ Z̄_{Γ̃;B} × ℙ^{dim_{+}(Γ̃)}.

While the moduli space 𝔐̃_{σ(Γ̃);B}(ℙ^{n}, d) is a blowup of ℙF_{σ(Γ̃);B}, none of the blowup loci intersects Z̃_{Γ̃;B}. Thus,

Z̃_{Γ̃;B} ⊂ 𝔐̃_{σ(Γ̃);B}(ℙ^{n}, d).

In fact,

Z̃_{Γ̃;B} ⊂ Z̃_{σ(Γ̃);B}(ℙ^{n}, d).

We put

Z̃Γ̃ = M̃_{Γ̃;P} × Z̃_{Γ̃;B}.

By the above, Z̃Γ̃ is a fixed point locus in 𝔐̃_{1,k}^{0}(ℙ^{n}, d). For example, in the case of the first diagram in Figure 6,

σ(Γ̃) = (7; {2}, {1, 3}),  M̃_{Γ̃;P} = M̃_{1,([7],{2})},
Z̄_{Γ̃;B} = (\overline{M}_{0,3}×\overline{M}_{0,2}^{6}×\overline{M}_{0,1}^{5}) × (\overline{M}_{0,3}^{2}×\overline{M}_{0,2}^{4}×\overline{M}_{0,1}^{3}) ≈ \{pt\};
rk F_{Γ̃;B} = rk F′_{Γ̃;B} = 3,  Z̃_{Γ̃;B} ≈ ℙ^{2},  Z̃Γ̃ ≈ M̃_{1,([7],{2})}×ℙ^{2}.

The weight of the G-action on the line L_{e;0} is 1/2 of the weight of the G-action on T_{p_{0}}ℙ_{p_{0},p_{1}}^{1}; see [MirSym, Sects 27.1 and 27.2].

We next describe the equivariant normal bundle N Z̃Γ̃ of Z̃Γ̃ in 𝔐̃_{1,k}^{0}(ℙ^{n}, d). Let

N_{\overline{𝔐}_{σ(Γ̃);B}(ℙ^{n},d)}Z̄_{Γ̃;B} ⟶ Z̄_{Γ̃;B}

be the normal bundle of Z̄_{Γ̃;B} in \overline{𝔐}_{σ(Γ̃);B}(ℙ^{n}, d). This normal bundle can easily be described using [MirSym, Section 27.4]. Let

F_{Γ̃;B}^{−} = F_{σ(Γ̃);B}/(F′_{Γ̃;B} ⊗L_{e;0}) ≈ ⨁_{e′∈Edg(v_{0})−Edg^{+}} L_{e′;0} ⊕ \begin{cases}L_{e;0}, & if 𝔡_{+}(Γ̃)=1; \\ \{0\}, & if 𝔡_{+}(Γ̃)≥2, \end{cases}

where e is an element of Edg_{+}. The normal bundle of Z̃_{Γ̃;B} in 𝔐̃_{σ(Γ̃);B}(ℙ^{n}, d) is given by

N_{𝔐̃_{σ(Γ̃);B}(ℙ^{n},d)}Z̃_{Γ̃;B} = N_{\overline{𝔐}_{σ(Γ̃);B}(ℙ^{n},d)}Z̄_{Γ̃;B} ⊕ γ_{dim_{+} Γ̃}^{∗}⊗L_{e;0}^{∗}⊗F_{Γ̃;B}^{−},

where γ_{dim_{+} Γ̃} ⟶ ℙ^{dim^{+} Γ̃} is the tautological line bundle. Since none of the exceptional divisors intersects Z̃_{Γ̃;B},

𝔼_{σ;B}|Z̃_{Γ̃;B} = γ_{dim_{+} Γ̃}⊗L_{e;0}.  (1.9)

Since the section D̃_{σ;B} is transverse to the zero set, the normal bundle of Z̃_{Γ̃;B} in Z̃_{σ(Γ̃);B}(ℙ^{n}, d) is

N_{Z̃_{σ(Γ̃);B}(ℙ^{n},d)}Z̃_{Γ̃;B} = N_{𝔐̃_{σ(Γ̃);B}(ℙ^{n},d)}Z̃_{Γ̃;B}/(γ_{dim_{+} Γ̃}^{∗}⊗L_{e;0}^{∗}⊗T_{μ(v_{0})}ℙ^{n})

by (1.6) and (1.7). Finally,

N Z̃Γ̃ = N_{Z̃_{σ(Γ̃);B}(ℙ^{n},d)}Z̃_{Γ̃;B} ⊕ 𝕃_{σ(Γ̃);P} ⊗γ_{dim_{+} Γ̃}⊗L_{e;0},

since the normal bundle of Z̃_{σ(Γ̃)}(ℙ^{n}, d) in 𝔐̃_{1,k}^{0}(ℙ^{n}, d) is 𝕃_{σ(Γ̃);P} ⊗γ_{σ(Γ̃);B}.

In order to compute the last number in (1.4), we also need to determine the restriction of the vector bundle Ṽ_{1,k}^{d} to Z̃Γ̃. By Theorem 1.2 and (1.9), there is a short exact sequence of vector bundles:

0 ⟶ Ṽ_{1,k}^{d}|Z̃Γ̃ ⟶ V_{σ(Γ̃);B}^{d}|Z̃_{Γ̃;B} ⟶ γ_{dim_{+} Γ̃}^{∗}⊗L_{e;0}^{∗}⊗L_{μ(v_{0})} ⟶ 0

over Z̃Γ̃. This exact sequence describes the euler class of the restriction of Ṽ_{1,k}^{d} to Z̃Γ̃.

If σ =(m; J_{P} , J_{B})∈A_{1}(d, k),

⟨c_{1}^{|m|+|J_{P} |}(𝕃_{σ;P}^{∗}), M̃_{σ;P} ⟩ = \frac{m^{|J_{P} |} · (m−1)!}{24},

by Corollary 1.2 in [Z7]. This is the only intersection number on M̃_{σ;P} needed for computing the last number in (1.4) and the integrals of the cohomology classes on \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d) that count elliptic curves in ℙ^{n} passing through specified constraints. For more general enumerative problems, such as counting curves with tangency conditions, as in [V1], and with singularities, as in [Z3], one would need to compute the intersection numbers of the form

⟨c_{1}^{β_{0}}(𝕃_{σ;P}^{∗}) · ∏_{l∈J_{P}} ψ_{l}^{β_{l}}, M̃_{σ;P} ⟩,  where  β_{0} + ∑_{l∈J_{P}} β_{l} = |m| + |J_{P} |.

Theorem 1.1 in [Z7] gives a recursive formula for such numbers. The recursion is on |m|+|J_{P} |, i.e. the total number of marked points. The starting data for the recursion is the well-known number ⟨ψ_{1}, \overline{M}_{1,1}⟩=1/24.

In the above discussion we ignored all automorphism groups. As in [MirSym, Chapter 27], the rational function for each refined decorated rooted tree Γ̃ obtained following the above algorithm and applying the localization theorem of [ABo] should be divided by the order of the appropriate automorphism group 𝔸Γ̃ :

|𝔸Γ̃| = |Aut(Γ̃)| · ∏_{e∈Edg−Edg_{0}} 𝔡(e).

For example, in the case of the first diagram in Figure 6,

|𝔸Γ̃| = 1 · (1^{3} · 2^{5} · 3^{3}) = 864.

2 Blowups of Moduli Spaces of Curves

2.1 Blowups and Subvarieties

In this section we construct blowups of certain moduli spaces of genus-one and genus-zero curves; see Subsections 2.3 and 2.4. The former appear in Subsection 4.3 as the first factor in the domain of the proper transforms of the immersion ι_{σ} of Subsection 1.2. The latter play the analogous role in Subsection 3.4, where we blow up certain moduli spaces of genus-zero maps. In turn, these last blowups describe the second factor of the domain of maps induced by ι_{σ} in Subsection 4.3; see Subsection 1.3 for more details.

We begin by introducing convenient terminology and reviewing standard facts from algebraic geometry. If \overline{M} is a smooth variety and Z is a smooth subvariety of \overline{M}, let

N\overline{M}Z ≡ T \overline{M}|_{Z}/T Z

be the normal bundle of Z in \overline{M}. We denote by

π_{Z}^{⊥}: T \overline{M}|_{Z} ⟶ N\overline{M}Z

the quotient projection map.

Definition 2.1 Let \overline{M} be a smooth variety.

(1) Smooth subvarieties X and Y of \overline{M} intersect properly if X ∩Y is a smooth subvariety of \overline{M} and

T (X ∩Y) = T X|_{X∩Y} ∩ T Y |_{X∩Y} .^{6}

(2) If Z is a smooth subvariety of \overline{M}, properly intersecting subvarieties X and Y of \overline{M} intersect

properly relative to Z if

π_{Z}^{⊥}(T (X ∩Y)|_{X∩Y ∩Z}) = π_{Z}^{⊥}(T X|_{X∩Y ∩Z}) ∩ π_{Z}^{⊥}(T Y |_{X∩Y ∩Z}) ⊂ N\overline{M}Z.

6In other words, the scheme-theoretic intersection of X and Y is smooth. If the set-theoretic intersection X ∩Y is smooth, the second part of this condition is also equivalent to the injectivity of the natural homomorphism

T X|_{X∩Y} /T (X ∩Y) ⟶ T \overline{M}/T Y.

For example, if X and Y are two smooth curves in a projective space that intersect without being tangent to each other, then X and Y intersect properly (but not transversally, unless the dimension of the projective space is 2). If X, Y, and Z are three distinct concurrent lines that lie a plane, then they intersect properly pairwise, but X and Y do not intersect properly relative to Z.

Definition 2.2 If \overline{M} is a smooth variety, a collection \{\overline{M}_{ρ}\}_{ρ∈A} of smooth subvarieties is properly intersecting if \overline{M}_{ρ_{1}} and \overline{M}_{ρ_{2}} intersect properly relative to \overline{M}_{ρ_{3}} for all ρ_{1}, ρ_{2}, ρ_{3} ∈A.

If Z is a smooth subvariety of \overline{M}, let

π: Bl_{Z} \overline{M} ⟶ \overline{M}

be the blowup of \overline{M} along Z. If X is a subvariety of \overline{M}, we denote by Pr_{Z}X the proper transform of X in Bl_{Z} \overline{M}, i.e. the closure of π^{−1}(X −Z) in Bl_{Z} \overline{M}.^{7} The next lemma follows from a local computation. (The local geometry of a proper intersection is particularly simple.)

Lemma 2.3 Let \overline{M} be a smooth variety.

(1) If X and Z are properly intersecting subvarieties of \overline{M}, then Pr_{Z}X is a smooth subvariety of Bl_{Z} \overline{M} and

Pr_{Z}X = Bl_{X∩Z}X.

(2) If X, Y, and Z are pairwise properly intersecting subvarieties of \overline{M} and X and Y intersect properly relative to Z, then Pr_{Z}X and Pr_{Z}Y are properly intersecting subvarieties of Pr_{Z} \overline{M} and

Pr_{Z}X ∩ Pr_{Z}Y = Pr_{Z}(X ∩Y).

(3) If X, Y, Z, and Z′ are pairwise properly intersecting subvarieties of \overline{M} and X and Y intersect properly relative to Z and Z′, then Pr_{Z}X and Pr_{Z}Y intersect properly relative to Pr_{Z}Z′.

Corollary 2.4 If \overline{M} is a smooth variety, \{\overline{M}_{ρ}\}_{ρ∈A} is a properly intersecting collection of subvarieties of \overline{M}, and ρ∈A, then \{Pr_{\overline{M}_{ρ}} \overline{M}_{ρ′}\}_{ρ′∈A−\{ρ\}} is a properly intersecting collection of subvarieties of Bl_{\overline{M}_{ρ}} \overline{M}.

Remark: By our definitions, properly intersecting subvarieties are necessarily smooth subvarieties of smooth varieties.

2.2 Moduli Spaces of Genus-One and Zero Curves

In this subsection, we describe natural subvarieties of moduli spaces of genus-one and -zero curves and natural bundle sections over these moduli spaces. These bundle sections and their twisted versions introduced in the next two subsections are used in Subsections 3.4 and 4.3 to describe the structure of the proper transforms of \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d). Below we also state the now-standard facts about these objects that are used in the next two subsections.

7For the purposes of these definitions we do not require that \overline{M} and X be smooth.

i_{1}

i_{2} i_{3}

i_{4} i_{5}

i_{6}

i_{7}

i_{8} i_{9}

i_{1}

i_{2}

i_{3} i_{4}

i_{5} i_{6}

i_{7} i_{8} i_{9}

1 0

0

0

I_{P} ={i_{1}, i_{2}}
K ={1, 2, 3}
I_{1} ={i_{3}, i_{4}}
I_{2} ={i_{5}, i_{6}}
I_{3} ={i_{7}, i_{8}, i_{9}}

Figure 7: A typical element of \overline{M}_{1,ρ}

If I is a finite set, let

\begin{aligned}A_{1}(I) & = \{(I_{P} , \{I_{k}: k ∈K\}): K ≠∅; I = ⨆_{k∈\{P \}⊔K} I_{k}; |I_{k}|≥2 ∀ k ∈K\}; \\ A_{0}(I) & = \{(I_{P} , \{I_{k}: k ∈K\}): K ≠∅; I = ⨆_{k∈\{P \}⊔K} I_{k}; |I_{k}|≥2 ∀ k ∈K; |K|+|I_{P} |≥2\}. \end{aligned}  (2.1)

If ρ = (I_{P} , \{I_{k} : k ∈ K\}) is an element of \{(I, ∅)\}⊔A_{1}(I), we denote by M_{1,ρ} the subset of \overline{M}_{1,I} consisting of the stable curves C such that

(i) C is a union of a smooth torus and |K| projective lines, indexed by K;

(ii) each line is attached directly to the torus;

(iii) for each k ∈K, the marked points on the line corresponding to k are indexed by I_{k}. Let \overline{M}_{1,ρ} be the closure of M_{1,ρ} in \overline{M}_{1,I}. Figure 7 illustrates this definition, from the points of view of symplectic topology and of algebraic geometry. In the first diagram, each circle represents a sphere, or ℙ^{1}. In the second diagram, the irreducible components of C are represented by curves, and the integer next to each component shows its genus. Similarly, if

ρ=(I_{P} , {I_{k}: k ∈K}) ∈ {(I, ∅)}⊔A_{0}(I),

let M_{0,ρ} be the subset of \overline{M}_{0,\{0\}⊔I} consisting of the stable curves C such that (i) the components of C are indexed by the set {P }⊔K; (ii) for each k ∈K, the component C_{k} of C is attached directly to C_{P}; (iii) for each k ∈K, the marked points on C_{k} are indexed by I_{k}.

We denote by \overline{M}_{0,ρ} the closure of M_{0,ρ} in \overline{M}_{0,\{0\}⊔I}. This definition is illustrated in Figure 8. In this case, we do not indicate the genus of the irreducible components in the second diagram, as all of the curves are rational.

The next lemma follows from the fact that for any nodal curve, the deformations of the nodes are independent. More precisely, in the dual to the first-order deformation space of a nodal curve, the vectors corresponding to the smoothings of each node are linearly independent.

Lemma 2.5 If g =0, 1 and I is a finite set, the collection \{\overline{M}_{g,ρ}\}_{ρ∈A_{g}(I)} is properly intersecting.

We define a partial ordering on the sets A_{g}(I) for g =0, 1 by setting

ρ′ ≡(I′_{P} , {I′_{k}: k ∈K′}) ≺ ρ≡(I_{P} , {I_{k}: k ∈K})  (2.2)

if ρ′ ≠ ρ and there exists a map φ: K ⟶ K′ such that I_{k} ⊂ I′_{φ(k)} for all k ∈ K. This condition means that the elements of M_{ρ′} can be obtained from the elements of M_{ρ} by moving more points

0 i_{1} i_{2} i_{3} i_{4}  i_{5} i_{6} i_{7} i_{8}  i_{1} 0 i_{2} i_{3} i_{4} i_{5} i_{6} i_{7} i_{8}  \begin{matrix}I_{P} =\{i_{1}\} \\ K =\{1, 2, 3\} \\ I_{1} =\{i_{2}, i_{3}\} \\ I_{2} =\{i_{4}, i_{5}\} \\ I_{3} =\{i_{6}, i_{7}, i_{8}\} \end{matrix}

Figure 8: A typical element of \overline{M}_{0,ρ}

onto the bubble components or combining the bubble components; see Figure 9. In the g =0 case, we define the bubble components to be the components not containing the marked point 0.

In the blowup constructions of the next two subsections we will twist certain line bundles over moduli spaces of curves and homomorphisms between them. In the rest of this subsection we describe the relevant starting data.

For each i∈I, let L_{i} ⟶\overline{M}_{1,I} be the universal tangent line bundle at the marked point labeled i. Let 𝔼 ⟶ \overline{M}_{1,I} be the Hodge line bundle of holomorphic differentials. The natural pairing of tangent vectors with cotangent vectors induces a section

s_{i} ∈ Γ(\overline{M}_{1,I}; Hom(L_{i}, 𝔼^{∗})).

Explicitly,

{s_{i}([C; w])}([C, ψ]) = ψ_{x_{i}(C)}w if
[C]∈\overline{M}_{1,I}, [C, w]∈L_{i}|_{C} =T_{x_{i}(C)}C, [C, ψ]∈𝔼|_{C} =H^{0}(C; T^{∗}C),

and x_{i}(C)∈C is the marked point on C labeled by i.

In the genus-zero case, the line bundle L_{0} ⟶ \overline{M}_{0,\{0\}⊔I} will be one of the substitutes for 𝔼. We note that for every p∈ℙ^{1}, there is a natural isomorphism between the tangent space T_{p}ℙ^{1} of ℙ^{1} at p and the space of holomorphic differentials H^{0}(ℙ^{1}; T^{∗}ℙ^{1}⊗O(2p)) on ℙ^{1} that have a pole of order two at p. More precisely, let w be a meromorphic function on ℙ^{1} such that p is the only zero of w and this zero is a simple one. We can then view w as a coordinate around p in ℙ^{1}. Every tangent vector v ∈T_{p}ℙ^{1} can be written as

v = c_{w}(v)\frac{∂}{∂w} , c_{w}(v) ∈ ℂ.

i_{1}_{i_{2}} ^{i_{3}} _{i_{4}} ^{i_{5}} ^{i_{6}}_{i_{7}} _{i_{8}} _{i_{9}}_{1}  ^{0}_{0} ≺ ^{i_{1}}_{i_{2}} _{i_{3}} _{i_{4}} ^{i_{5}} ^{i_{6}}_{i_{7}} _{i_{8}} _{i_{9}}_{1}  ^{0}_{0}  ^{i_{1}}_{i_{2}} ^{i_{3}} _{i_{4}} ^{i_{5}} ^{i_{6}}_{i_{7}} _{i_{8}}_{i_{9}}_{1}  ^{0}  ^{i_{1}}_{i_{2}} ^{i_{3}}  ^{i_{4}}_{i_{5}} _{i_{6}}_{i_{7}} _{i_{8}} _{i_{9}}_{1}  ^{0}_{0}_{0}

Figure 9: Examples of partial ordering (2.2)

We define the isomorphism

ψ: T_{p}ℙ^{1} ⟶ H^{0}(ℙ^{1}; T^{∗}ℙ^{1}⊗O(2p))  by  v ⟶ ψ_{v} = \frac{c_{w}(v) dw}{w^{2}} .

If w′ is another meromorphic function on ℙ^{1} such that p is the only zero of w′ and this zero is a simple one, then

w′ = \frac{w}{αw+β} ⟹ dw′ = \frac{β dw}{(αw+β)^{2}} , c_{w′}(v) = \frac{c_{w}(v)}{β} ⟹ \frac{c_{w′}(v) dw′}{w′^{2}} = \frac{c_{w}(v) dw}{w^{2}} .

Thus, the isomorphism ψ is well-defined. If i∈I, we define the section

s_{i} ∈ Γ(\overline{M}_{0,\{0\}⊔I}; Hom(L_{i}, L_{0}^{∗})) by \{s_{i}([C; w])\}([C, v]) = ψ_{v}|_{x_{i}(C)}w
if [C]∈\overline{M}_{0,\{0\}⊔I}, [C, w]∈L_{i}|_{C} =T_{x_{i}(C)}C, [C, v]∈L_{0}|_{C} =T_{x_{0}(C)}C.

(2.3)

We note that in both cases the section s_{i} vanishes precisely on the curves for which the point i lies on a bubble component. In fact,

s_{i}^{−1}(0) = ∑_{ρ∈B_{g}(I;i)} \overline{M}_{1,ρ},  where  B_{g}(I; i) = \{(I_{P} , \{I_{B}\})∈A_{g}(I): i∈I_{B}\}.  (2.4)

2.3 A Blowup of a Moduli Space of Genus-One Curves

Let I and J be finite sets such that I is nonempty. In this subsection, we construct a blowup

π_{1,(I,J)} : M̃_{1,(I,J)} ⟶ \overline{M}_{1,I⊔J}

of the moduli space \overline{M}_{1,I⊔J}, |I|+1 line bundles

𝔼̃, L̃_{i} ⟶ M̃_{1,(I,J)}, i∈I,

and |I| nowhere vanishing sections

s̃_{i} ∈ Γ(M̃_{1,(I,J)}; Hom(L̃_{i}, 𝔼̃^{∗})), i∈I.

Since the sections s̃_{i} do not vanish, all |I|+1 bundles L̃_{i} and 𝔼̃^{∗} are explicitly isomorphic. They will be denoted by 𝕃 and called the universal tangent line bundle.

The smooth variety M̃_{1,(I,J)} is obtained by blowing up some of the subvarieties \overline{M}_{1,ρ}, defined in the previous subsection, and their proper transforms in an order consistent with the partial ordering ≺. The line bundle 𝔼̃ is the sum of the Hodge line bundle 𝔼 and all exceptional divisors. For each given i ∈ I, L̃_{i} is the tangent line bundle L_{i} for the marked point i minus some of these divisors. The section s̃_{i} is induced from the pairing s_{i} of the previous subsection.

With I and J as above and A_{g}(I ⊔J) as in (2.1), let

A_{g}(I, J) = {((I_{P} ⊔J_{P}), {I_{k}⊔J_{k}: k ∈K})∈A_{g}(I ⊔J): I_{k} ≠∅ ∀ k ∈K}.  (2.5)

We note that if ρ∈A_{g}(I⊔J), then ρ∈A_{g}(I, J) if and only if every bubble component of an element of M_{ρ} carries at least one element of I. Furthermore,

B_{g}(I ⊔J; i) ⊂ A_{g}(I, J)  ∀ i∈I.  (2.6)

If |I|+|J|≥2, with respect to the partial ordering ≺ the set A_{1}(I, J) has a unique minimal element:

ρ_{min} ≡ (∅, {I ⊔J}).

Let < be an ordering on A_{1}(I, J) extending the partial ordering ≺. We denote the corresponding maximal element by ρ_{max}. If ρ∈A_{1}(I, J), we put

ρ−1 = \begin{cases}max\{ρ′ ∈A_{1}(I, J): ρ′ <ρ\}, & if ρ≠ρ_{min}; \\ 0, & if ρ=ρ_{min}, \end{cases}  (2.7)

where the maximum is taken with respect to the ordering <.

We now describe the starting data for the inductive blowup procedure involved in constructing the space M̃_{1,(I,J)} and the line bundle 𝕃 over M̃_{1,(I,J)}. Let

\overline{M}_{1,(I,J)}^{0} = \overline{M}_{1,I⊔J}, 𝔼_{0} = 𝔼 ⟶ \overline{M}_{1,(I,J)}^{0}, and \overline{M}_{1,ρ}^{0} = \overline{M}_{1,ρ} ∀ ρ∈A_{1}(I, J).

For each i∈I, let

L_{0,i} = L_{i} ⟶ \overline{M}_{1,(I,J)}^{0}  and  s_{0,i} = s_{i} ∈ Γ(\overline{M}_{1,(I,J)}^{0}; Hom(L_{0,i}, 𝔼_{0}^{∗})).

By (2.4),

s_{0,i}^{−1}(0) = ∑_{ρ^{∗}∈B_{1}(I⊔J;i)} \overline{M}_{1,ρ^{∗}}^{0}.

Suppose ρ∈A_{1}(I, J) and we have constructed

(I1) a blowup π_{ρ−1} : \overline{M}_{1,(I,J)}^{ρ−1} ⟶\overline{M}_{1,(I,J)}^{0} of \overline{M}_{1,(I,J)}^{0} such that π_{ρ−1} is an isomorphism outside of the preimages of the spaces \overline{M}_{1,ρ′}^{0} with ρ′ ≤ρ−1;

(I2) line bundles L_{ρ−1,i} ⟶\overline{M}_{1,(I,J)}^{ρ−1} for i∈I and 𝔼_{ρ−1} ⟶\overline{M}_{1,(I,J)}^{ρ−1}; (I3) sections s_{ρ−1,i} ∈Γ(\overline{M}_{1,(I,J)}^{ρ−1}; Hom(L_{ρ−1,i}, 𝔼_{ρ−1}^{∗})) for i∈I.

For each ρ^{∗} >ρ−1, let \overline{M}_{1,ρ^{∗}}^{ρ−1} be the proper transform of \overline{M}_{1,ρ^{∗}}^{0} in \overline{M}_{1,(I,J)}^{ρ−1}. We assume that (I4) the collection \{\overline{M}_{1,ρ^{∗}}^{ρ−1}\}_{ρ^{∗}∈A_{1}(I,J),ρ^{∗}>ρ−1} is properly intersecting; (I5) for all i∈I,

s_{ρ−1,i}^{−1}(0) = ∑_{ρ^{∗}∈B_{1}(I⊔J;i),ρ^{∗}>ρ−1} \overline{M}_{1,ρ^{∗}}^{ρ−1}.

The assumption (I5) means that we will gradually be killing the zero locus of the section s_{i}. We note that all five assumptions are satisfied if ρ−1 is replaced by 0.

If ρ is as above, let

π̃_{ρ}: \overline{M}_{1,(I,J)}^{ρ} ⟶ \overline{M}_{1,(I,J)}^{ρ−1}

be the blowup of \overline{M}_{1,(I,J)}^{ρ−1} along \overline{M}_{1,ρ}^{ρ−1}. We denote by \overline{M}_{1,ρ}^{ρ} the corresponding exceptional divisor. If ρ^{∗} >ρ, let \overline{M}_{1,ρ^{∗}}^{ρ} ⊂\overline{M}_{1,(I,J)}^{ρ} be the proper transform of \overline{M}_{1,ρ^{∗}}^{ρ−1}. If

ρ = (I_{P} ⊔J_{P} , {I_{k}⊔J_{k}: k ∈K})  (2.8)

and i∈I, we put

L_{ρ,i} = \begin{cases}π̃_{ρ}^{∗}L_{ρ−1,i}, & if i∉I_{P}; \\ π̃_{ρ}^{∗}L_{ρ−1,i} ⊗ O(−\overline{M}_{1,ρ}^{ρ}), & if i∈I_{P}; \end{cases}  𝔼_{ρ} = π̃_{ρ}^{∗} 𝔼_{ρ−1} ⊗ O(\overline{M}_{1,ρ}^{ρ}).  (2.9)

The section π̃_{ρ}^{∗}s_{ρ−1,i} induces a section

s̃_{ρ,i} ∈ Γ(\overline{M}_{1,(I,J)}^{ρ}; Hom(L_{ρ,i}, π̃_{ρ}^{∗}𝔼_{ρ−1}^{∗})).

This section vanishes along \overline{M}_{1,ρ}^{ρ}, by the inductive assumption (I5) if i ∉ I_{P}. Thus, s̃_{ρ,i} induces a section

s_{ρ,i} ∈ Γ(\overline{M}_{1,(I,J)}^{ρ}; Hom(L_{ρ,i}, 𝔼_{ρ}^{∗})).

We have now described the inductive step of the procedure. It is immediate that the requirements (I1)-(I3) and (I5) are satisfied, with ρ−1 replaced by ρ, are satisfied. Corollary 2.4 and the assumption (I4) imply that the assumption (I4) with ρ−1 replaced by ρ is also satisfied.

We conclude the blowup construction after the ρ_{max} step. Let

M̃_{1,(I,J)} = \overline{M}_{1,(I,J)}^{ρ_{max}};  𝔼̃ = 𝔼_{ρ_{max}};  L̃_{i} = L_{ρ_{max},i}, s̃_{i} = s_{ρ_{max},i}  ∀ i∈I.

By (I5), with ρ−1 replaced by ρ_{max}, and (2.6), the section s̃_{i} does not vanish. We note that by (I1), the stratum

M_{1,(I,J)} ⊂ \overline{M}_{1,(I,J)}

consisting of the smooth curves is a Zariski open subset of \overline{M}_{1,(I,J)}^{ρ} for all ρ∈\{0\}⊔A_{1}(I, J).

By the next lemma, different extensions of the partial order ≺ to an order < on A_{1}(I, J) correspond to blowing up along disjoint subvarieties in different orders. Thus, the end result of the above blowup construction is well-defined, i.e. independent of the choice of the ordering < extending the partial ordering ≺.

Lemma 2.6 Suppose ρ, ρ′ ∈A_{1}(I, J) are such that ρ⊀ρ′ and ρ′ ⊀ρ. If ρ≠ρ′, then the spaces \overline{M}ρ̃ ; 1,ρ and \overline{M}ρ̃ ; 1,ρ′ are disjoint for some ρ̃≺ρ, ρ′.

Proof: (1) Suppose

ρ = (I_{P} ⊔J_{P} , {I_{k}⊔J_{k}: k ∈K})  and  ρ′ = (I′_{P} ⊔J′_{P} , {I′_{k}⊔J′_{k}: k ∈K′}).

For each k ∈K and k′ ∈K′, let

\begin{aligned}ρ_{k} & = ((I −I_{k})⊔(J −J_{k}), \{I_{k}⊔J_{k}\}) ∈ A_{1}(I, J) and \\ ρ′_{k′} & = ((I −I′_{k′})⊔(J −J′_{k′}), \{I′_{k′} ⊔J′_{k′}\}) ∈ A_{1}(I, J). \end{aligned}

By definition, \overline{M}_{1,ρ_{k}}^{0} and \overline{M}_{1,ρ′_{k′}}^{0} are divisors in \overline{M}_{1,(I,J)}^{0} =\overline{M}_{1,I⊔J},

\overline{M}_{1,ρ}^{0} = ⋂_{k∈K} \overline{M}_{1,ρ_{k}}^{0},  and  \overline{M}_{1,ρ′}^{0} = ⋂_{k′∈K′} \overline{M}_{1,ρ′_{k′}}^{0} .

Furthermore, if \overline{M}_{1,ρ_{k}}^{0} ∩\overline{M}_{1,ρ′_{k′}}^{0} ≠∅, then either

I_{k}⊔J_{k} ⊂ I′_{k′} ⊔J′_{k′},  or  I_{k}⊔J_{k} ⊃ I′_{k′} ⊔J′_{k′},  or  (I_{k}⊔J_{k}) ∩ (I′_{k′} ⊔J′_{k′}) = ∅.

(2) Suppose \overline{M}_{1,ρ}^{0}∩\overline{M}_{1,ρ′}^{0} ≠∅. By the above, there exist decompositions

K = K_{+} ⊔ K_{0} ⊔ ⨆_{l′∈K′_{+}}K_{l′}  and  K′ = K′_{+} ⊔ K′_{0} ⊔ ⨆_{l∈K_{+}}K′_{l}

and a bijection φ: K_{0} ⟶K′_{0} such that

I_{k}⊔J_{k} ⊊ I′_{l′} ⊔J′_{l′} ∀ k ∈K_{l′}, l′ ∈K′_{+}, I_{l}⊔J_{l} ⊋ I′_{k′} ⊔J′_{k′} ∀ k′ ∈K′_{l}, l ∈K_{+},
and I_{k}⊔J_{k} = I′_{φ(k)}⊔J′_{φ(k)} ∀ k ∈K_{0}.

We note that the subsets K_{+} and K′_{+} of K and K′ are nonempty. For example, if K_{+} were empty, then we would have ρ′ ≺ρ, contrary to our assumptions. Let

ρ̃ = (Ĩ_{P} ⊔J̃_{P} , {Ĩ_{k}⊔J̃_{k}: k ∈K_{0}⊔K_{+}⊔K′_{+}}) ∈ A_{1}(I, J)

be given by

Ĩ_{P} ⊔ J̃_{P} = (I_{P} ∩I′_{P}) ⊔ (J_{P} ∩J′_{P}),  Ĩ_{k}⊔J̃_{k} = \begin{cases}I_{k}⊔J_{k}, & if k ∈K_{0}⊔K_{+}; \\ I′_{k}⊔J′_{k}, & if k ∈K′_{+}. \end{cases}

For example, if ρ corresponds to the second diagram on the right side of Figure 9 and ρ′ corresponds to either the first or the third diagram on the right side, then ρ̃ corresponds to the diagram on the left side of Figure 9. By definition, ρ̃≺ρ, ρ′. Furthermore,

\overline{M}_{1,ρ}^{0}∩\overline{M}_{1,ρ′}^{0} ⊂ \overline{M}_{1,ρ̃}^{0}.

Thus, by Lemma 2.5, Corollary 2.4, and (2) of Lemma 2.3,

\overline{M}ρ̃ ; 1,ρ ∩ \overline{M}ρ̃ ; 1,ρ′ ⊂ \overline{M}ρ̃ ; 1,(I,J)

is the closure of the empty set.

2.4 A Blowup of a Moduli Space of Genus-Zero Curves

Suppose ℵ is a nonempty finite set and ϱ = (I_{l}, J_{l})_{l∈ℵ} is a tuple of finite sets such that I_{l} ≠ ∅ and |I_{l}|+|J_{l}|≥2 for all l ∈ℵ. Let

\overline{M}_{0,ϱ} = ∏_{l∈ℵ}\overline{M}_{0,\{0\}⊔I_{l}⊔J_{l}}  and  F_{ϱ} = ⨁_{l∈ℵ} π_{l}^{∗}L_{0} ⟶ \overline{M}_{0,ϱ},

where L_{0} ⟶\overline{M}_{0,\{0\}⊔I_{l}⊔J_{l}} is the universal tangent line bundle for the marked point 0 and

π_{l}: \overline{M}_{0,ϱ} ⟶ \overline{M}_{0,\{0\}⊔I_{l}⊔J_{l}}

is the projection map. In this subsection, we construct a blowup

π_{0,ϱ}: M̃_{0,ϱ} ⟶ ℙF_{ϱ}

of the projective bundle ℙF_{ϱ} over \overline{M}_{0,ϱ}. We also construct line bundles

𝔼̃, L̃_{(l,i)} ⟶ M̃_{0,ϱ}, i∈I_{l}, l ∈ℵ,

and nowhere vanishing sections

s̃_{(l,i)} ∈ Γ(M̃_{0,ϱ}; Hom(L̃_{(l,i)}, 𝔼̃^{∗})), i∈I_{l}, l ∈ℵ.

In particular, all line bundles L̃_{(l,i)} and 𝔼̃^{∗} are explicitly isomorphic. They will be denoted by 𝕃 and called the universal tangent line bundle.

Similarly to the previous subsection, the smooth variety M̃_{0,ϱ} is obtained by blowing up the subvarieties \overline{M}_{0,ρ} defined below and their proper transforms in an order consistent with a natural partial ordering ≺. The line bundle 𝔼̃ is the sum of the tautological line bundle

γ_{ϱ} ⟶ ℙF_{ϱ}

and all exceptional divisors. For every l ∈ ℵ and i ∈ I_{l}, L̃_{(l,i)} is π_{l}^{∗}L_{i} minus some of these divisors. The section s̃_{(l,i)} is induced from the pairings s_{i} of Subsection 2.2.

With ϱ as above and A_{0}(I_{l}, J_{l}) as in (2.5), let

\begin{aligned}A_{0}(ϱ) = \{(ℵ_{+}, (ρ_{l})_{l∈ℵ}): & ℵ_{+} ⊂ℵ, ℵ_{+} ≠∅; ρ_{l} ∈\{(I_{l}⊔J_{l}, ∅)\}⊔A_{0}(I_{l}, J_{l}) ∀ l ∈ℵ; \\ & ρ_{l} =(I_{l}⊔J_{l}, ∅) ∀ l ∈ℵ−ℵ_{+}; (ℵ_{+}, (ρ_{l})_{l∈ℵ})≠(ℵ, (I_{l}⊔J_{l}, ∅)_{l∈ℵ})\}. \end{aligned} (2.10)

We define a partial ordering on A_{0}(ϱ) by setting

ρ′ ≡(ℵ′_{+}, (ρ′_{l})_{l∈ℵ}) ≺ ρ≡(ℵ_{+}, (ρ_{l})_{l∈ℵ})  (2.11)

if ρ′ ≠ρ, ℵ′_{+} ⊂ℵ_{+}, and for every l ∈ℵ either ρ′_{l} =ρ_{l}, ρ′_{l} ≺ρ_{l}, or ρ′_{l} =(I_{l}⊔J_{l}, ∅). Let < be an ordering on A_{0}(ϱ) extending the partial ordering ≺. We denote the corresponding minimal and maximal elements of A_{0}(ϱ) by ρ_{min} and ρ_{max}, respectively. If ρ∈A_{0}(ϱ), we define

ρ−1 ∈ {0}⊔A_{0}(ϱ)

as in (2.7).

If ρ∈A_{0}(ϱ) is as in (2.11), let

\overline{M}_{0,ρ} = ∏_{l∈ℵ} \overline{M}_{0,ρ_{l}}, F_{ρ} = ⨁_{l∈ℵ^{+}} π_{l}^{∗}L_{0}|_{\overline{M}_{0,ρ}} ⊂ F_{ϱ},

and  M̃_{0,ρ}^{0} =ℙF_{ρ} ⊂ M̃_{0,ϱ}^{0} ≡ℙF_{ϱ}.

+ + +  \begin{matrix}ℵ=\{1, 2, 3\} \\ |I_{1}⊔J_{1}|=2 \\ |I_{2}⊔J_{2}|=|I_{3}⊔J_{3}|=3 \\ ℵ_{+} =\{1, 2, 3\} \\ ρ_{1} =(I_{1}⊔J_{1}, ∅) \\ ρ_{2} =(I_{2}⊔J_{2}, ∅) \\ ρ_{3} =(I_{3}⊔J_{3}, ∅) \end{matrix}  − + +  \begin{matrix}ℵ=\{1, 2, 3\} \\ |I_{1}⊔J_{1}|=2 \\ |I_{2}⊔J_{2}|=|I_{3}⊔J_{3}|=3 \\ ℵ_{+} =\{2, 3\} \\ ρ_{1} =(I_{1}⊔J_{1}, ∅) \\ ρ_{2} ≠(I_{2}⊔J_{2}, ∅) \\ ρ_{3} =(I_{3}⊔J_{3}, ∅) \end{matrix}

Figure 10: Typical elements of M̃_{0,ϱ}^{0} and M̃_{0,ρ}^{0}

The spaces M̃_{0,ϱ}^{0} and M̃_{0,ρ}^{0} can be represented by diagrams as in Figure 10. The trees of circles attached to the vertical lines correspond to the tuples ρ_{l}, with conventions as in the first, symplectictopology, diagram in Figure 8. For each such tree, the marked point 0 is the point on the line. We indicate the elements of ℵ_{+} ⊂ℵ with plus signs next to these points. Note that by (2.10), every dot on a vertical line for which the corresponding tree of circles contains more than one circle must be labeled with a plus sign. From Lemma 2.5, we immediately obtain

Lemma 2.7 Suppose ℵ is a nonempty finite set and ϱ=(I_{l}, J_{l})_{l∈ℵ} is a tuple of finite sets such that I_{l} ≠ ∅ and |I_{l}|+|J_{l}| ≥ 2 for all l ∈ ℵ. If A_{0}(ϱ) is as above, the collection {M̃_{0,ρ}^{0}}_{ρ∈A_{0}(ϱ)} is properly intersecting.

We now describe the starting data for the sequential blowup construction of this subsection. Let

𝔼_{0} =γ_{ϱ} ⟶ M̃_{0,ϱ}^{0} =ℙF_{ϱ}  and  L_{0,(l,i)} =π_{0,ϱ}^{∗}π_{l}^{∗}L_{i} ⟶ M̃_{0,ϱ}^{0} ∀ i∈I_{l}, l ∈ℵ.

We take

s_{0,(l,i)} ∈ Γ(M̃_{0,ϱ}^{0}; Hom(L_{0,(l,i)}, 𝔼_{0}^{∗}))

to be the section induced by π_{0,ϱ}^{∗}π_{l}^{∗}s_{i}, with s_{i} defined by (2.3). It follows immediately from (2.4) that

s_{0,(l,i)}^{−1}(0) = ∑_{ρ^{∗}∈B_{0}(ϱ;l,i)} M̃_{0,ρ^{∗}}^{0}, where

\begin{matrix}B_{0}(ϱ; l, i) = \{(ℵ_{+}, (ρ_{l′})_{l′∈ℵ})∈A_{0}(ϱ): ℵ_{+} =ℵ−\{l\} and ρ_{l′} =(I_{l′} ⊔J_{l′}, ∅) ∀l′ ∈ℵ, or \\ ℵ_{+} =ℵ, ρ_{l} ∈B_{0}(I_{l}⊔J_{l}; i), ρ_{l′} =(I_{l′} ⊔J_{l′}, ∅) ∀l′ ∈ℵ−\{l\} & \}. \end{matrix}

The rest of the construction proceeds as in Subsection 2.3. The analogue of (2.9) now is

L_{ρ,(l,i)} = \begin{cases}π̃_{ρ}^{∗}L_{ρ−1,(l,i)}, & if l ∉ℵ_{+} or ρ_{l} ≠(I_{l}⊔J_{l}, ∅), i∉I_{l,P}; \\ π̃_{ρ}^{∗}L_{ρ−1,(l,i)} ⊗ O(−M̃_{0,ρ}^{ρ}), & otherwise; \end{cases}  (2.12)
𝔼_{ρ} = π̃_{ρ}^{∗} 𝔼_{ρ−1}⊗O(M̃_{0,ρ}^{ρ}).  (2.13)

As before, we take

M̃_{0,ϱ} = M̃_{0,ϱ}^{ρ_{max}};  𝔼̃ = 𝔼_{ρ_{max}};
L̃_{(l,i)} = L_{ρ_{max},(l,i)} and s̃_{(l,i)} = s_{ρ_{max},(l,i)} ∀ i∈I_{l}, l ∈ℵ.

The analogue of the inductive assumption (I5) insures that each section s̃_{(l,i)} does not vanish. The statement and the proof of Lemma 2.6 remain valid in the present setting, with only minor changes. Thus, the end result of the above blowup construction is again well-defined, i.e. independent of the choice of the ordering < extending the partial ordering ≺.

3 A Blowup of a Moduli Space of Genus-Zero Maps

3.1 Blowups and Immersions

In this section we construct blowups of certain moduli spaces of genus-zero maps; see Subsections 3.3 and 3.4. As outlined in Subsection 1.3, these blowups appear in Subsection 4.3 as the second factor in the domain of the immersions induced by the immersions ι_{σ} of Subsection 1.2.

As in Section 2, we begin by introducing convenient terminology and reviewing standard facts from algebraic geometry. If \overline{𝔐} is a variety, we denote its Zariski tangent space and its tangent cone by T \overline{𝔐} and T C\overline{𝔐}, respectively. If X is a smooth variety (but not necessarily equidimensional), we recall that a morphism ι_{X}: X ⟶\overline{𝔐} is an immersion if the differential of ι_{X},

dι_{X}: T X ⟶ ι_{X}^{∗}T C\overline{𝔐},

is injective at every point of X. Let

Im^{s} ι_{X} ≡ \{p∈\overline{𝔐}: |ι_{X}^{−1}(p)|≥2\}  and  N_{ι_{X}} ≡ ι_{X}^{∗}T C\overline{𝔐}/Im dι_{X}

be the singular locus of ι_{X} and the normal cone of ι_{X} in \overline{𝔐}, respectively. We denote by

π_{ι_{X}}^{⊥}: ι_{X}^{∗}T C\overline{𝔐} ⟶ N_{ι_{X}}

the projection map. If Z is a subvariety of \overline{𝔐}, let

ι_{Z}: Z ⟶ \overline{𝔐}

the inclusion map.

Definition 3.1 Let \overline{𝔐} be a variety.

(1) An immersion ι_{X} : X ⟶ \overline{𝔐} is properly self-intersecting if for all x_{1}, x_{2} ∈ X such that ι_{X}(x_{1})=ι_{X}(x_{2}) and sufficiently small neighborhoods U_{1} of x_{1} and U_{2} of x_{2} in X

T C_{ι_{X}(x_{1})}(ι_{X}(U_{1})∩ι_{X}(U_{2})) = Im dι_{X}|_{x_{1}} ∩ Im dι_{X}|_{x_{2}} ⊂ T C_{ι_{X}(x_{1})}\overline{𝔐}. ^{8}

(2) If ι_{X}: X ⟶\overline{𝔐} and ι_{Y} : Y ⟶\overline{𝔐} are immersions such that ι_{X} is properly self-intersecting, ι_{X}

is properly self-intersecting relative to ι_{Y} if for all x_{1}, x_{2} ∈X and y ∈Y such that

ι_{X}(x_{1}) = ι_{X}(x_{2}) = ι_{Y} (y)

and for all sufficiently small neighborhoods U_{1} of x_{1} and U_{2} of x_{2} in X,

π_{ι_{Y}}^{⊥} |_{y}(T C_{ι_{Y} (y)}(ι_{X}(U_{1})∩ι_{X}(U_{2}))) = π_{ι_{Y}}^{⊥} |_{y}Im dι_{X}|_{x_{1}} ∩ π_{ι_{Y}}^{⊥} |_{y}Im dι_{X}|_{x_{2}} ⊂ N_{ι_{Y}} |_{y}.

8We emphasize that intersections are taken to be set-theoretic intersections unless otherwise noted.

This definition generalizes Definition 2.1; see the paragraph following the latter for some examples.

Definition 3.2 If \overline{𝔐} is a variety, a collection \{ι_{ϱ} : X_{ϱ} ⟶ \overline{𝔐}\}_{ϱ∈A} of immersions is properly self-intersecting if for all ρ_{1}, ρ_{2}, ρ_{3} ∈ A the immersion ι_{ρ_{1}} ⊔ ι_{ρ_{2}} is properly self-intersecting relative to ι_{ρ_{3}}.

The next lemma follows from a local computation. (The local geometry of a proper self-intersection is particularly simple.)

Lemma 3.3 Suppose \overline{𝔐} is a variety and Z is a smooth subvariety of \overline{𝔐}.

(1) If ι_{X} : X ⟶ \overline{𝔐} is an immersion such that the immersion ι_{X} ⊔ι_{Z} : X ⊔Z ⟶ \overline{𝔐} is properly self-intersecting, then ι_{X} lifts to an immersion

Pr_{Z}ι_{X}: Bl_{ι_{X}^{−1}(Z)}X ⟶ Bl_{Z} \overline{𝔐}  s.t.  Im Pr_{Z}ι_{X} = Pr_{Z} Im ι_{X}.

(2) If in addition ι_{X} is properly self-intersecting relative to ι_{Z}, then Pr_{Z}ι_{X} is properly selfintersecting and

Im^{s} Pr_{Z}ι_{X} = Pr_{Z} Im^{s} ι_{X}.

(3) If in addition ι_{Y} : Y ⟶ \overline{𝔐} is an immersion such that ι_{X} ⊔ ι_{Y} ⊔ι_{Z} is properly self-intersecting and ι_{X} is properly self-intersecting relative to ι_{Y}, then Pr_{Z}ι_{X} is properly self-intersecting relative to Pr_{Z}ι_{Y}. Furthermore,

{Pr_{Z}ι_{X}}^{−1}(Pr_{Z}Im ι_{Y} ) = Pr_{ι_{X}^{−1}(Z)}ι_{X}^{−1}(Im ι_{Y}).

Remark: Since we always require that the blowup locus be smooth, an implicit conclusion of (1) of Lemma 3.3 is that ι_{X}^{−1}(Z) is a smooth subvariety of X; this is immediate from the local situation. Note that X itself is smooth, as it is the domain of the immersion ι_{X}.

Corollary 3.4 If \overline{𝔐} is a variety, \{ι_{ϱ}: X_{ϱ} ⟶ \overline{𝔐}\}_{ϱ∈A} is a properly self-intersecting collection of immersions, and ϱ ∈ A is such that ι_{ϱ} is an embedding, then {Pr_{Im ι_{ϱ}}ι_{ϱ′}}_{ϱ′∈A−{ϱ}} is a properly self-intersecting collection of immersions into Bl_{Im ι_{ϱ}} \overline{𝔐}.

Like Lemma 3.3, the next lemma follows from a local computation, using the simple geometry of a proper self-intersection.

Lemma 3.5 Suppose \overline{𝔐} is a smooth variety, Z is a smooth subvariety of \overline{𝔐}, ι_{X}: X ⟶ \overline{𝔐} is an immersion such that the immersion ι_{X} ⊔ι_{Z} is properly self-intersecting. Let

ι_{X}^{−1}(Z) = ⨆_{ϱ∈A} Z_{ϱ}

be the decomposition of ι_{X}^{−1}(Z) into path components. If there exist a splitting

N_{ι_{X}} = ⨁_{i∈I} L_{i} ⟶ X

and a subset I_{ϱ} of I for each ϱ∈A such that

ι_{X}|_{Z_{ϱ}}^{∗}T Z/T Z_{ϱ} = ⨁_{i∈I−I_{ϱ}}L_{i}|_{Z_{ϱ}}  ∀ ϱ∈A,  (3.1)

then

N_{Pr_{Z}ι_{X}} = ⨁_{i∈I} (π^{∗}L_{i} ⊗ ⨂_{i∈I_{ϱ}} O(−E_{ϱ})),

where E_{ϱ} is the component of the exceptional divisor for the blowup π: Bl_{ι_{X}^{−1}(Z)}X ⟶X that projects onto Z_{ϱ}.

We note that by (1) of Definition 3.1, the homomorphism

ι_{X}|_{Z_{ϱ}}^{∗}T Z/T Z_{ϱ} ⟶ N_{ι_{X}} ≡ι_{X}^{∗}T \overline{𝔐}/Im dι_{X}

induced by the inclusions is injective. Thus, we can identify ι_{X}|_{Z_{ϱ}}^{∗}T Z/T Z_{ϱ} with a subbundle of N_{ι_{X}}, as we have done in Lemma 3.5.

3.2 Moduli Spaces of Genus-Zero Maps

In this subsection, we describe natural subvarieties of the moduli space of genus-zero maps and a natural bundle section over them. This bundle section induces other bundle sections, introduced in the next two subsections, that are used in the blowup construction of Subsection 4.3 to describe the structure of the proper transforms of \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d); see Subsection 1.3 for more details. Below we also state two well-known facts in the Gromov-Witten theory, Lemmas 3.6 and 3.7, and a more recent result, Lemma 3.8.

If d∈ℤ^{+} and J is a finite set, let

A_{0}(d, J) = {(m; J_{P} , J_{B}): m∈ℤ^{+}, m≤d; J =J_{P} ⊔J_{B}, m+|J_{P} |≥2};  (3.2)
\overline{𝔐}_{0,(0,J)}(ℙ^{n}, d) = \overline{𝔐}_{0,\{0\}⊔J}(ℙ^{n}, d).

If σ = (m; J_{P} , J_{B}) is an element of A_{0}(d, J), let 𝔐_{0,σ}(ℙ^{n}, d) be the subset of \overline{𝔐}_{0,\{0\}⊔J}(ℙ^{n}, d) consisting of the stable maps [Σ, u] such that

(i) the components of Σ are Σ_{i} =ℙ^{1} with i∈{P }⊔[m];

(ii) u|_{Σ_{P}} is constant and the marked points on Σ_{P} are indexed by the set {0}⊔J_{P}; (iii) for each i∈[m], Σ_{i} is attached to Σ_{P} and u|_{Σ_{i}} is not constant.

We denote by \overline{𝔐}_{0,σ}(ℙ^{n}, d) the closure of 𝔐_{0,σ}(ℙ^{n}, d) in \overline{𝔐}_{0,\{0\}⊔J}(ℙ^{n}, d). Figure 11 illustrates this definition, from the points of view of symplectic topology and of algebraic geometry. In the first diagram, each disk represents a sphere, and we shade the components on which the map u is nonconstant. In the second diagram, the irreducible components of Σ are represented by lines, and the integer next to each component shows the degree of u on that component. In both cases, we indicate the marked points lying on the component Σ_{P} only.

We define a partial ordering on the set A_{0}(d, J) by setting

σ′ ≡(m′; J′_{P} , J′_{B}) ≺ σ ≡(m; J_{P} , J_{B})  if σ′ ≠σ, m′ ≤m, J′_{P} ⊂J_{P} .  (3.3)

0 j_{1}  j_{1} 0  d_{1} d_{2} d_{3}  \begin{matrix}m=3, J_{P} =\{j_{1}\} \\ d_{1}, d_{2}, d_{3} >0 \\ d_{1}+d_{2}+d_{3} =d \end{matrix}

Figure 11: A typical element of \overline{𝔐}_{0,σ}(ℙ^{n}, d)

Similarly to Subsection 2.2, this condition means that the elements of 𝔐_{0,σ′}(ℙ^{n}, d) can be obtained from the elements of 𝔐_{0,σ}(ℙ^{n}, d) by moving more points onto the bubble components or combining the bubble components; see Figure 12. As in the g =0 case of Subsection 2.2, the bubble components are the components not containing the marked point 0.

Lemma 3.6 If σ_{1}, σ_{2} ∈A_{0}(d, J), σ_{1} ≠σ_{2} σ_{1} ⊀σ_{2}, and σ_{2} ⊀σ_{1}, then

\overline{𝔐}_{0,σ_{1}}(ℙ^{n}, d) ∩ \overline{𝔐}_{0,σ_{2}}(ℙ^{n}, d) ⊂ \overline{𝔐}_{0,σ̃(σ_{1},σ_{2})}(ℙ^{n}, d),
where  σ̃(σ_{1}, σ_{2}) = max {σ′ ∈A_{0}(d, J): σ′ ≺σ_{1}, σ_{2}}.

If σ̃(σ_{1}, σ_{2}) is not defined, \overline{𝔐}_{0,σ_{1}}(ℙ^{n}, d) and \overline{𝔐}_{0,σ_{2}}(ℙ^{n}, d) are disjoint.

For example, if σ_{1} and σ_{2} correspond to the two diagrams on the right side of Figure 12, then σ̃(σ_{1}, σ_{2}) corresponds to the diagram on the left side of Figure 12. Lemma 3.6 is immediate from the definition of the topology on \overline{𝔐}_{0,\{0\}⊔J}(ℙ^{n}, d). It can also be easily deduced from [P, Subsect. 3.2] by an argument similar to the proof of Lemma 2.6.

If σ =(m; J_{P} , J_{B}) is an element of A_{0}(d, J), let

\overline{𝔐}_{σ;B}(ℙ^{n}, d) ⊂ ∏_{i∈[m]} ⨆_{d_{i}>0,J_{i}⊂J_{B}} \overline{𝔐}_{0,\{0\}⊔J_{i}}(ℙ^{n}, d_{i})  and

π_{i}: \overline{𝔐}_{σ;B}(ℙ^{n}, d) ⟶ ⨆_{d_{i}>0,J_{i}⊂J_{B}} \overline{𝔐}_{0,\{0\}⊔J_{i}}(ℙ^{n}, d_{i}), i∈[m],

0 ≺  _{0} _{j_{1}}  _{0}

Figure 12: Examples of partial ordering (3.3) be as in Subsection 1.2. Since each of the spaces \overline{𝔐}_{0,\{0\}⊔J_{i}}(ℙ^{n}, d_{i}) is smooth and each of the evaluation maps

ev_{0} : \overline{𝔐}_{0,\{0\}⊔J_{i}}(ℙ^{n}, d_{i}) ⟶ ℙ^{n}

is a submersion, the space \overline{𝔐}_{σ;B}(ℙ^{n}, d) is smooth. We denote by

ι_{σ}: \overline{M}_{0,\{0\}⊔[m]⊔J_{P}} ×\overline{𝔐}_{σ;B}(ℙ^{n}, d) ⟶ \overline{𝔐}_{0,σ}(ℙ^{n}, d) ⊂ \overline{𝔐}_{0,\{0\}⊔J}(ℙ^{n}, d)  (3.4)

the natural node-identifying map. It descends to an immersion

ῑ_{σ}: (\overline{M}_{0,\{0\}⊔[m]⊔J_{P}} ×\overline{𝔐}_{σ;B}(ℙ^{n}, d))/S_{m} ⟶ \overline{𝔐}_{0,\{0\}⊔J}(ℙ^{n}, d).

Let

π_{P} , π_{B}: \overline{M}_{0,\{0\}⊔[m]⊔J_{P}} ×\overline{𝔐}_{σ;B}(ℙ^{n}, d) ⟶ \overline{M}_{0,\{0\}⊔[m]⊔J_{P}} , \overline{𝔐}_{σ;B}(ℙ^{n}, d)

be the natural projection maps.

Lemma 3.7 If d ∈ ℤ^{+} and J is a finite set, the collections {ι_{σ}}_{σ∈A_{0}(d,J)} and {ῑ_{σ}}_{σ∈A_{0}(d,J)} of immersions are properly self-intersecting. If σ ∈A_{0}(d, J) is as in (3.3),

Im^{s} ῑ_{σ} ⊂ ⋃_{σ′≺σ}\overline{𝔐}_{0,σ′}(ℙ^{n}, d)  and  N_{ι_{σ}} = ⨁_{i∈[m]} π_{P}^{∗} L_{i}⊗π_{B}^{∗}π_{i}^{∗}L_{0}.

If in addition σ′ ∈A_{0}(d, J), σ′ ≺σ, and σ′ is as in (3.3), then

ι_{σ}^{−1}(\overline{𝔐}_{0,σ′}(ℙ^{n}, d)) = ( ⋃_{ρ∈A_{0}(σ;σ′)} \overline{M}_{0,ρ}) × \overline{𝔐}_{σ;B}(ℙ^{n}, d), where

A_{0}(σ; σ′) = {ρ=(I_{P} ⊔J′_{P} , {I_{k}⊔J_{k}: k ∈K})∈A_{0}([m], J_{P}): |K|+|I_{P} |=m′}

and A_{0}([m], J_{P}) and \overline{M}_{0,ρ} are as in Subsection 2.2. Finally, if ρ∈A_{0}(σ; σ′) is as above,

ι_{σ}|_{\overline{M}_{0,ρ}×\overline{𝔐}_{σ;B}(ℙ^{n},d)}^{∗}T \overline{𝔐}_{0,σ′}(ℙ^{n}, d)/T (\overline{M}_{0,ρ}×\overline{𝔐}_{σ;B}(ℙ^{n}, d)) = ⨁_{i∈[m]−I_{P}} π_{P}^{∗} L_{i}⊗π_{B}^{∗}π_{i}^{∗}L_{0}.

The first claim in the second sentence and the claim of the third sentence in Lemma 3.7 follow immediately from the definition of the topology on \overline{𝔐}_{0,\{0\}⊔J}(ℙ^{n}, d). The remaining claims are also restatements of standard facts in GW-theory; they all follow from the description of the tangent bundle of \overline{𝔐}_{0,\{0\}⊔J}(ℙ^{n}, d) in [MirSym, (27.6)].

We finish this subsection by describing a natural bundle section

D_{0} ∈ Γ(\overline{𝔐}_{0,\{0\}⊔J}(ℙ^{n}, d), Hom(L_{0}; ev_{0}^{∗}T ℙ^{n}))

which plays a central role in the rest of the paper. An element [b] ∈ \overline{𝔐}_{0,\{0\}⊔J}(ℙ^{n}, d) consists of a prestable nodal curve Σ with marked points and a map u: Σ ⟶ ℙ^{n}. One of the marked points is labeled by 0. We denote it by x_{0}(b). We define D_{0} by

D_{0}|_{b} = du|_{x_{0}(b)} : T_{x_{0}(b)}Σ ⟶ T_{ev_{0}(b)}ℙ^{n}.

If 𝔘⟶\overline{𝔐}_{0,\{0\}⊔J}(ℙ^{n}, d) is the universal curve and ev: 𝔘⟶ℙ^{n} is the natural evaluation map, then D_{0}|_{b} is simply the restriction of dev|_{x_{0}(b)} to the vertical tangent bundle of 𝔘. The bundle section D_{0} vanishes identically along the subvarieties \overline{𝔐}_{0,σ}(ℙ^{n}, d) with σ ∈A_{0}(d, J).

Lemma 3.8 If d ∈ ℤ^{+} and J is a finite set, the section D_{0} is transverse to the zero set on the complement of the subvarieties \overline{𝔐}_{0,σ}(ℙ^{n}, d) with σ ∈A_{0}(d, J). Furthermore, for every

σ ≡(m; J_{P} , J_{B}) ∈ A_{0}(d, J),

the differential of D_{0},

∇D_{0} : N_{ι_{σ}} ⟶ ι_{σ}^{∗} Hom(L_{0}, ev_{0}^{∗}T ℙ^{n}) = π_{P}^{∗} L_{0}^{∗}⊗π_{B}^{∗}ev_{0}^{∗}T ℙ^{n},

in the normal direction to the immersion ι_{σ} is given by

∇D_{0}|_{π_{P}^{∗} L_{i}⊗π_{B}^{∗}π_{i}^{∗}L_{0}} = π_{P}^{∗} s_{i}⊗π_{B}^{∗}π_{i}^{∗}D_{0} ∀ i∈[m],

where s_{i} is the homomorphism defined in Subsection 2.2.

The first claim of the lemma is an immediate consequence of the fact that

H^{1}(Σ; u^{∗}T ℙ^{n}⊗O(−2z)) = {0}

for every genus-zero stable map (Σ, u) and a smooth point z ∈ Σ such that the restriction of u to the irreducible component of Σ containing z is not constant. The second statement of the lemma follows from Theorem 2.8 in [Z1].

3.3 Initial Data

If ℵ and J are finite sets and d is positive integer, let

\begin{aligned}\overline{𝔐}_{0,(ℵ,J)}(ℙ^{n}, d) & = \{(b_{l})_{l∈ℵ} ∈ ∏_{l∈ℵ} \overline{𝔐}_{0,\{0\}⊔J_{l}}(ℙ^{n}, d_{l}): d_{l} ∈ℤ^{+}, ∑_{l∈ℵ} d_{l} =d; ⨆_{l∈ℵ} J_{l} =J; \\ & ev_{0}(b_{l})=ev_{0}(b_{l′}) ∀ l, l′ ∈ℵ\}; \end{aligned}

\begin{aligned}𝔐_{0,(ℵ,J)}(ℙ^{n}, d) & = \{(b_{l})_{l∈ℵ} ∈ ∏_{l∈ℵ} 𝔐_{0,\{0\}⊔J_{l}}(ℙ^{n}, d_{l}): d_{l} ∈ℤ^{+}, ∑_{l∈ℵ} d_{l} =d; ⨆_{l∈ℵ} J_{l} =J; \\ & ev_{0}(b_{l})=ev_{0}(b_{l′}) ∀ l, l′ ∈ℵ\}, \end{aligned}

where 𝔐_{0,\{0\}⊔J_{l}}(ℙ^{n}, d_{l}) is the subset of \overline{𝔐}_{0,\{0\}⊔J_{l}}(ℙ^{n}, d_{l}) consisting of stable maps with smooth domains. For each l ∈ℵ, let

π_{l}: \overline{𝔐}_{0,(ℵ,J)}(ℙ^{n}, d) ⟶ ⨆_{d_{l}>0,J_{l}⊂J} \overline{𝔐}_{0,\{0\}⊔J_{l}}(ℙ^{n}, d_{l})

be the projection map. We put

F_{(ℵ,J)} = ⨁_{l∈ℵ} π_{l}^{∗}L_{0},

where L_{0} ⟶ \overline{𝔐}_{0,\{0\}⊔J_{l}}(ℙ^{n}, d_{l}) is the universal tangent line bundle for the marked point 0. In the next subsection, we construct a blowup

π_{0,(ℵ,J)} : 𝔐̃_{0,(ℵ,J)}(ℙ^{n}, d) ⟶ ℙF_{(ℵ,J)}

of the projective bundle ℙF_{(ℵ,J)} over \overline{𝔐}_{0,(ℵ,J)}(ℙ^{n}, d) and a line bundle

𝔼̃ ⟶ 𝔐̃_{0,(ℵ,J)}(ℙ^{n}, d).

We also describe a natural bundle section

D̃_{(ℵ,J)} ∈ Γ(𝔐̃_{0,(ℵ,J)}(ℙ^{n}, d); 𝔼̃^{∗}⊗π_{0,(ℵ,J)}^{∗}π_{ℙF_{(ℵ,J)}}^{∗}ev_{0}^{∗}T ℙ^{n}),

where

π_{ℙF_{(ℵ,J)}} : ℙF_{(ℵ,J)} ⟶ \overline{𝔐}_{0,(ℵ,J)}(ℙ^{n}, d)

is the bundle projection map. This section is transverse to the zero set.

Similarly to Subsection 2.4, the smooth variety 𝔐̃_{0,(ℵ,J)}(ℙ^{n}, d) is obtained by blowing up the subvarieties 𝔐̃_{0,ϱ}^{0}(ℙ^{n}, d) defined below and their proper transforms in an order consistent with a natural partial ordering ≺. The line bundle 𝔼̃ is the sum of the tautological line bundle

γ_{(ℵ,J)} ⟶ ℙF_{(ℵ,J)}

and all exceptional divisors. The section D̃_{(ℵ,J)} is induced from the sections π_{l}^{∗}D_{0}, with l ∈ℵ, where D_{0} is as in Subsection 3.2.

If ℵ, J, and d are as above, let

\begin{matrix}A_{0}(ℵ; d, J) = \{((σ_{l})_{l∈ℵ}, J_{B}): (σ_{l}, ∅)∈\{(0, ∅)\}⊔A_{0}(d_{l}, J_{l,P}), (σ_{l})_{l∈ℵ} ≠(0)_{l∈ℵ}; \\ ∑_{l∈ℵ} d_{l} =d, J = J_{B} ⊔ ⨆_{l∈ℵ} J_{l,P} & \}. \end{matrix}

We define a partial ordering ≺ on A_{0}(ℵ; d, J) by setting

ϱ′ ≡((σ′_{l})_{l∈ℵ}, J′_{B}) ≺ ϱ≡((σ_{l})_{l∈ℵ}, J_{B})  (3.5)

if ϱ′ ≠ ϱ and for every l ∈ ℵ either σ′_{l} = σ_{l}, (σ′_{l}, ∅) ≺ (σ_{l}, ∅), or σ′_{l} = 0. If ϱ ∈ A_{0}(ℵ; d, J) is as in (3.5), we put

ℵ_{P} (ϱ) = {l ∈ℵ: σ_{l} ≠0}  and  ℵ_{S}(ϱ) = {l ∈ℵ: σ_{l} =0}.

Here P and S stand for the subsets of principal and secondary elements of ℵ, respectively; see the next paragraph. Note that

ϱ′ ≺ ϱ ⟹ ℵ_{P} (ϱ′) ⊂ ℵ_{P} (ϱ),  ℵ_{P} (ϱ) ≠ ∅ ∀ ϱ∈A_{0}(ℵ; d, J),  and
ϱ = ((m_{l}; J_{l,P})_{l∈ℵ_{P} (ϱ)}, (0)_{l∈ℵ_{S}(ϱ)}, J_{B})  (3.6)

for some m_{l} and J_{l,P}. Choose an ordering < on A_{0}(ℵ; d, J) extending the partial ordering ≺. We denote the corresponding minimal and maximal element by ϱ_{min} and ϱ_{max}, respectively. For every ϱ∈A_{0}(ℵ; d, J), define

ϱ−1 ∈ {0}⊔A_{0}(ℵ; d, J)

as in (2.7).

If ϱ∈A_{0}(ℵ; d, J) is as in (3.5), let

\begin{matrix}\overline{𝔐}_{0,ϱ}(ℙ^{n}, d) = \{(b_{l})_{l∈ℵ} ∈ ∏_{l∈ℵ} \overline{𝔐}_{0,(σ_{l},J_{l,B})}(ℙ^{n}, d_{l}): ∑_{l∈ℵ} d_{l} =d; ⨆_{l∈ℵ} J_{l,B} =J_{B}; \\ ev_{0}(b_{l_{1}})=ev_{0}(b_{l_{2}}) ∀ l_{1}, l_{2} ∈ℵ & \} ⊂ \overline{𝔐}_{0,(ℵ,J)}(ℙ^{n}, d). \end{matrix}

With •=P, S, we define

F_{ϱ;•} = ⨁_{l∈ℵ_{•}(ϱ)}π_{l}^{∗}L_{0}|_{\overline{𝔐}_{0,ϱ}(ℙ^{n},d)} ⊂ F_{(ℵ,J)}|_{\overline{𝔐}_{0,ϱ}(ℙ^{n},d)}.

Let

𝔐̃_{0,ϱ}^{0}(ℙ^{n}, d) = ℙF_{ϱ;P} ⊂ 𝔐̃_{0,(ℵ,J)}^{0}(ℙ^{n}, d)≡ℙF_{(ℵ,J)}.

From Lemma 3.6, we immediately obtain Lemma 3.9 If ϱ_{1}, ϱ_{2} ∈A_{0}(ℵ; d, J), ϱ_{1} ≠ϱ_{2} ϱ_{1} ⊀ϱ_{2}, and ϱ_{2} ⊀ϱ_{1}, then

𝔐̃_{0,ϱ_{1}}^{0}(ℙ^{n}, d) ∩ 𝔐̃_{0,ϱ_{2}}^{0}(ℙ^{n}, d) ⊂ 𝔐̃_{0,ϱ̃(ϱ_{1},ϱ_{2})}^{0}(ℙ^{n}, d),

where ϱ̃(ϱ_{1}, ϱ_{2}) = max {ϱ′ ∈A_{0}(ℵ; d, J): ϱ′ ≺ϱ_{1}, ϱ_{2}}.

If ϱ̃(ϱ_{1}, ϱ_{2}) is not defined, 𝔐̃_{0,ϱ_{1}}^{0}(ℙ^{n}, d) and 𝔐̃_{0,ϱ_{2}}^{0}(ℙ^{n}, d) are disjoint. With ϱ as (3.6), let

ϱ_{P} = ([m_{l}], J_{l,P} )_{l∈ℵ_{P} (ϱ)}, ℵ_{B}(ϱ) = ℵ_{S}(ϱ) ⊔ ⨆_{l∈ℵ_{P} (ϱ)} [m_{l}], J_{B}(ϱ) = J_{B}, and G_{ϱ} = ∏_{l∈ℵ_{P} (ϱ)} S_{m_{l}}.

With M̃_{0,ϱ_{P}}^{0} as in Subsection 2.4, we denote by

ι_{0,ϱ}: M̃_{0,ϱ_{P}}^{0} × \overline{𝔐}_{0,(ℵ_{B}(ϱ),J_{B}(ϱ))}(ℙ^{n}, d) ⟶ 𝔐̃_{0,ϱ}^{0}(ℙ^{n}, d) ⊂ 𝔐̃_{0,(ℵ,J)}^{0}(ℙ^{n}, d)

the natural node-identifying map induced by the immersions ι_{(σ_{l},J_{l,B})} in (3.4). It descends to an immersion

ῑ_{0,ϱ}: (M̃_{0,ϱ_{P}}^{0} ×\overline{𝔐}_{0,(ℵ_{B}(ϱ),J_{B}(ϱ))}(ℙ^{n}, d))/G_{ϱ} ⟶ 𝔐̃_{0,(ℵ,J)}^{0}(ℙ^{n}, d).

Let

π_{P} , π_{B}: M̃_{0,ϱ_{P}}^{0} × \overline{𝔐}_{0,(ℵ_{B}(ϱ),J_{B}(ϱ))}(ℙ^{n}, d) ⟶ M̃_{0,ϱ_{P}}^{0} , \overline{𝔐}_{0,(ℵ_{B}(ϱ),J_{B}(ϱ))}(ℙ^{n}, d)

be the projection maps. For the rest of this section, as well as for Section 4, we take

\overline{𝔐}_{0,(ℵ,J)} = \overline{𝔐}_{0,(ℵ,J)}(ℙ^{n}, d), 𝔐_{0,(ℵ,J)} = 𝔐_{0,(ℵ,J)}(ℙ^{n}, d), 𝔐̃_{0,(ℵ,J)}^{0} = 𝔐̃_{0,(ℵ,J)}^{0}(ℙ^{n}, d) ∀ (ℵ, J); 𝔐̃_{0,ϱ}^{0} = 𝔐̃_{0,ϱ}^{0}(ℙ^{n}, d) ∀ ϱ∈A_{0}(ℵ; d, J).

Lemma 3.10 If ℵ and J are finite sets and d∈ℤ^{+}, the collections

{ι_{0,ϱ}}_{ϱ∈A_{0}(ℵ;d,J)}  and  {ῑ_{0,ϱ}}_{ϱ∈A_{0}(ℵ;d,J)}

of immersions are properly self-intersecting. If

ϱ^{∗} ≡((m_{l}^{∗}; J_{l,P}^{∗})_{l∈ℵ_{P} (ϱ^{∗})}, (0)_{l∈ℵ_{S}(ϱ^{∗})}, J_{B}^{∗}) ∈ A_{0}(ℵ; d, J),

then

Im^{s} ῑ_{0,ϱ^{∗}} ⊂ ⋃_{ϱ′≺ϱ^{∗}}𝔐̃_{0,ϱ′}^{0} and

N_{ι_{0,ϱ^{∗}}} = ⨁_{l∈ℵ_{P} (ϱ^{∗})} ⨁_{i∈[m_{l}^{∗}]}π_{P}^{∗} L_{0,(l,i)}⊗π_{B}^{∗}π_{(l,i)}^{∗}L_{0} ⊕ π_{P}^{∗} 𝔼_{0}^{∗}⊗ π_{B}^{∗} ⨁_{l∈ℵ_{S}(ϱ^{∗})} π_{l}^{∗}L_{0},

where 𝔼_{0} and L_{0,(l,i)} are as in Subsections 2.4. If ϱ, ϱ^{∗} ∈A_{0}(ℵ; d, J), ϱ is as (3.6), and ϱ≺ϱ^{∗}, then

ι_{0,ϱ^{∗}}^{−1}(𝔐̃_{0,ϱ}^{0}) = ( ⋃_{ρ∈A_{0}(ϱ^{∗};ϱ)} M̃_{0,ρ}^{0}) × \overline{𝔐}_{0,(ℵ_{B}(ϱ^{∗}),J_{B}(ϱ^{∗}))},

where

\begin{matrix}A_{0}(ϱ^{∗}; ϱ) = \{ρ=(ℵ_{P} (ϱ), (I_{l,P}^{∗} ⊔J_{l,P} , \{I_{l,k}^{∗}⊔J_{l,k}^{∗}: k ∈K_{l}^{∗}\})_{l∈ℵ_{P} (ϱ^{∗})})∈A_{0}(ϱ_{P}^{∗}): \\ |K_{l}^{∗}|+|I_{l,P}^{∗} |=m_{l} ∀ l ∈ℵ_{P} (ϱ^{∗}) & \} \end{matrix}

and A_{0}(ϱ_{P}^{∗}) and M̃_{0,ρ}^{0} are as in Subsection 2.4. Finally, if ρ∈A_{0}(ϱ^{∗}; ϱ) is as above,

\begin{matrix}ι_{0,ϱ^{∗}} |_{M̃_{0,ρ}^{0}×\overline{𝔐}_{0,(ℵ_{B} (ϱ^{∗}),J_{B} (ϱ^{∗}))}}^{∗}T \overline{𝔐}_{0,ϱ}/T (M̃_{0,ρ}^{0}×\overline{𝔐}_{0,(ℵ_{B}(ϱ^{∗}),J_{B}(ϱ^{∗}))}) \\ = ⨁_{l∈ℵ_{P} (ϱ^{∗})−ℵ_{P} (ϱ)} ⨁_{i∈[m_{l}^{∗}]}π_{P}^{∗} L_{0,(l,i)}⊗π_{B}^{∗}π_{(l,i)}^{∗}L_{0} ⊕ ⨁_{l∈ℵ_{P} (ϱ)} ⨁_{i∈[m_{l}^{∗}]−I_{l,P}} π_{P}^{∗} L_{0,(l,i)}⊗π_{B}^{∗}π_{(l,i)}^{∗}L_{0}. \end{matrix}

The normal bundle N_{ι_{0,ϱ^{∗}}} for the immersion ι_{0,ϱ^{∗}} splits into horizontal and vertical bundles:

N_{ι_{0,ϱ^{∗}}} = N_{ι_{0,ϱ^{∗}}}^{⊥} ⊕ N_{ι_{0,ϱ^{∗}}}^{⊤} .

It is immediate from the definitions that

N_{ι_{0,ϱ^{∗}}}^{⊤} = ι_{0,ϱ^{∗}}^{∗}(γ_{(ℵ,J)}^{∗}⊗F_{ϱ;S}) = π_{P}^{∗} 𝔼_{0}^{∗}⊗ π_{B}^{∗} ⨁_{l∈ℵ_{S}(ϱ^{∗})} π_{l}^{∗}L_{0}.

The horizontal normal bundle N_{ι_{0,ϱ^{∗}}}^{⊥} is the pullback of the normal bundle for the node identifying immersion

\overline{M}_{0,ϱ_{P}^{∗}} × \overline{𝔐}_{0,(ℵ_{B}(ϱ^{∗}),J_{B}(ϱ^{∗}))} ⟶ \overline{𝔐}_{0,ϱ^{∗}}(ℙ^{n}, d) ⊂ \overline{𝔐}_{0,(ℵ,J)}

induced by the immersions ι_{(σ_{l}^{∗},J_{l,B}^{∗})} in (3.4) by the bundle projection map π_{ℙF_{ϱ;P}}. The normal bundle for this immersion is the sum of component-wise normal bundles given by Lemma 3.7. The remaining claims of Lemma 3.10 follow easily from the corresponding statements of Lemma 3.7 as well.

We note that for every ϱ^{∗} ∈A_{0}(ℵ; d, J),

A_{0}(ϱ_{P}^{∗}) = ⨆_{ϱ≺ϱ^{∗}}A_{0}(ϱ^{∗}; ϱ).

Furthermore, if ϱ_{1}, ϱ_{2} ∈A_{0}(ℵ; d, J) are such that ϱ_{1}, ϱ_{2} ≺ϱ^{∗}, then

ρ_{1}^{∗} ∈A_{0}(ϱ^{∗}; ϱ_{1}), ρ_{2}^{∗} ∈A_{0}(ϱ^{∗}; ϱ_{2}), ρ_{1}^{∗} ≺ρ_{2}^{∗}  ⟹  ϱ_{1} ≺ϱ_{2}.

Thus, we can choose an ordering < on A_{0}(ϱ_{P}^{∗}) extending the partial ordering ≺ of Subsection 2.4 such that

ϱ_{1} <ϱ_{2}, ρ_{1}^{∗} ∈A_{0}(ϱ^{∗}; ϱ_{1}), ρ_{2}^{∗} ∈A_{0}(ϱ^{∗}; ϱ_{2})  ⟹  ρ_{1}^{∗} <ρ_{2}^{∗},

whenever ϱ_{1}, ϱ_{2} ∈ A_{0}(ℵ; d, J) are such that ϱ_{1}, ϱ_{2} ≺ ϱ^{∗}. In the next subsection, we will refer to the blowup construction of Subsection 2.4 corresponding to such an ordering.

Via the projection maps π_{l}, the bundle sections D_{0} of Subsection 3.2 induce a linear bundle map

D_{(ℵ,J)} : F_{(ℵ,J)} ⟶ ev_{0}^{∗}T ℙ^{n}

over \overline{𝔐}_{0,(ℵ,J)}. In turn, this homomorphism induces a bundle section

D̃_{0} ∈ Γ(𝔐̃_{0,(ℵ,J)}^{0}; 𝔼_{0}^{∗}⊗π_{ℙF_{(ℵ,J)}}^{∗}ev_{0}^{∗}T ℙ^{n}), where 𝔼_{0} = γ_{(ℵ,J)} ⟶ 𝔐̃_{0,(ℵ,J)}^{0}.

This section vanishes identically on the subvarieties 𝔐̃_{0,ϱ}^{0} of 𝔐̃_{0,(ℵ,J)}^{0} with ϱ∈A_{0}^{∗}(ℵ; d, J).

Lemma 3.11 The section D̃_{0} is transverse to the zero set on the complement of the subvarieties 𝔐̃_{0,ϱ^{∗}}^{0} with ϱ^{∗} ∈ A_{0}(ℵ; d, J). Furthermore, for every ϱ^{∗} ∈ A_{0}(ℵ; d, J) as in Lemma 3.10, the differential of D̃_{0},

∇D̃_{0} : N_{ι_{0,ϱ^{∗}}} ⟶ ι_{0,ϱ^{∗}}^{∗}(𝔼_{0}^{∗}⊗π_{ℙF_{(ℵ,J)}}^{∗}ev_{0}^{∗}T ℙ^{n}) = π_{P}^{∗} 𝔼_{0}^{∗}⊗π_{B}^{∗}ev_{0}^{∗}T ℙ^{n},

in the normal direction to the immersion ι_{0,ϱ^{∗}} is given by

∇D̃_{0}|_{π_{P}^{∗} L_{0,(l,i)}⊗π_{B}^{∗}π_{(l,i)}^{∗}L_{0}} = π_{P}^{∗} s_{0,(l,i)}⊗π_{B}^{∗}π_{(l,i)}^{∗}D_{0} ∀ i∈[m_{l}^{∗}], l ∈ℵ_{P} (ϱ^{∗}),
and ∇D̃_{0}|_{N_{ι_{0,ϱ^{∗}}}^{⊤}} = π_{P}^{∗} id⊗π_{B}^{∗}D_{(ℵ_{B}(ϱ^{∗}),J_{B}(ϱ^{∗}))},

where s_{0,(l,i)} is the homomorphism defined in Subsection 2.4.

This lemma follows immediately from Lemma 3.8.

3.4 Inductive Construction

We are now ready to describe the inductive assumptions for our construction of the blowup

π_{0,(ℵ,J)} : 𝔐̃_{0,(ℵ,J)}^{ϱ_{max}} ≡𝔐̃_{0,(ℵ,J)}(ℙ^{n}, d) ⟶ 𝔐̃_{0,(ℵ,J)}^{0} ≡𝔐̃_{0,(ℵ,J)}^{0}(ℙ^{n}, d).

Suppose ϱ∈A_{0}(ℵ; d, J) and we have constructed (I1) a blowup π_{ϱ−1} : 𝔐̃_{0,(ℵ,J)}^{ϱ−1} ⟶ 𝔐̃_{0,(ℵ,J)}^{0} such that π_{ϱ−1} is an isomorphism outside of the

preimage of the spaces 𝔐̃_{0,ϱ′}^{0} with ϱ′ ≤ϱ−1; (I2) a line bundle 𝔼_{ϱ−1} ⟶𝔐̃_{0,(ℵ,J)}^{ϱ−1};

(I3) a section D̃_{ϱ−1} ∈ Γ(𝔐̃_{0,(ℵ,J)}^{ϱ−1}; 𝔼_{ϱ−1}^{∗}⊗π_{ϱ−1}^{∗}π_{ℙF_{(ℵ,J)}}^{∗}ev_{0}^{∗}T ℙ^{n}).

For each ϱ^{∗} >ϱ−1, let

𝔐̃_{0,ϱ^{∗}}^{ϱ−1} ≡ 𝔐̃_{0,ϱ^{∗}}^{ϱ−1}(ℙ^{n}, d) ⊂ 𝔐̃_{0,(ℵ,J)}^{ϱ−1}

be the proper transform of 𝔐̃_{0,ϱ^{∗}}^{0} in 𝔐̃_{0,(ℵ,J)}^{ϱ−1}. We assume that

(I4) the section D̃_{ϱ−1} is transverse to the zero set on the complement of the subvarieties 𝔐̃_{0,ϱ^{∗}}^{ϱ−1} with ϱ^{∗} >ϱ−1 and vanishes identically along these subvarieties;

(I5) if ϱ_{1}, ϱ_{2} ∈A_{0}(ℵ; d, J) are such that ϱ_{1} ≠ϱ_{2}, ϱ_{1} ⊀ϱ_{2}, ϱ_{2} ⊀ϱ_{1}, and ϱ−1<ϱ_{1}, ϱ_{2}, then

𝔐̃_{0,ϱ_{1}}^{ϱ−1} ∩ 𝔐̃_{0,ϱ_{2}}^{ϱ−1} \begin{cases}⊂ 𝔐̃_{0,ϱ̃(ϱ_{1},ϱ_{2})}^{ϱ−1}, & if ϱ̃(ϱ_{1}, ϱ_{2})>ϱ−1; \\ = ∅, & otherwise, \end{cases}

where ϱ̃(ϱ_{1}, ϱ_{2}) is as in Lemma 3.10.

We also assume that for all ϱ^{∗} ∈A_{0}(ℵ; d, J) such that ϱ^{∗} >ϱ−1:

(I6) the domain of the G_{ϱ^{∗}}-invariant immersion ι_{ϱ−1,ϱ^{∗}} induced by ι_{0,ϱ^{∗}} is

M̃_{0,ϱ_{P}^{∗}}^{ρ_{ϱ^{∗}}(ϱ−1)} × \overline{𝔐}_{0,(ℵ_{B}(ϱ^{∗}),J_{B}(ϱ^{∗}))}, where

ρ_{ϱ^{∗}}(ϱ−1) = \begin{cases}max \{ρ∈A_{0}(ϱ^{∗}; ϱ′): ϱ′ ≤ϱ−1, ϱ′ ≺ϱ^{∗}\}, & \begin{matrix}if ∃ϱ′ ∈A_{0}(ℵ; d, J) \\ s.t. ϱ′ ≤ϱ−1, ϱ′ ≺ϱ^{∗}; \end{matrix} \\ 0, & otherwise; \end{cases}

(I7) if ϱ′ ∈A_{0}(ℵ; d, J) is such that ϱ−1<ϱ′ ≺ϱ^{∗}, then

ι_{ϱ−1,ϱ^{∗}}^{−1}(𝔐̃_{0,ϱ′}^{ϱ−1}) = ( ⋃_{ρ∈A_{0}(ϱ^{∗};ϱ′)} M̃_{0,ρ}^{ρ_{ϱ^{∗}}(ϱ−1)}) × \overline{𝔐}_{0,(ℵ_{B}(ϱ^{∗}),J_{B}(ϱ^{∗}))};

(I8) Im^{s} ῑ_{ϱ−1,ϱ^{∗}} ⊂ ⋃_{ϱ−1<ϱ′≺ϱ^{∗}} 𝔐̃_{0,ϱ′}^{ϱ−1}, where

ῑ_{ϱ−1,ϱ^{∗}}: (M̃_{0,ϱ_{P}^{∗}}^{ρ_{ϱ^{∗}}(ϱ−1)}×\overline{𝔐}_{0,(ℵ_{B}(ϱ^{∗}),J_{B}(ϱ^{∗}))})/G_{ϱ^{∗}} ⟶ 𝔐̃_{0,(ℵ,J)}^{ϱ−1},

is the immersion map induced by ι_{ϱ−1,ϱ^{∗}}.

Furthermore, we assume that (I9) the collections {ι_{ϱ−1,ϱ^{∗}}}_{ϱ^{∗}∈A_{0}(ℵ;d,J),ϱ^{∗}>ϱ−1} and {ῑ_{ϱ−1,ϱ^{∗}}}_{ϱ−1∈A_{0}(ℵ;d,J),ϱ^{∗}>ϱ−1} of immersions are properly self-intersecting.

Finally, for all ϱ^{∗} ∈A_{0}(ℵ; d, J) such that ϱ^{∗} >ϱ−1:

(I10) ι_{ϱ−1,ϱ^{∗}}^{∗}𝔼_{ϱ−1} = π_{P}^{∗} 𝔼_{ρ_{ϱ^{∗}}(ϱ−1)}, where

π_{P} , π_{B}: M̃_{0,ϱ_{P}^{∗}}^{ρ_{ϱ^{∗}}(ϱ−1)} × \overline{𝔐}_{0,(ℵ_{B}(ϱ^{∗}),J_{B}(ϱ^{∗}))} ⟶ M̃_{0,ϱ_{P}^{∗}}^{ρ_{ϱ^{∗}}(ϱ−1)}, \overline{𝔐}_{0,(ℵ_{B}(ϱ^{∗}),J_{B}(ϱ^{∗}))}

are the two projection maps; (I11) if ϱ^{∗} is as in Lemma 3.10, then the normal bundle for the immersion ι_{ϱ−1,ϱ^{∗}} is given by

\begin{aligned}N_{ι_{ϱ−1,ϱ^{∗}}} & = N_{ι_{ϱ−1,ϱ^{∗}}}^{⊥} ⊕N_{ι_{ϱ−1,ϱ^{∗}}}^{⊤} \\ & ≡ ⨁_{l∈ℵ_{P} (ϱ^{∗})} ⨁_{i∈[m_{l}^{∗}]}π_{P}^{∗} L_{ρ_{ϱ^{∗}}(ϱ−1),(l,i)}⊗π_{B}^{∗}π_{(l,i)}^{∗}L_{0} ⊕ π_{P}^{∗} 𝔼_{ρ_{ϱ^{∗}}(ϱ−1)}^{∗}⊗π_{B}^{∗} ⨁_{l∈ℵ_{S}(ϱ^{∗})} π_{l}^{∗}L_{0}, \end{aligned}

where L_{ρ_{ϱ^{∗}}(ϱ−1),(l,i)}, 𝔼_{ρ_{ϱ^{∗}}(ϱ−1)} ⟶M̃_{0,ϱ_{P}^{∗}}^{ρ_{ϱ^{∗}}(ϱ−1)} are the line bundles constructed in Subsection 2.4;

(I12) the differential of D̃_{ϱ−1},

∇D̃_{ϱ−1} : N_{ι_{ϱ−1,ϱ^{∗}}} ⟶ ι_{ϱ−1,ϱ^{∗}}^{∗}(𝔼_{ϱ−1}^{∗}⊗π_{ϱ−1}^{∗}π_{ℙF_{(ℵ,J)}}^{∗}ev_{0}^{∗}T ℙ^{n}) = π_{P}^{∗} 𝔼_{ρ_{ϱ^{∗}}(ϱ−1)}^{∗}⊗π_{B}^{∗}ev_{0}^{∗}T ℙ^{n},

in the normal direction to the immersion ι_{ϱ−1,ϱ^{∗}} is given by

∇D̃_{ϱ−1}|_{π_{P}^{∗} L_{ρ_{ϱ^{∗}} (ϱ−1),(l,i)}⊗π_{B}^{∗}π_{(l,i)}^{∗}L_{0}} = π_{P}^{∗} s_{ρ_{ϱ^{∗}}(ϱ−1),(l,i)}⊗π_{B}^{∗}π_{(l,i)}^{∗}D_{0} ∀ i∈[m_{l}^{∗}], l ∈ℵ_{P} (ϱ^{∗})
and ∇D̃_{ϱ−1}|_{N_{ι_{ϱ−1,ϱ^{∗}}}^{⊤}} = π_{P}^{∗} id⊗π_{B}^{∗}D_{(ℵ_{B}(ϱ^{∗}),J_{B}(ϱ^{∗}))},

where s_{ρ_{ϱ^{∗}}(ϱ−1),(l,i)} is the homomorphism defined in Subsection 2.4.

By the inductive assumption (I4), the loci on which the sections D̃_{ϱ} fail to be transverse to the zero set shrink and eventually disappear. For each ϱ, the behavior of D̃_{ϱ} in the directions normal to the “bad” locus is described by (I12). By the inductive assumption (I5), if ϱ_{1} and ϱ_{2} are noncomparable elements of (A_{0}(ℵ; d, J), ≺), the proper transforms of 𝔐̃_{0,ϱ_{1}}^{0} and 𝔐̃_{0,ϱ_{2}}^{0} become disjoint by the time either is ready to be blown up for any ordering < extending the partial ordering ≺. Similarly to Subsections 2.3 and 2.4, (I5) will imply that the end result of the present blowup construction is independent of the choice of an extension <. By (I6), our blowup construction modifies each immersion ι_{0,σ^{∗}} by changing the first factor of the domain according to the blowup construction of Subsection 2.4, until a proper transform of the image of ι_{0,σ^{∗}} is to be blown up; see below. By (I8), by the time this happens the immersion ῑ_{0,σ^{∗}} induced by ι_{0,σ^{∗}} transforms into an embedding. Thus, all blowup loci are smooth.

We note that all of the assumptions (I1)-(I12) are satisfied if ϱ−1 is replaced by 0. In particular, (I5) is a restatement of Lemma 3.9, while (I4) and (I12) are the two parts of Lemma 3.11. The statements (I7)-(I11), with ϱ−1 replaced by 0, are contained in Lemma 3.10.

If ϱ∈A_{0}(ℵ; d, J) is as above, let

π̃_{ϱ}: 𝔐̃_{0,(ℵ,J)}^{ϱ} ⟶ 𝔐̃_{0,(ℵ,J)}^{ϱ−1}

be the blowup of 𝔐̃_{0,(ℵ,J)}^{ϱ−1} along 𝔐̃_{0,ϱ}^{ϱ−1}, which is a smooth subvariety by the inductive assump-

tion (I8). We denote the exceptional divisor for this blowup by 𝔐̃_{0,ϱ}^{ϱ}. If ϱ^{∗} >ϱ, let 𝔐̃_{0,ϱ^{∗}}^{ϱ} ⊂𝔐̃_{0,(ℵ,J)}^{ϱ}

be the proper transform of 𝔐̃_{0,ϱ^{∗}}^{ϱ−1}. We put

𝔼_{ϱ} = π̃_{ϱ}^{∗}𝔼_{ϱ−1} ⊗ O(𝔐̃_{0,ϱ}^{ϱ}).  (3.7)

The section π̃_{ϱ}^{∗}D̃_{ϱ−1} vanishes identically along the divisor 𝔐̃_{0,ϱ}^{ϱ}. Thus, it induces a section

D̃_{ϱ} ∈ Γ(𝔐̃_{(ℵ,J)}^{ϱ}; 𝔼_{ϱ}^{∗}⊗π_{ϱ}^{∗}π_{ℙF_{(ℵ,J)}}^{∗}ev_{0}^{∗}T ℙ^{n}),

where π_{ϱ} = π_{ϱ−1}∘ π̃_{ϱ}.

The inductive assumptions (I1)-(I3), with ϱ−1 replaced by ϱ, are clearly satisfied, while (I5), (I8), and (I9) follow from (2) of Lemma 3.3 and Corollary 3.4. On the other hand, by (I6), the domain of the immersion ι_{ϱ−1,ϱ} is

M̃_{0,ϱ_{P}}^{ρ_{ϱ}(ϱ−1)} × \overline{𝔐}_{0,(ℵ_{B}(ϱ),J_{B}(ϱ))} = M̃_{0,ϱ_{P}} × \overline{𝔐}_{0,(ℵ_{B}(ϱ),J_{B}(ϱ))},

where M̃_{0,ϱ_{P}} ⟶M̃_{0,ϱ_{P}}^{0} is the blowup constructed in Subsection 2.4. By (I11), the normal bundle for the immersion ι_{ϱ−1,ϱ} is given by

N_{ι_{ϱ−1,ϱ}} = ⨁_{l∈ℵ_{P} (ϱ)} ⨁_{i∈[m_{l}]}π_{P}^{∗} L_{ρ_{ϱ}(ϱ−1),(l,i)}⊗π_{B}^{∗}π_{(l,i)}^{∗}L_{0} ⊕ π_{P}^{∗} 𝔼_{ρ_{ϱ}(ϱ−1)}^{∗}⊗π_{B}^{∗} ⨁_{l∈ℵ_{S}(ϱ)}π_{l}^{∗}L_{0}

= ⨁_{l∈ℵ_{P} (ϱ)} ⨁_{i∈[m_{l}]}π_{P}^{∗} 𝕃⊗π_{B}^{∗}π_{(l,i)}^{∗}L_{0} ⊕ π_{P}^{∗} 𝕃⊗π_{B}^{∗} ⨁_{l∈ℵ_{S}(ϱ)}π_{l}^{∗}L_{0}

= π_{P}^{∗} 𝕃 ⊗ π_{B}^{∗}F_{(ℵ_{B}(ϱ),J_{B}(ϱ))},

where 𝕃 ⟶ M̃_{0,ϱ_{P}} is the universal tangent line bundle constructed in Subsection 2.4. We also note that by (I10),

ι_{ϱ−1,ϱ}^{∗}(𝔼_{ϱ−1}^{∗}⊗π_{ℙF_{(ℵ,J)}}^{∗}ev_{0}^{∗}T ℙ^{n}) = π_{P}^{∗} 𝕃 ⊗ π_{B}^{∗}π_{ℙF_{(ℵ_{B} (ϱ),J_{B} (ϱ))}}^{∗}ev_{0}^{∗}T ℙ^{n}.

By (I12), the differential of D̃_{ϱ−1} in the normal direction to the immersion ι_{ϱ−1,ϱ} is given by

∇D̃_{ϱ−1} = π_{P}^{∗} id ⊗ π_{B}^{∗}D_{(ℵ_{B}(ϱ),J_{B}(ϱ))}.

Thus, if

ι_{ϱ,ϱ}: ℙN_{ι_{ϱ−1,ϱ}} ≈ M̃_{0,ϱ_{P}} × 𝔐̃_{0,(ℵ_{P} (ϱ),J_{B}(ϱ))}^{0} ⟶ 𝔐̃_{0,ϱ}^{ϱ} ⊂ 𝔐̃_{0,(ℵ,J)}^{ϱ}

is the immersion induced by ι_{ϱ−1,ϱ}, then

\begin{aligned}ι_{ϱ,ϱ}^{∗}D̃_{ϱ} =π_{B}^{∗}D̃_{0} & ∈ Γ(M̃_{0,ϱ_{P}} ×𝔐̃_{0,(ℵ_{P} (ϱ),J_{B}(ϱ))}^{0}; ι_{ϱ,ϱ}^{∗}(𝔼_{ϱ}^{∗}⊗π_{ϱ}^{∗}π_{ℙF_{(ℵ,J)}}^{∗}ev_{0}^{∗}T ℙ^{n})) \\ & = Γ(M̃_{0,ϱ_{P}} ×𝔐̃_{0,(ℵ_{B}(ϱ),J_{B}(ϱ))}^{0}; π_{B}^{∗}(γ_{(ℵ_{B}(ϱ),J_{B}(ϱ))}^{∗}⊗π_{ℙF_{(ℵ_{B} (ϱ),J_{B} (ϱ))}}^{∗}ev_{0}^{∗}T ℙ^{n})). \end{aligned}

Lemmas 3.10 and 3.11 thus imply that the restriction of the section D̃_{ϱ} to the exceptional divisor 𝔐̃_{0,ϱ}^{ϱ} is transverse to the zero set away from the subvarieties 𝔐̃_{0,ϱ^{∗}}^{ϱ} with ϱ^{∗} > ϱ. Thus, by the inductive assumption (I4) as stated above, (I4) is satisfied with ϱ−1 replaced by ϱ.

We now verify that the remaining inductive assumptions are satisfied. If ϱ<ϱ^{∗}, but ϱ⊀ϱ^{∗},

ρ_{ϱ^{∗}}(ϱ) = ρ_{ϱ^{∗}}(ϱ−1)  and  \overline{𝔐}_{0,ϱ^{∗}}^{ϱ−1} ∩ \overline{𝔐}_{0,ϱ}^{ϱ−1} = ∅,

by definition and by (I5), respectively. It then follows that

ι_{ϱ,ϱ^{∗}} = ι_{ϱ−1,ϱ^{∗}},  𝔐̃_{0,ϱ^{∗}}^{ϱ} ∩𝔐̃_{0,ϱ′}^{ϱ} = 𝔐̃_{0,ϱ^{∗}}^{ϱ−1}∩𝔐̃_{0,ϱ′}^{ϱ−1} ∀ϱ′ >ϱ,
ι_{ϱ,ϱ^{∗}}^{∗}𝔼_{ϱ} = ι_{ϱ−1,ϱ^{∗}}^{∗}𝔼_{ϱ−1}, N_{ι_{ϱ,ϱ^{∗}}} = N_{ι_{ϱ−1,ϱ^{∗}}} , and ∇D̃_{ϱ} = ∇D̃_{ϱ−1}.

Thus, the inductive assumptions (I6), (I7), and (I10)-(I12), as stated above, imply the corresponding statements with ϱ−1 replaced by ϱ.

Suppose that ϱ≺ϱ^{∗}. By (I6) and (1) of Lemma 3.3, the domain of the immersion ι_{ϱ,ϱ^{∗}} induced by the immersion ι_{ϱ−1,ϱ^{∗}} is the blowup of

M̃_{0,ϱ_{P}^{∗}}^{ρ_{ϱ^{∗}}(ϱ−1)} × \overline{𝔐}_{0,(ℵ_{B}(ϱ^{∗}),J_{B}(ϱ^{∗}))}

along the preimage of \overline{𝔐}_{0,ϱ}^{ϱ−1} under ι_{ϱ−1,ϱ^{∗}} in

π_{ρ_{ϱ^{∗}}(ϱ−1)}×id: M̃_{0,ϱ_{P}^{∗}}^{ρ_{ϱ^{∗}}(ϱ−1)}×\overline{𝔐}_{0,(ℵ_{B}(ϱ^{∗}),J_{B}(ϱ^{∗}))} ⟶ M̃_{0,ϱ_{P}^{∗}}^{0} ×\overline{𝔐}_{0,(ℵ_{B}(ϱ^{∗}),J_{B}(ϱ^{∗}))}.

By (I7), this preimage is

( ⋃_{ρ∈A_{0}(ϱ^{∗};ϱ)} M̃_{0,ρ}^{ρ_{ϱ^{∗}}(ϱ−1)}) × \overline{𝔐}_{0,(ℵ_{B}(ϱ^{∗}),J_{B}(ϱ^{∗}))}.

By the last paragraph of Subsection 2.4 and the second paragraph after Lemma 3.10,

M̃_{0,ρ_{1}}^{ρ_{ϱ^{∗}}(ϱ−1)} ∩ M̃_{0,ρ_{2}}^{ρ_{ϱ^{∗}}(ϱ−1)} = ∅ ∀ ρ_{1}, ρ_{2} ∈A_{0}(ϱ^{∗}; ϱ), ρ_{1} ≠ρ_{2}.

Thus, by the construction of Subsection 2.4, the blowup of M̃_{0,ϱ_{P}^{∗}}^{ρ_{ϱ^{∗}}(ϱ−1)} along

⋃_{ρ∈A_{0}(ϱ^{∗};ϱ)} M̃_{0,ρ}^{ρ_{ϱ^{∗}}(ϱ−1)}

is M̃_{0,ϱ_{P}^{∗}}^{ρ_{ϱ^{∗}}(ϱ)}, as needed for the inductive statement (I6), with ϱ−1 replaced by ϱ. For the same reasons, (I10), (2.13), and (3.7) imply that

\begin{aligned}ι_{ϱ,ϱ^{∗}}^{∗}𝔼_{ϱ} & = ι_{ϱ−1,ϱ^{∗}}^{∗}𝔼_{ϱ−1} ⊗ ι_{ϱ,ϱ^{∗}}^{∗}O(𝔐̃_{0,ϱ}^{ϱ}) \\ & = π_{P}^{∗} 𝔼_{ρ_{ϱ^{∗}}(ϱ−1)} ⊗ ⨂_{ρ∈A_{0}(ϱ^{∗};ϱ)} π_{P}^{∗} O(M̃_{0,ρ}^{ρ}) = π_{P}^{∗} 𝔼_{ρ_{ϱ^{∗}}(ϱ)}. \end{aligned}

Thus, the inductive statement (I10), with ϱ−1 replaced by ϱ, is satisfied. The assumption (I7) is checked similarly, using (3) of Lemma 3.3.

We next determine the normal bundle for the immersion ι_{ϱ,ϱ^{∗}}. The restrictions of the line bundles L_{ρ_{ϱ^{∗}}(ϱ−1),(l,i)} and 𝔼_{ρ_{ϱ^{∗}}(ϱ−1)} to the complement of the exceptional divisors in M̃_{0,ϱ_{P}^{∗}}^{ρ_{ϱ^{∗}}(ϱ−1)} are π_{ρ_{ϱ^{∗}}(ϱ−1)}^{∗}L_{0,(l,i)} and π_{ρ_{ϱ^{∗}}(ϱ−1)}^{∗}𝔼_{0}, by the construction of Subsection 2.4. Thus, by the last statement of Lemma 3.10, (I11), and the inductive assumptions (I1) above and in Subsection 2.4,

\begin{matrix}ι_{ϱ−1,ϱ^{∗}} |_{M̃_{0,ρ}^{ρ_{ϱ^{∗}} (ϱ−1)}×\overline{𝔐}_{0,(ℵ_{B} (ϱ^{∗}),J_{B} (ϱ^{∗}))}}^{∗}T \overline{𝔐}_{0,ϱ}^{ϱ−1}/T (M̃_{0,ρ}^{ρ_{ϱ^{∗}}(ϱ−1)}×\overline{𝔐}_{0,(ℵ_{B}(ϱ^{∗}),J_{B}(ϱ^{∗}))}) \\ = ⨁_{l∈ℵ_{P} (ϱ^{∗})−ℵ_{P} (ϱ)} ⨁_{i∈[m_{l}^{∗}]}π_{P}^{∗} L_{ρ_{ϱ^{∗}}(ϱ−1),(l,i)}⊗π_{B}^{∗}π_{(l,i)}^{∗}L_{0} ⊕ ⨁_{l∈ℵ_{P} (ϱ)} ⨁_{i∈[m_{l}^{∗}]−I_{l,P}} π_{P}^{∗} L_{ρ_{ϱ^{∗}}(ϱ−1),(l,i)}⊗π_{B}^{∗}π_{(l,i)}^{∗}L_{0} \end{matrix}

for all ρ∈A_{0}(ϱ^{∗}; ϱ) as in the statement of Lemma 3.10. Let

I_{P} (ρ) = {(l, i): l ∈ℵ_{P} (ϱ), i∈I_{l,P} }.

From Lemma 3.5, we then obtain

N_{ι_{ϱ,ϱ^{∗}}} = ⨁_{l∈ℵ_{P} (ϱ^{∗})} ⨁_{i∈[m_{l}^{∗}]}((π_{P}^{∗} L_{ρ_{ϱ^{∗}}(ϱ−1),(l,i)}⊗π_{B}^{∗}π_{(l,i)}^{∗}L_{0})⊗π_{P}^{∗} O( − ∑_{ρ∈A_{0}(ϱ^{∗};ϱ),(l,i)∈I_{P} (ρ)} M̃_{0,ρ}^{ρ}))

⊕ π_{P}^{∗} 𝔼_{ρ_{ϱ^{∗}}(ϱ−1)}^{∗}⊗π_{B}^{∗} ⨁_{l∈ℵ_{S}(ϱ^{∗})} π_{l}^{∗}L_{0}⊗π_{P}^{∗} O( − ∑_{ρ∈A_{0}(ϱ^{∗};ϱ)} M̃_{0,ρ}^{ρ}))

= ⨁_{l∈ℵ_{P} (ϱ^{∗})} ⨁_{i∈[m_{l}^{∗}]}(π_{P}^{∗} L_{ρ_{ϱ^{∗}}(ϱ),(l,i)}⊗π_{B}^{∗}π_{(l,i)}^{∗}L_{0}) ⊕ π_{P}^{∗} 𝔼_{ρ_{ϱ^{∗}}(ϱ)}^{∗}⊗π_{B}^{∗} ⨁_{l∈ℵ_{S}(ϱ^{∗})} π_{l}^{∗}L_{0}.

The last equality above follows from (2.12) by the same argument as in the previous paragraph. We have thus verified that the inductive assumption (I11), with ϱ−1 replaced by ϱ, is satisfied. Finally, the inductive assumption (I12) and the continuity of the two bundle sections involved in the identity in (I12), with ϱ−1 replaced by ϱ, imply (I12) with ϱ−1 replaced by ϱ.

We conclude this construction after the blowup at the ϱ_{max} step. Let

𝔐̃_{0,(ℵ,J)}(ℙ^{n}, d) = 𝔐̃_{0,(ℵ,J)}^{ϱ_{max}},  𝔼̃ = 𝔼_{ϱ_{max}},  D̃_{(ℵ,J)} = D̃_{ϱ_{max}}.

By the inductive assumption (I4), applied with ϱ−1 replaced by ϱ_{max}, the section D̃_{(ℵ,J)} is transverse to the zero set. As in the previous two subsections, the final result of this blowup construction is independent of the order < chosen to extend the partial ordering ≺ on A_{0}(ℵ; d, J), as can be seen from (I5).

4 A Blowup of a Moduli Space of Genus-One Maps

4.1 Idealized Blowups and Immersions

In this section we describe the main blowup construction of this paper. This is the sequential idealized blowup construction for \overline{𝔐}_{1,k}(ℙ^{n}, d) with the initial data and the inductive step specified in Subsections 4.2 and 4.3, respectively. This construction is outlined in Subsections 1.2 and 1.3.

In contrast to the situations in Sections 2 and 3, the variety \overline{𝔐}_{1,k}(ℙ^{n}, d) is singular. In order to describe the structure of \overline{𝔐}_{1,k}(ℙ^{n}, d), we introduce the notion of idealized normal bundle for an immersion. Recall that the domain of an immersion is assumed to be a smooth variety.

Definition 4.1 Suppose \overline{𝔐} is a variety and ι_{X}: X ⟶M is an immersion. An idealized normal bundle for the immersion ι_{X} is a vector bundle N_{ι_{X}}^{ide} over X such that N_{ι_{X}} ⊂N_{ι_{X}}^{ide}.

Remark: An idealized normal bundle is of course not unique; an idealized normal bundle plus any other vector bundle is still an idealized normal bundle. If the image of ι_{X} is an irreducible component of \overline{𝔐}, an idealized normal bundle of the smallest possible rank still need not be unique; it can be twisted by any divisor in X disjoint from the preimage under ι_{X} of the other components of \overline{𝔐}. For each of the immersions we encounter in the next subsection, there is a natural choice for N_{ι_{X}}^{ide}. These idealized normal bundles also transform in a natural way under blowups and proper immersions, as described in Lemma 4.3 below.

Suppose \overline{𝔐} is a variety, Z is a smooth subvariety of \overline{𝔐}, and N_{ι_{Z}}^{ide} is an idealized normal bundle for the embedding ι_{Z} of Z into \overline{𝔐}. Let

E_{Z} ≡ℙN_{ι_{Z}} ⊂ Bl_{Z} \overline{𝔐}

be the exceptional divisor for the blowup of \overline{𝔐} along Z. We denote by Bl_{Z}^{ide}\overline{𝔐} the variety obtained by identifying Bl_{Z} \overline{𝔐} with

E_{Z}^{ide} ≡ ℙN_{ι_{Z}}^{ide}

along E_{Z}. We will call

π^{ide} : Bl_{Z}^{ide}\overline{𝔐} ⟶ \overline{𝔐}  and  E_{Z}^{ide} ⊂ Bl_{Z}^{ide}\overline{𝔐}

the idealized blowup of \overline{𝔐} along Z and the idealized exceptional divisor for π^{ide}, re-

spectively. (Caution: the idealized exceptional divisor is not necessarily a divisor!) More generally, we will call

π: 𝔐̃ ⟶ \overline{𝔐}

an idealized blowup of \overline{𝔐} if π is a composition of idealized blowups along smooth subvarieties. In practice, idealized blowup is simply a convenient term. In the inductive assumption (I1) in Subsection 4.3 below, it can be replaced by morphism of varieties, as the remaining inductive assumptions describe all the relevant properties of this morphism. Let

γ_{Z} ⟶ E_{Z}^{ide}

be the tautological line bundle. Note that the normal bundle of E_{Z} ⊂E_{Z}^{ide} in

Pr_{Z} \overline{𝔐} = Bl_{Z} \overline{𝔐}

is γ_{Z}|_{E_{Z}}. This observation implies the first statement of Lemma 4.3 below.

Our strategy is as follows. We begin with a space with a properly self-intersecting collection of immersions, each with an idealized normal bundle. These are the immersions ι_{σ} with σ ∈A_{1}(d, k) defined in Subsection 1.2; their images are the subvarieties \overline{𝔐}_{1,σ}(ℙ^{n}, d) of \overline{𝔐}_{1,k}(ℙ^{n}, d). The idealized normal bundle for the immersion ι_{σ} is the direct sum of the deformation spaces of the nodes between the contracted genus-one curve and the non-contracted genus-zero curves that are identified by ι_{σ}. At each stage, one of our immersions is an embedding, and we blow it up, replacing it with its idealized exceptional divisor. The exceptional divisor of the blowup of the main component is the intersection of the new main component with the idealized exceptional divisor. Then after each step, we have a new properly self-intersecting collection of immersions. Moreover, there is a natural idealized normal bundle for each of the proper transforms of the immersions we have “yet to blow up”.

We now say this more explicitly. Lemmas 4.2 and 4.3 below are direct extensions of Corollary 3.4 and Lemma 3.5. The first lemma states that if we have a properly self-intersecting collection of immersions, one of which is an embedding, then upon blowing up the embedding, we still have a properly self-intersecting collection of immersions. It is immediate from the definition of “properly self-intersecting”, by checking in local coordinates.

The second part of the second lemma follows from Lemma 3.5 with only one change. Instead of writing

N_{ι_{X}} = ⨁_{i∈I} L_{i}  and  N_{Pr_{Z}ι_{X}} = ⨁_{i∈I} (π^{∗}L_{i} ⊗ ⨂_{i∈I_{ϱ}} O(−E_{ϱ}))

as in the statement of Lemma 3.5, we are saying that if we have a natural inclusion N_{ι_{X}} ⊂⨁_{i∈I} L_{i}, then we get a natural inclusion

N_{Pr_{Z}ι_{X}} ⊂ ⨁_{i∈I} (π^{∗}L_{i} ⊗ ⨂_{i∈I_{ϱ}} O(−E_{ϱ})).

The vector bundles on the right are the original idealized normal bundle and the new idealized normal bundle, respectively.

Lemma 4.2 Suppose \overline{𝔐} is a variety, \{ι_{σ}: X_{σ} ⟶\overline{𝔐}\}_{σ∈A} is a properly self-intersecting collection of immersions, and σ ∈ A is such that ι_{σ} is an embedding. If N_{ι_{σ}}^{ide} is an idealized normal bundle for ι_{σ}, then

{Pr_{Im ι_{σ}}ι_{σ′}}_{σ′∈A−{σ}} ∪ {ι_{E_{Imισ}^{ide}} }

is a properly self-intersecting collection of immersions into Bl_{Im ι_{σ}}^{ide} \overline{𝔐}.

Lemma 4.3 If \overline{𝔐} is a variety, Z is a smooth subvariety of \overline{𝔐}, and N_{ι_{Z}}^{ide} is an idealized normal bundle for ι_{Z}, then

N_{ι_{E{ide \atop Z}}}^{ide} = γ_{Z}

is an idealized normal bundle for the immersion ι_{E_{Z}^{ide}}. Suppose in addition that ι_{X}, A, Z_{ϱ}, and E_{ϱ} are as in Lemma 3.5 and N_{ι_{X}}^{ide} is an idealized normal bundle for ι_{X}. If there exist a splitting

N_{ι_{X}}^{ide} = ⨁_{i∈I} L_{i} ⟶ X

and a subset I_{ϱ} of I for each ϱ∈A such that (3.1) holds, then

N_{Pr_{Z}ι_{X}}^{ide} = ⨁_{i∈I} (π^{∗}L_{i} ⊗ ⨂_{i∈I_{ϱ}} O(−E_{ϱ}))

is an idealized normal bundle for the immersion Pr_{Z}ι_{X}.

Definition 4.4 Suppose \overline{𝔐} is a variety, ι_{X}: X ⟶\overline{𝔐} is an immersion, \overline{𝔐}^{0} is a subvariety in \overline{𝔐}, and T C\overline{𝔐}^{0} ⊂T \overline{𝔐} is the tangent cone of \overline{𝔐}^{0} in \overline{𝔐} (T C\overline{𝔐}^{0} not necessarily reduced). The subvariety

\overline{𝔐}^{0} is proper relative to ι_{X} if

dι_{X} T Cι_{X}^{−1}(\overline{𝔐}^{0}) = ι_{X}^{∗}T C\overline{𝔐}^{0} ∩ Im dι_{X} ⊂ ι_{X}^{∗}T \overline{𝔐}

and the map

ι_{X}^{∗}T C\overline{𝔐}^{0}|_{ι_{X}^{−1}(\overline{𝔐}^{0})}/Im dι_{X}|_{T Cι_{X}^{−1}(\overline{𝔐}^{0})} ⟶ ι_{X}^{∗}T \overline{𝔐}/Im dι_{X} ⊂ N_{ι_{X}}^{ide}  (4.1)

induced by inclusions is injective, with its image being reduced.

The left-hand side of (4.1) denotes the family of cones over ι_{X}^{−1}(\overline{𝔐}^{0}) such that for each x∈ι_{X}^{−1}(\overline{𝔐}^{0})

ι_{X}^{∗}T C\overline{𝔐}^{0}|_{ι_{X}^{−1}(\overline{𝔐}^{0})}/Im dι_{X}|_{T Cι_{X}^{−1}(\overline{𝔐}^{0})}|_{x}

is the quotient by the minimal vector subspace of Im dι_{X}|_{x} = dι_{X}(T_{x}X) containing the cone Im dι_{X}|_{T_{x}Cι_{X}^{−1}(\overline{𝔐}^{0})}. If T Cι_{X}^{−1}(\overline{𝔐}^{0}) is a vector bundle, the two conditions in Definition 4.4 are equivalent.

If \overline{𝔐}^{0} is a subvariety of \overline{𝔐} which is proper relative to an immersion ι_{X}: X ⟶\overline{𝔐}, we denote by

N_{ι_{X}|\overline{𝔐}^{0}} ⊂ ι_{X}^{∗}T \overline{𝔐}/Im dι_{X} ⊂ N_{ι_{X}}^{ide}

the image of the homomorphism (4.1). We will call N_{ι_{X}|\overline{𝔐}^{0}} the normal cone of ι_{X}|_{ι_{X}^{−1}(\overline{𝔐}^{0})}

in \overline{𝔐}^{0}.

Lemma 4.5 Suppose \overline{𝔐} is a variety, ι_{X} : X ⟶ \overline{𝔐} is an immersion with an idealized normal bundle N_{ι_{X}}^{ide}, \overline{𝔐}^{0} is a subvariety of \overline{𝔐} which is proper relative to ι_{X}, and

Z ⊂ \overline{Z} ≡ι_{X}^{−1}(\overline{𝔐}^{0})

is such that N_{ι_{X}|\overline{𝔐}^{0}} is the closure of N_{ι_{X}|\overline{𝔐}^{0}}|_{Z} in N_{ι_{X}}^{ide}.

(1) If X is a smooth subvariety of \overline{𝔐}, then Pr_{X} \overline{𝔐}^{0} is proper relative to the immersion ι_{E_{X}^{ide}},

E_{X}^{ide} ∩ Pr_{X} \overline{𝔐}^{0} ⊂ E_{X}

is the closure of ℙN_{ι_{X}|\overline{𝔐}^{0}}|_{Z} in E_{X}^{ide}, and

N_{ι_{E{ide \atop X}}|Pr_{X} \overline{𝔐}^{0}} = γ_{X}|_{E_{X}^{ide}∩Pr_{X} \overline{𝔐}^{0}}.

(2) If Z is a smooth subvariety of \overline{𝔐} disjoint from ι_{X}(Z) and N_{ι_{Z}}^{ide} is an idealized normal bundle for ι_{Z}, then Pr_{Z} \overline{𝔐}^{0} is a proper subvariety of Bl_{Z}^{ide}\overline{𝔐} relative to the immersion Pr_{Z}ι_{X} and N_{Pr_{Z}ι_{X}|Pr_{Z} \overline{𝔐}^{0}} is the closure of N_{ι_{X}|\overline{𝔐}^{0}}|_{Z} in N_{Pr_{Z}ι_{X}}^{ide} .

The first part of (1) essentially follows from the universal property of blowing up: if \overline{𝔐} is blown up along Z, then the proper transform of \overline{𝔐}^{0} in \overline{𝔐} (the scheme-theoretic closure of \overline{𝔐}^{0} −Z in the blowup) is the blowup of \overline{𝔐}^{0} along \overline{𝔐}^{0}∩Z, and the normal bundle to the exceptional divisor in Bl_{\overline{𝔐}^{0}∩Z} \overline{𝔐}^{0} is the restriction of the normal bundle of the exceptional divisor in Bl_{Z} \overline{𝔐}. The statement (1) is the etale-local version of this. Part (2) is clear by working in local coordinates.

4.2 Preliminaries

In this subsection, we state a number of known facts concerning the moduli space \overline{𝔐}_{1,k}(ℙ^{n}, d) that insure that the inductive requirements of the next subsection are satisfied at the initial stage of the inductive construction. Lemmas 4.6-4.9, with the exception of one statement, are well-known in Gromov-Witten theory and are obtained similarly to Lemmas 3.6 and 3.7. We show that the last statement of Lemma 4.7 is simply a reinterpretation of a standard fact concerning moduli spaces of stable maps.

Let (A_{1}(d, k), ≺) be the partially ordered set of triples described in Subsection 1.2. It has a unique minimal element and a unique maximal element:

σ_{min} = (1; ∅, [k])  and  σ_{max} = (d; [k], ∅).

Let < be an order on A_{1}(d, k) extending the partial ordering ≺. For every σ ∈A_{1}(d, k), we define

σ−1 ∈ {0}⊔A_{1}(d, k)

as in (2.7). For each element σ =(m; J_{P} , J_{B}) of A_{1}(d, k), let

\overline{𝔐}_{1,σ}^{0} ≡\overline{𝔐}_{1,σ}(ℙ^{n}, d) ⊂ \overline{𝔐}_{1,k}^{0} ≡\overline{𝔐}_{1,k}(ℙ^{n}, d)

be the subvarieties defined in Subsection 1.2.

Warning: Note that \overline{𝔐}_{1,k}^{0} denotes the entire moduli space \overline{𝔐}_{1,k}(ℙ^{n}, d) and not the main component \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d). Similarly to Sections 2 and 3, the superscript 0 indicates the 0th stage in the blowup process.

Lemma 4.6 If σ_{1} = (m_{1}; J_{1;P} , J_{1;B}) and σ_{2} = (m_{2}; J_{2;P} , J_{2;B}) are elements of A_{1}(d, k), σ_{1} ≠ σ_{2}, σ_{1} ⊀σ_{2}, and σ_{2} ⊀σ_{1}, then

\overline{𝔐}_{1,σ_{1}}^{0} ∩ \overline{𝔐}_{1,σ_{2}}^{0} ⊂ \overline{𝔐}_{1,σ̃(σ_{1},σ_{2})}^{0}, where
σ̃(σ_{1}, σ_{2}) = (min(m_{1}, m_{2}); J_{1;P} ∩J_{2;P} , J_{1;B} ∪J_{2;B}).

With σ as above, we define

I_{P} (σ) = ℵ_{B}(σ) = [m],  J_{P} (σ) = J_{P} ,  J_{B}(σ) = J_{B},  G_{σ} = S_{m}.

As in Subsection 1.2, we denote by

ι_{0,σ}: \overline{M}_{1,(I_{P} (σ),J_{P} (σ))}^{0} × \overline{𝔐}_{0,(ℵ_{B}(σ),J_{B}(σ))} ⟶ \overline{𝔐}_{1,σ}^{0} ⊂ \overline{𝔐}_{1,k}^{0},
where \overline{𝔐}_{0,(ℵ_{B}(σ),J_{B}(σ))} = \overline{𝔐}_{0,(ℵ_{B}(σ),J_{B}(σ))}(ℙ^{n}, d),

the natural node-identifying map and by

ῑ_{0,σ}: (\overline{M}_{1,(I_{P} (σ),J_{P} (σ))}^{0}×\overline{𝔐}_{0,(ℵ_{B}(σ),J_{B}(σ))})/G_{σ} ⟶ \overline{𝔐}_{1,k}^{0}

the induced immersion. Let

π_{P} , π_{B}: \overline{M}_{1,(I_{P} (σ),J_{P} (σ))}^{0}×\overline{𝔐}_{0,(ℵ_{B}(σ),J_{B}(σ))} ⟶ \overline{M}_{1,(I_{P} (σ),J_{P} (σ))}^{0}, \overline{𝔐}_{0,(ℵ_{P} (σ),J_{B}(σ))}

be the two projection maps.

Lemma 4.7 If d, n ∈ ℤ^{+} and k ∈ ℤ̄^{+}, the collections {ι_{0,σ}}_{σ∈A_{1}(d,k)} and {ῑ_{0,σ}}_{σ∈A_{1}(d,k)} of immersions are properly self-intersecting. If σ^{∗} =(m^{∗}; J_{P}^{∗} , J_{B}^{∗})∈A_{1}(d, k),

Im^{s} ῑ_{0,σ^{∗}} ⊂ ⋃_{σ′≺σ^{∗}}\overline{𝔐}_{1,σ′}  and  N_{ι_{0,σ^{∗}}}^{ide} = ⨁_{i∈[m^{∗}]}π_{P}^{∗} L_{i}⊗π_{B}^{∗}π_{i}^{∗}L_{0}

is an idealized normal bundle for ι_{0,σ^{∗}}.

We deduce the last claim of this lemma from the deformation-obstruction exact sequence (24.2) in [MirSym] as follows. Suppose

[Σ, u] = ι_{0,σ^{∗}}([Σ_{P}]×[Σ_{B}, u_{B}]) ∈ \overline{𝔐}_{1,σ^{∗}}^{0}, where
[Σ_{B}, u_{B}] = ([Σ_{i}, u_{i}])_{i∈[m^{∗}]} ∈ \overline{𝔐}_{0,(ℵ_{B}(σ^{∗}),J_{B}(σ^{∗}))}.

By [MirSym, (24.2)], there exists a natural homomorphism

j_{Σ,u}: T \overline{𝔐}_{1,k}(ℙ^{n}, d)|_{[Σ,u]} = Def(Σ, u) ⟶ Def(Σ),

where Def(Σ, u) and Def(Σ) denote the deformations of the stable-map pair (Σ, u) and the deformations of the curve Σ (with its marked points), respectively. As [Σ, u] is considered as the image of [Σ_{P}]×[Σ_{B}, u_{B}] under ι_{0,σ^{∗}}, there are m^{∗} distinguished nodes of Σ. These are the nodes of Σ that do not correspond to either the nodes of Σ_{P} or the nodes of any of the curves Σ_{i} with i∈[m^{∗}]; see Figure 13. Let

Def(Σ_{P} , Σ_{B}) ⊂ Def(Σ)

be the deformations of Σ that do not smooth out the distinguished nodes of Σ. Since the smoothing of a given node of Σ is parametrized by the tensor product of the tangent lines to the two branches of Σ at the node, we have an exact sequence

0 ⟶ Def(Σ_{P} , Σ_{B}) ⟶ Def(Σ) ⟶^{j_{Σ}} N_{ι_{0,σ^{∗}}}^{ide} |_{[Σ,u]} ⟶ 0.

We denote by

Def(Σ_{P} , (Σ_{B}, u_{B})) ⊂ T \overline{𝔐}_{1,k}(ℙ^{n}, d)|_{[Σ,u]} = Def(Σ, u)

the kernel of the map

j_{Σ}∘ j_{Σ,u}: Def(Σ, u) ⟶ N_{ι_{0,σ^{∗}}}^{ide} |_{[Σ_{P}]×[Σ_{B},u_{B}]}.

The space Def(Σ_{P} , (Σ_{B}, u_{B})) consists of deformations of (Σ, u) that do not smooth out the m^{∗} distinguished nodes of Σ. Thus,

Def(Σ_{P} , (Σ_{B}, u_{B})) ≈ Def(Σ_{P}) ⊕ Def(Σ_{B}, u_{B})
= T \overline{M}_{1,(I_{P} (σ^{∗}),J_{P} (σ^{∗}))}^{0}|_{[Σ_{P}]} ⊕ T \overline{𝔐}_{0,(ℵ_{B}(σ^{∗}),J_{B}(σ^{∗}))}|_{[Σ_{B},u_{B}]}.

The isomorphism from the right-hand side to the left-hand side is given by dι_{0,σ^{∗}}. Thus, the homomorphism j_{Σ}∘j_{Σ,u} induces an injection

N_{ι_{0,σ^{∗}}} |_{[Σ,u]} ≡ T C\overline{𝔐}_{1,k}(ℙ^{n}, d)|_{[Σ,u]}/Im dι_{0,σ^{∗}} ⟶ N_{ι_{0,σ^{∗}}}^{ide} |_{[Σ_{P}]×[Σ_{B},u_{B}]},

as needed.

1 2 3

2 3

× ι_{0,σ^{∗}}

distinguished nodes

Figure 13: A point in the domain of ι_{0,σ^{∗}} and its image in \overline{𝔐}_{1,k}(ℙ^{n}, d)

Lemma 4.8 If d, n, k, and σ^{∗} are as in Lemma 4.7, σ ∈A_{1}(d, k) is as above, and σ ≺σ^{∗}, then

ι_{0,σ^{∗}}^{−1}(\overline{𝔐}_{1,σ}^{0}) = ( ⋃_{ρ∈A_{P} (σ^{∗};σ)} \overline{M}_{1,ρ}^{0}) × \overline{𝔐}_{0,(ℵ_{B}(σ^{∗}),J_{B}(σ^{∗}))}, where

A_{P} (σ^{∗}; σ) = {ρ=(I_{P} ⊔J_{P} , {I_{k}⊔J_{k}: k ∈K})∈A_{1}(I_{P} (σ^{∗}), J_{P} (σ^{∗})): |K|+|I_{P} |=m} and A_{1}(I_{P} (σ^{∗}), J_{P} (σ^{∗})) and \overline{M}_{1,ρ}^{0} ≡ \overline{M}_{1,ρ} are as in Subsection 2.2. Furthermore, if ρ ∈ A_{P} (σ^{∗}; σ) is as above,

ι_{0,σ^{∗}} |_{\overline{M}_{1,ρ}^{0}×\overline{𝔐}_{0,(ℵ_{B} (σ^{∗}),J_{B} (σ^{∗}))}}^{∗}T \overline{𝔐}_{1,σ}^{0}/T (\overline{M}_{1,ρ}^{0}×\overline{𝔐}_{0,(ℵ_{B}(σ^{∗}),J_{B}(σ^{∗}))}) = ⨁_{i∈I_{P} (σ^{∗})−I_{P}} π_{P}^{∗} L_{i}⊗π_{B}^{∗}π_{i}^{∗}L_{0}.

Lemma 4.9 If d, n, k, σ, and σ^{∗} are as above, then

ι_{0,σ}^{−1}(\overline{𝔐}_{1,σ^{∗}}^{0}) = \overline{M}_{1,(I_{P} (σ),J_{P} (σ))}^{0} × ( ⋃_{ϱ∈A_{B}(σ;σ^{∗})} \overline{𝔐}_{0,ϱ}), where

A_{B}(σ; σ^{∗}) = {ϱ=((σ_{l})_{l∈ℵ_{B}(σ)}, J_{B})∈A_{0}(ℵ_{B}(σ); d, J_{B}(σ)): |ℵ_{B}(ϱ)|=m^{∗}},

and A_{0}(ℵ_{B}(σ); d, J_{B}(σ)), ℵ_{B}(ϱ), and \overline{𝔐}_{0,ϱ} ≡\overline{𝔐}_{0,ϱ}(ℙ^{n}, d) are as in Subsection 3.3. Furthermore, if ϱ∈A_{B}(σ; σ^{∗}) is as above,

ι_{0,σ}|_{\overline{M}_{1,(I_{P} (σ),J_{P} (σ))}^{0}×\overline{𝔐}_{0,ϱ}}^{∗}T \overline{𝔐}_{1,σ^{∗}}^{0}/T (\overline{M}_{1,(I_{P} (σ),J_{P} (σ))}^{0}×\overline{𝔐}_{0,ϱ}) = ⨁_{i∈ℵ_{P} (ϱ)}π_{P}^{∗} L_{i}⊗π_{B}^{∗}π_{i}^{∗}L_{0},

where ℵ_{P} (ϱ)⊂ℵ_{B}(σ) is as in Subsection 3.3. We note that for every σ^{∗} ∈A_{1}(d, k),

A_{1}(I_{P} (σ^{∗}), J_{P} (σ^{∗})) = ⨆_{σ≺σ^{∗}}A_{P} (σ^{∗}; σ).

Furthermore, if σ_{1}, σ_{2} ∈A_{1}(d, k) are such that σ_{1}, σ_{2} ≺σ^{∗}, then ρ_{1} ∈A_{P} (σ^{∗}; σ_{1}), ρ_{2} ∈A_{P} (σ^{∗}; σ_{2}), ρ_{1} ≺ρ_{2} ⟹ σ_{1} ≺σ_{2}.

Thus, we can choose an ordering < on A_{1}(I_{P} (σ^{∗}), J_{P} (σ^{∗})) extending the partial ordering ≺ of Subsection 2.3 such that

σ_{1} <σ_{2}, ρ_{1} ∈A_{P} (σ^{∗}; σ_{1}), ρ_{2} ∈A_{P} (σ^{∗}; σ_{2})  ⟹  ρ_{1} <ρ_{2},

whenever σ_{1}, σ_{2} ∈ A_{1}(d, k) are such that σ_{1}, σ_{2} ≺ σ^{∗}. In the next subsection, we will refer to the blowup construction of Subsection 2.3 corresponding to such an ordering.

Similarly, if σ′ ∈A_{1}(d, k),

A_{0}(ℵ_{B}(σ′); d, J_{B}(σ′)) = ⨆_{σ′≺σ}A_{B}(σ′; σ).

Furthermore, if σ_{1}, σ_{2} ∈A_{1}(d, k) are such that σ′ ≺σ_{1}, σ_{2}, then

ϱ_{1} ∈A_{B}(σ′; σ_{1}), ϱ_{2} ∈A_{B}(σ′; σ_{2}), ϱ_{1} ≺ϱ_{2}  ⟹  σ_{1} ≺σ_{2}.

Thus, we can choose an ordering < on A_{0}(ℵ_{B}(σ′); d, J_{B}(σ′)) extending the partial ordering ≺ of Subsection 3.3 such that

σ_{1} <σ_{2}, ϱ_{1} ∈A_{B}(σ′; σ_{1}), ϱ_{2} ∈A_{B}(σ′; σ_{2})  ⟹  ϱ_{1} <ϱ_{2},

whenever σ_{1}, σ_{2} ∈ A_{1}(d, k) are such that σ′ ≺ σ_{1}, σ_{2}. In the next subsection, we will refer to the blowup construction of Subsection 3.4 corresponding to such an ordering.

We denote by \overline{𝔐}_{1,(0)}^{0} the main component \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d) of the moduli space \overline{𝔐}_{1,k}(ℙ^{n}, d). If σ ∈ A_{1}(d, k), we put

\overline{Z}_{σ}^{0} = ι_{0,σ}^{−1}(\overline{𝔐}_{1,(0)}^{0}) ≡ ι_{0,σ}^{−1}(\overline{𝔐}_{1,(0)}^{0}∩ \overline{𝔐}_{1,σ}^{0});
Z_{σ}^{0} = ι_{0,σ}^{−1}(\overline{𝔐}_{1,(0)}^{0}∩ 𝔐_{1,σ}) ⊂ \overline{Z}_{σ}^{0}, where 𝔐_{1,σ} = 𝔐_{1,σ}(ℙ^{n}, d).

We denote by N \overline{Z}_{σ}^{0} ⊂ N_{ι_{0,σ}}^{ide} the normal cone N_{ι_{0,σ}|\overline{𝔐}_{1,(0)}^{0}} for ι_{0,σ}|\overline{Z}{0 \atop σ} in \overline{𝔐}_{1,(0)}^{0}. Its structure is described in Lemma 4.10 below. Let

D_{0,σ} ∈ Γ(\overline{M}_{1,(I_{P} (σ),J_{P} (σ))}^{0}×\overline{𝔐}_{0,(ℵ_{B}(σ),J_{B}(σ))}; Hom(N_{ι_{0,σ}}^{ide}, π_{P}^{∗} 𝔼_{0}^{∗}⊗π_{B}^{∗}ev_{0}^{∗}T ℙ^{n}))

be the section defined by

D_{0,σ}|_{π_{P}^{∗} L_{i}⊗π_{B}^{∗}π_{i}^{∗}L_{0}} = π_{P}^{∗} s_{0,i}⊗π_{B}^{∗}π_{i}^{∗}D_{0}, ∀ i∈[m],

where s_{0,i} and D_{0} are as in Subsections 2.3 and 3.2, respectively.

Lemma 4.10 For all σ ∈A_{1}(d, k), \overline{𝔐}_{1,(0)}^{0} is a proper subvariety of \overline{𝔐}_{1,k}^{0} relative to the immersions ι_{0,σ} and ῑ_{0,σ}. Furthermore,

Z_{σ}^{0} = {b∈M_{1,I_{P} (σ)⊔J_{P} (σ)}×𝔐_{0,(ℵ_{B}(σ),J_{B}(σ))} : ker D_{0,σ}|_{b} ≠{0}}
and N \overline{Z}_{σ}^{0}|_{Z_{σ}^{0}} = ker D_{0,σ}|_{Z_{σ}^{0}}.

Finally, \overline{Z}_{σ}^{0} is the closure of Z_{σ}^{0} in \overline{M}_{1,(I_{P} (σ),J_{P} (σ))}^{0} ×\overline{𝔐}_{0,(ℵ_{B}(σ),J_{B}(σ))} and N \overline{Z}_{σ}^{0} is the closure of N \overline{Z}_{σ}^{0}|_{Z_{σ}^{0}} in N_{ι_{0,σ}}^{ide}.

This lemma is a consequence of [Z4, Theorem 2.3] and related results. In particular, the first claim in the second sentence of Lemma 4.10 is a special case of the first statement of [Z4, Theorem 2.3]. The second claim is nearly a special case of the last statement of [Z4, Theorem 2.3], but some additional comments are required. [Z4, Theorem 2.3] by itself is a purely topological statement, as it describes the topological structure of a neighborhood of each stratum of ι_{0,σ}(\overline{Z}_{σ}^{0}) in \overline{𝔐}_{1,(0)}^{0}. On the other hand, by Subsection 4.1 in [Z2], N \overline{Z}_{σ}^{0}|_{Z_{σ}^{0}} is contained in ker D_{0,σ}. The second claim in the second sentence of Lemma 4.10 can then be obtained from a dimension count and a comparison of the gluing construction used in the proof of [Z4, Theorem 2.3] with the analysis of limiting behavior in [Z2, Subsect. 4.1]. This comparison implies that the gluing parameter in the analytic construction of [Z4] agrees to the first two orders in the zero limit with the smoothing parameter in algebraic geometry. Thus, N \overline{Z}_{σ}^{0}|_{Z_{σ}^{0}} must be equal to ker D_{σ,0}. These considerations also imply

the first claim of Lemma 4.10. Alternatively, suppose that d ≤ n. If the moduli space \overline{𝔐}_{1,σ}^{0} is nonempty, then m≤n and thus for a Zariski open subset Z_{σ;1} of Z_{σ}^{0}

1 ≤ dim N \overline{Z}_{σ}^{0}|_{Z_{σ;1}} = 1 = dim ker D_{0,σ}|_{Z_{σ;1}}  ⟹  N \overline{Z}_{σ}^{0}|_{Z_{σ;1}} = ker D_{0,σ}|_{Z_{σ;1}}.  (4.2)

Since D_{0,σ} is transverse to the zero set over Z_{σ}^{0}, the second claim in the second sentence of the lemma follows from (4.2), if d≤n. The general case follows from the observation that

\overline{𝔐}_{1,σ}(ℙ^{n}, d) = \{[Σ, u]∈\overline{𝔐}_{1,σ}(ℙ^{n+d}, d): u(Σ)⊂ℙ^{n}\}

and the d≤n case.

The first claim in the last sentence of Lemma 4.10 can be obtained by combining the first statement of [Z4, Theorem 2.3], the m=1 case of [Z1, Theorem 2.8], and the Implicit Function Theorem. It also follows immediately from the last claim of Lemma 4.10. The latter can be deduced from [Z4, Theorem 2.3] as follows. Suppose first that m≤n. In this case, [Z1, Theorem 2.8] implies that \overline{Z}_{σ}^{0} admits a stratification

\overline{Z}_{σ}^{0} = Z_{σ;1} ⊔ ⨆_{α∈A} Z_{σ;α}

such that Z_{σ;1} is a Zariski open subset of \overline{Z}_{σ}^{0},

Z_{σ;1} ⊂ Z_{σ}^{0}, dim N \overline{Z}_{σ}^{0}|_{b} =1 ∀ b∈Z_{σ;1},
and  max \{dim N \overline{Z}_{σ}^{0}|_{b}: b∈Z_{σ;α}\} ≤ codim\overline{Z}{0 \atop σ}Z_{σ;α} ∀ α∈A;  (4.3)

see the next paragraph. Let

Z̃_{σ}^{0} = ℙN \overline{Z}_{σ}^{0} ⊂ ℙN_{ι_{0,σ}}^{ide} |\overline{Z}{0 \atop σ}

be the exceptional divisor for the blowup of \overline{𝔐}_{1,(0)}^{0} along \overline{𝔐}_{1,σ}^{0}. Since all irreducible components of Z̃_{σ}^{0} must be of the same dimension, Z̃_{σ}^{0} must be the closure of Z̃_{σ}^{0}|_{Z_{σ}^{0}} by (4.3). This closure property remains valid even if we do not assume that m≤n for the following reason. Let pt∈ℙ^{n+d} be any point not contained in ℙ^{n}. Let

π: ℙ^{n+d} − {pt} ⟶ ℙ^{n}

be the corresponding linear projection. It induces projection maps

φ: \{[Σ, u]∈\overline{𝔐}_{1,k}^{0}(ℙ^{n+d}, d): pt∉u(Σ)\} ⟶ \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d) and
φ̃: {[Σ, u; v]∈Z̃_{σ}^{0}(ℙ^{n+d}, d): pt∉u(Σ)} ⟶ Z̃_{σ}^{0}(ℙ^{n}, d).

The latter map takes Z̃_{σ}^{0}(ℙ^{n+d}, d)|_{Z_{σ}^{0}(ℙ^{n+d},d)} to Z̃_{σ}^{0}(ℙ^{n}, d)|_{Z_{σ}^{0}(ℙ^{n},d)}. Since the closure of

Z̃_{σ}^{0}(ℙ^{n+d}, d)|_{Z_{σ}^{0}(ℙ^{n+d},d)}

contains Z̃_{σ}^{0}(ℙ^{n}, d), it follows that so does the closure of Z̃_{σ}^{0}(ℙ^{n}, d)|_{Z_{σ}^{0}(ℙ^{n},d)}. This observation implies the last claim of Lemma 4.10.

We conclude this subsection by briefly describing the stratification mentioned above. A stratum 𝔐_{Γ_{B}} of \overline{𝔐}_{0,(ℵ_{B}(σ),J_{B}(σ))} corresponds to a tuple Γ_{B} ≡ (Γ_{B;l})_{l∈ℵ_{B}(σ)} of dual graphs, all of which are trees. The vertices of Γ_{B;l} correspond to the irreducible components of the domain of the stable map b_{l} in the definition of \overline{𝔐}_{0,(ℵ_{B}(σ),J_{B}(σ))} at the beginning of Subsection 3.3. Each vertex v of Γ_{B;l} is labeled by a nonnegative integer, which specifies the degree of the stable map b_{l} on the corresponding component Σ_{v}. There is an edge in Γ_{B;l} between two vertices if and only if the two corresponding components of the domain share a node. In addition, there are tails attached at some vertices of Γ_{B;l}, which are labeled by the indexing set for marked points of the map b_{l}, i.e. J_{l,P} in the notation of Subsection 3.3. Let v_{l}^{∗} be the vertex of Γ_{B;l} to which the tail corresponding to the marked point 0 is attached. If the degree of v_{l}^{∗} is positive, let

χ_{l}(Γ_{B}) ≡ χ_{l}(Γ_{B;l}) = {v_{l}^{∗}}.

Otherwise, denote by χ_{l}(Γ_{B}) the set of positive-degree vertices of Γ_{B;l} that are not separated from v_{l}^{∗} by a positive-degree vertex. Suppose

b≡(b_{l})_{l∈ℵ_{B}(σ)} ∈ 𝔐_{Γ_{B}} ≡ \overline{𝔐}_{0,(ℵ_{B}(σ),J_{B}(σ))} ∩ ∏_{l∈ℵ_{B}(σ)} 𝔐_{Γ_{B;l}},  with  b_{l} =[Σ_{l}, u_{l}]

as in the paragraph preceding Lemma 3.8. If l ∈ℵ_{B}(σ) and v =v_{l}^{∗}, let

Im D_{v}|_{b} = Im D_{0}|_{b_{l}} ≡ Im du_{l}|_{x_{0}(b_{l})} ⊂ T_{ev_{0}(b)}ℙ^{n}.

If v is a vertex of Γ_{B;l} different from v_{l}^{∗}, we denote by Im D_{v}|_{b} the image of d{u_{l}|_{Σ_{v}}} at the node of Σ_{v} corresponding to the edge of Γ_{B;l} that leaves v on the unique path from v on v_{l}^{∗} in Γ_{B;l}. Note that if v ∈χ_{l}(Γ_{B}), the image of this node under u_{l} is ev_{0}(b). We set

χ(Γ) = ⨆_{l∈ℵ_{B}(σ)}χ_{l}(Γ_{B}).

With b as above, let

codim D|_{b} = |χ(Γ_{B})| − dim Span{Im D_{v}|_{b}: v ∈χ_{l}(Γ_{B}), l ∈ℵ_{B}(σ)}.

For each pair α=(Γ_{B}, μ), where μ∈ℤ^{+} is such that

max (1, |χ(Γ_{B})|−n) ≤ μ ≤ |χ(Γ_{B})|,  (4.4)

we put

Z_{Γ_{B};α} = {b∈𝔐_{Γ_{B}}: codim D|_{b} =μ}.

By the first statement of [Z4, Theorem 2.3],

\overline{Z}_{σ}^{0} = ⨆_{α} Z_{σ;α},  where  Z_{σ;α} = \overline{M}_{1,I_{P} (σ)⊔J_{P} (σ)}×Z_{Γ_{B};α}.

The disjoint union is taken over all pairs α = (Γ, μ) as described above. From transversality as in the first claim of Lemma 3.8, it is easy to see that

codim_{𝔐_{Γ_{B}}}Z_{Γ_{B};α} = (n − (|χ(Γ_{B})|−μ))μ
≥ n − (|χ(Γ_{B})|−μ);

(4.5)

see the end of [Z5, Subsect. 2.3], for example. The above inequality follows from the first inequality in (4.4). By (4.5), if m=|ℵ_{B}(σ)|≤n,

codim\overline{Z}{0 \atop σ}Z_{σ;α} = codim_{𝔐_{Γ_{B}}}Z_{Γ_{B};α} + codim_{\overline{𝔐}_{0,(ℵ_{B} (σ),J_{B} (σ))}}𝔐_{Γ_{B}}
− codim_{\overline{M}_{1,I_{P} (σ)⊔J_{P} (σ)}×\overline{𝔐}_{0,(ℵ_{B} (σ),J_{B} (σ))}} \overline{Z}_{σ}^{0}
≥ (n−|χ(Γ_{B})|+μ) + (|χ(Γ_{B})|−|ℵ_{B}(σ)|) − (n−|ℵ_{B}(σ)|+1) = μ−1.

On the other hand, by the last statement of [Z4, Theorem 2.3],

max \{dim N \overline{Z}_{σ}^{0}|_{b}: b∈Z_{σ;α}\} = μ.

We conclude that

max \{dim N \overline{Z}_{σ}^{0}|_{b}: b∈Z_{σ;α}\} ≤ codim\overline{Z}{0 \atop σ}Z_{σ;α} + 1.

The equality holds if and only if μ = 1 and Γ_{B} is a tuple of one-vertex graphs, i.e. the image of M_{1,I_{P} (σ)⊔J_{P} (σ)}×𝔐_{Γ_{B}} under ι_{0,σ} is contained in 𝔐_{1,σ}. This observation concludes the proof of the stratification claim made in the previous paragraph.

4.3 Inductive Construction

This subsection is the analogue of Subsection 3.4 in the present situation. Suppose σ ∈ A_{1}(d, k) and we have constructed

(I1) an idealized blowup π_{σ−1} : \overline{𝔐}_{1,k}^{σ−1} ⟶\overline{𝔐}_{1,k}^{0} such that π_{σ−1} is an isomorphism outside of the preimages of the subvarieties \overline{𝔐}_{1,σ′}^{0} with σ′ ≤σ−1;

(I2) for each σ′ ∈\{(0)\}⊔A_{1}(d, k), a subvariety \overline{𝔐}_{1,σ′}^{σ−1} of \overline{𝔐}_{1,k}^{σ−1} such that

\overline{𝔐}_{1,k}^{σ−1} = \overline{𝔐}_{1,(0)}^{σ−1} ∪ ⋃_{σ′∈A_{1}(d,k)} \overline{𝔐}_{1,σ′}^{σ−1}, π_{σ−1}(\overline{𝔐}_{1,σ′}^{σ−1}) = \overline{𝔐}_{1,σ′}^{0} ∀ σ′ ∈\{(0)\}⊔A_{1}(d, k),

and \overline{𝔐}_{1,σ^{∗}}^{σ−1} is the proper transform of \overline{𝔐}_{1,σ^{∗}}^{0} for σ^{∗} =(0) and for all σ^{∗} ∈A_{1}(d, k) such that σ^{∗} >σ−1. We assume that

(I3) for all σ_{1}, σ_{2} ∈A_{1}(d, k) such that σ_{1} ≠σ_{2}, σ_{1} ⊀ σ_{2}, σ_{2} ⊀ σ_{1}, and σ−1<σ_{1}, σ_{2},

\overline{𝔐}_{1,σ_{1}}^{σ−1} ∩ \overline{𝔐}_{1,σ_{2}}^{σ−1} \begin{cases}⊂ \overline{𝔐}_{1,σ̃(σ_{1},σ_{2})}^{σ−1}, & if σ̃(σ_{1}, σ_{2})>σ−1; \\ = ∅, & otherwise, \end{cases}

where σ̃(σ_{1}, σ_{2}) is as in Lemma 4.6.

We also assume that for every σ′ ∈A_{1}(d, k) such that σ′ ≤σ−1:

(I4) \overline{𝔐}_{1,σ′}^{σ−1} is the image of a G_{σ′}-invariant immersion

ι_{σ−1,σ′}: M̃_{1,(I_{P} (σ′),J_{P} (σ′))} × 𝔐̃_{0,(ℵ_{B}(σ′),J_{B}(σ′))}^{ϱ_{σ′}(σ−1)} ⟶ \overline{𝔐}_{1,k}^{σ−1}, where

ϱ_{σ′}(σ−1) = \begin{cases}max \{ϱ∈A_{B}(σ′; σ^{∗}): σ′ ≺σ^{∗} ≤σ−1\}, & if ∃σ^{∗} ∈A_{1}(d, k) s.t. σ′ ≺σ^{∗} ≤σ−1; \\ 0, & otherwise, \end{cases}

and 𝔐̃_{0,(ℵ_{B}(σ′),J_{B}(σ′))}^{ϱ_{σ′}(σ−1)} ≡𝔐̃_{0,(ℵ_{B}(σ′),J_{B}(σ′))}^{ϱ_{σ′}(σ−1)}(ℙ^{n}, d) is the blowup of 𝔐̃_{0,(ℵ_{B}(σ′),J_{B}(σ′))}^{0}(ℙ^{n}, d) constructed in Subsection 3.4;

(I5) if σ^{∗} ∈A_{1}(d, k) is such that σ−1<σ^{∗} and σ′ ≺σ^{∗}, then

ι_{σ−1,σ′}^{−1}(\overline{𝔐}_{1,σ^{∗}}^{σ−1}) = M̃_{1,(I_{P} (σ′),J_{P} (σ′))} × ( ⋃_{ϱ∈A_{B}(σ′;σ^{∗})} 𝔐̃_{0,ϱ}^{ϱ_{σ′}(σ−1)}),

where 𝔐̃_{0,ϱ}^{ϱ_{σ′}(σ−1)} ≡ 𝔐̃_{0,ϱ}^{ϱ_{σ′}(σ−1)}(ℙ^{n}, d) is the subvariety of 𝔐̃_{0,(ℵ_{B}(σ′),J_{B}(σ′))}^{ϱ_{σ′}(σ−1)} described in Subsection 3.4;

(I6) an idealized normal bundle for the immersion ι_{σ−1,σ′} is given by

N_{ι_{σ−1,σ′}}^{ide} = π_{P}^{∗} 𝕃 ⊗ π_{B}^{∗}π_{ϱ_{σ′}(σ−1)}^{∗}γ_{(ℵ_{B}(σ′),J_{B}(σ′))},

where

π_{B}, π_{P} : M̃_{1,(I_{P} (σ′),J_{P} (σ′))} × 𝔐̃_{0,(ℵ_{B}(σ′),J_{B}(σ′))}^{ϱ_{σ′}(σ−1)} ⟶ M̃_{1,(I_{P} (σ′),J_{P} (σ′))}, 𝔐̃_{0,(ℵ_{B}(σ′),J_{B}(σ′))}^{ϱ_{σ′}(σ−1)}

are the two projection maps and 𝕃 ⟶ M̃_{1,(I_{P} (σ′),J_{P} (σ′))} is the universal tangent line bundle of Subsection 2.3;

(I7) \overline{Z}_{σ′}^{σ−1} ≡ι_{σ−1,σ′}^{−1}(\overline{𝔐}_{1,(0)}^{σ−1}) is the closure of

Z_{σ′}^{σ−1} ≡ M̃_{1,(I_{P} (σ′),J_{P} (σ′))} × (D̃_{ϱ_{σ′}(σ−1)}^{−1}(0) − ⋃_{ϱ_{σ′}(σ−1)<ϱ^{ϱ∈A_{B} (ℵ_{B} (σ′);d,J_{B} (σ′))}} 𝔐̃_{0,ϱ}^{ϱ_{σ′}(σ−1)})

in M̃_{1,(I_{P} (σ′),J_{P} (σ′))}×𝔐̃_{0,(ℵ_{B}(σ′),J_{B}(σ′))}^{ϱ_{σ′}(σ−1)} and

N \overline{Z}_{σ′}^{σ−1} ≡ N_{ι_{σ−1,σ′}|\overline{𝔐}_{1,(0)}^{σ−1}} = N_{ι_{σ−1,σ′}}^{ide} |\overline{Z}{σ−1 \atop σ′}

is the normal cone for ι_{σ−1,σ′}|\overline{𝔐}_{1,(0)}^{σ−1} in \overline{𝔐}_{1,k}^{σ−1}; (I8) the immersion map

ῑ_{σ−1,σ′}: (M̃_{1,(I_{P} (σ′);J_{P} (σ′))}×𝔐̃_{0,(ℵ_{B}(σ′),J_{B}(σ′))}^{ϱ_{σ′}(σ−1)})/G_{σ′} ⟶ \overline{𝔐}_{1,k}^{σ−1}

induced by ι_{σ−1,σ′} is an embedding.

Furthermore, we assume that for every σ^{∗} ∈A_{1}(d, k) such that σ^{∗} >σ−1:

(I9) the domain of the G_{σ^{∗}}-invariant immersion ι_{σ−1,σ^{∗}} induced by ι_{0,σ^{∗}} is

\overline{M}_{1,(I_{P} (σ^{∗}),J_{P} (σ^{∗}))}^{ρ_{σ^{∗}}(σ−1)} × \overline{𝔐}_{0,(ℵ_{B}(σ^{∗}),J_{B}(σ^{∗}))}, where

ρ_{σ^{∗}}(σ−1) = \begin{cases}max \{ρ∈A_{P} (σ^{∗}; σ′): σ′ ≤σ−1, σ′ ≺σ^{∗}\}, & \begin{matrix}if ∃σ′ ∈A_{1}(d, k) \\ s.t. σ′ ≤σ−1, σ′ ≺σ^{∗}; \end{matrix} \\ 0, & otherwise, \end{cases}

and \overline{M}_{1,(I_{P} (σ^{∗}),J_{P} (σ^{∗}))}^{ρ_{σ^{∗}}(σ−1)} ⟶\overline{M}_{1,I_{P} (σ^{∗})⊔J_{P} (σ^{∗})} is the blowup constructed in Subsection 2.3; (I10) if σ′ ∈A_{1}(d, k) is such that σ−1<σ′ ≺σ^{∗}, then

ι_{σ−1,σ^{∗}}^{−1}(𝔐̃_{1,σ′}^{σ−1}) = ( ⋃_{ρ∈A_{P} (σ^{∗};σ′)} \overline{M}_{1,ρ}^{ρ_{σ^{∗}}(σ−1)}) × \overline{𝔐}_{0,(ℵ_{B}(σ^{∗}),J_{B}(σ^{∗}))};

(I11) if σ^{∗} is as in Lemma 4.7, an idealized normal bundle for the immersion ι_{σ−1,σ^{∗}} is given by

N_{ι_{σ−1,σ^{∗}}}^{ide} = ⨁_{i∈[m^{∗}]} π_{P}^{∗} L_{ρ_{σ^{∗}}(σ−1),i}⊗π_{B}^{∗}π_{i}^{∗}L_{0},

where

π_{P} , π_{B}: \overline{M}_{1,(I_{P} (σ^{∗}),J_{P} (σ^{∗}))}^{ρ_{σ^{∗}}(σ−1)}×\overline{𝔐}_{0,(ℵ_{B}(σ^{∗}),J_{B}(σ^{∗}))} ⟶ \overline{M}_{1,(I_{P} (σ^{∗}),J_{P} (σ^{∗}))}^{ρ_{σ^{∗}}(σ−1)}, \overline{𝔐}_{0,(ℵ_{B}(σ^{∗}),J_{B}(σ^{∗}))}

are the two projection maps and L_{ρ_{σ^{∗}}(σ−1),i} ⟶\overline{M}_{1,(I_{P} (σ^{∗}),J_{P} (σ^{∗}))}^{ρ_{σ^{∗}}(σ−1)} is the line bundle constructed in Subsection 2.3;

(I12) \overline{Z}_{σ^{∗}}^{σ−1} ≡ ι_{σ−1,σ^{∗}}^{−1}(\overline{𝔐}_{1,(0)}^{σ−1}) is the closure of Z_{σ^{∗}}^{0} in \overline{M}_{1,(I_{P} (σ^{∗}),J_{P} (σ^{∗}))}^{ρ_{σ^{∗}}(σ−1)}×\overline{𝔐}_{0,(ℵ_{B}(σ^{∗}),J_{B}(σ^{∗}))} and the normal cone

N \overline{Z}_{σ^{∗}}^{σ−1} ≡ N_{ι_{σ−1,σ^{∗}}|\overline{𝔐}_{1,(0)}^{σ−1}}

for ι_{σ−1,σ^{∗}}|\overline{Z}_{σ^{∗}}^{σ−1} is the closure of N \overline{Z}_{σ^{∗}}^{0}|_{Z_{σ^{∗}}^{0}} in N_{ι_{σ−1,σ^{∗}}}^{ide} ;

(I13) Im^{s} ῑ_{σ−1,σ^{∗}} ⊂ ⋃_{σ−1<σ′≺σ^{∗}} 𝔐̃_{1,σ′}^{σ−1}, where

ῑ_{σ−1,σ^{∗}}: (\overline{M}_{1,(I_{P} (σ^{∗}),J_{P} (σ^{∗}))}^{ρ_{σ^{∗}}(σ−1)}×\overline{𝔐}_{0,(ℵ_{B}(σ^{∗}),J_{B}(σ^{∗}))})/G_{σ^{∗}} ⟶ \overline{𝔐}_{1,k}^{σ−1},

is the immersion map induced by ι_{σ−1,σ^{∗}}.

Finally, we assume that

(I14) the collections {ι_{σ−1,σ′}}_{σ′∈A_{1}(d,k)} and {ῑ_{σ−1,σ′}}_{σ′∈A_{1}(d,k)} of immersions are properly selfintersecting;

(I15) for all σ′ ∈ A_{1}(d, k), the subvariety \overline{𝔐}_{1,(0)}^{σ−1} of \overline{𝔐}_{1,k}^{σ−1} is proper relative to the immersions ι_{σ−1,σ′} and ῑ_{σ−1,σ′}.

By the inductive assumption (I3), if σ_{1} and σ_{2} are non-comparable elements of (A_{1}(d, k), ≺), the proper transforms of \overline{𝔐}_{1,σ_{1}}^{0} and \overline{𝔐}_{1,σ_{2}}^{0} become disjoint by the time either is ready to be blown up for any ordering < extending the partial ordering ≺. Similarly to the three blowup constructions encountered previously, (I3) will imply that the end result of the present blowup construction is independent of the choice of an extension <. By (I9), our blowup construction modifies each immersion ι_{0,σ^{∗}} by changing the first factor of the domain according to the blowup construction of Subsection 2.3, until a proper transform of the image of ι_{0,σ^{∗}} is to be blown up; see below. By (I11) and (I13), in the process, the singular locus of ι_{0,σ^{∗}} disappears and the first component in every summand of N_{ι_{0,σ^{∗}}}^{ide} gets twisted to 𝕃. In particular, all blowup loci are smooth. On the other hand, by the inductive assumptions (I7) and (I8), for σ′ ≤σ−1 the intersection of the proper transform of \overline{𝔐}_{1,(0)}^{0} with the proper transform of the exceptional divisor \overline{𝔐}_{1,σ′}^{σ′} is an embedding of a subvariety of a smooth variety. The singular locus of this subvariety is annihilated by the time the entire blowup construction is complete, according to the inductive assumptions (I7) above and the inductive assumption (I4) in Subsection 3.4. These assumptions imply that the proper transform of \overline{𝔐}_{1,(0)}^{0} after the final blowup step is smooth.

We note that all of the assumptions (I1)-(I15) are satisfied if σ−1 is replaced by 0. In particular, (I3) is a restatement of Lemma 4.6, while (I10)-(I15) are contained in Lemmas 4.7, 4.8, and 4.10.

If σ ∈A_{1}(d, k) is as above, let

π̃_{σ}: \overline{𝔐}_{1,k}^{σ} ⟶ \overline{𝔐}_{1,k}^{σ−1}

be the idealized blowup of \overline{𝔐}_{1,k}^{σ−1} along \overline{𝔐}_{1,σ}^{σ−1}, which is a smooth subvariety by the inductive assumption (I13). We denote the idealized exceptional divisor,

E_{\overline{𝔐}_{1,σ}^{σ−1}}^{ide} = ℙN ide ; ῑ_{σ−1,σ},

by \overline{𝔐}_{1,σ}^{σ}. For each σ′ ∈\{(0)\}⊔(A_{1}(d, k)−\{σ\}), we denote by

\overline{𝔐}_{1,σ′}^{σ} ⊂ Bl\overline{𝔐}{σ−1 \atop 1,σ} \overline{𝔐}_{1,k}^{σ−1} ⊂ \overline{𝔐}_{1,k}^{σ}

the proper transform of \overline{𝔐}_{1,σ′}^{σ−1}. Let π_{σ} =π_{σ−1}∘π̃_{σ}.

The inductive assumptions (I1) and (I2), with σ−1 replaced by σ, are clearly satisfied, while (I3), (I8) for σ′ ≠σ, and (I13)-(I15) follow from (2) of Lemma 3.3, Corollary 3.4, and Lemma 4.5. On the other hand, by (I9), the domain of the immersion ι_{σ−1,σ} is

\overline{M}_{1,(I_{P} (σ),J_{P} (σ))}^{ρ_{σ}(σ−1)}×\overline{𝔐}_{0,(ℵ_{B}(σ),J_{B}(σ))} = M̃_{1,(I_{P} (σ),J_{P} (σ))}×\overline{𝔐}_{0,(ℵ_{B}(σ),J_{B}(σ))}.

By (I11), the chosen idealized normal bundle for the immersion ι_{σ−1,σ} is given by

N_{ι_{σ−1,σ}}^{ide} = ⨁_{i∈[m]} π_{P}^{∗} L_{ρ_{σ}(σ−1),i}⊗π_{B}^{∗}π_{i}^{∗}L_{0} = π_{P}^{∗} 𝕃 ⊗ π_{B}^{∗}F_{(ℵ_{B}(σ),J_{B}(σ))}.  (4.6)

Thus, the domain of the immersion ι_{σ,σ} induced by ι_{σ−1,σ} is

ℙN_{ι_{σ−1,σ}}^{ide} = M̃_{1,(I_{P} (σ),J_{P} (σ))}×𝔐̃_{0,(ℵ_{B}(σ),J_{B}(σ))}^{0} = M̃_{1,(I_{P} (σ),J_{P} (σ))}×𝔐̃_{0,(ℵ_{B}(σ),J_{B}(σ))}^{ϱ_{σ}(σ)}.

By the first statement of Lemma 4.3, an idealized normal bundle for the embedding ι_{σ,σ} is the tautological line bundle over ℙN_{ι_{σ−1,σ}}^{ide}, i.e.

N_{ι_{σ,σ}}^{ide} = π_{P}^{∗} 𝕃⊗π_{B}^{∗}γ_{(ℵ_{B}(σ),J_{B}(σ))} = π_{P}^{∗} 𝕃⊗π_{B}^{∗}π_{ϱ_{σ}(σ)}^{∗}γ_{(ℵ_{B}(σ),J_{B}(σ))}.

Thus, the inductive assumptions (I4) and (I6), with σ′ = σ and σ−1 replaced by σ, are satisfied. The same is the case with (I8), since the map ῑ_{σ−1,σ} is an embedding by (I13).

We also note that by the first statement of Lemma 4.9, the inductive assumptions (I1) and (I2), and the last statement of Lemma 3.3,

ι_{σ−1,σ}^{−1}(\overline{𝔐}_{1,σ^{∗}}^{σ−1}) = M̃_{1,(I_{P} (σ),J_{P} (σ))} × ( ⋃_{ϱ∈A_{B}(σ;σ^{∗})} \overline{𝔐}_{0,ϱ})

for all σ^{∗} ∈A_{1}(d, k) such that σ ≺σ^{∗}. In addition, by the last statement of Lemma 4.9

ι_{σ−1,σ}|_{M̃_{1,(I_{P} (σ),J_{P} (σ))}×\overline{𝔐}_{0,ϱ}}^{∗}T \overline{𝔐}_{1,σ^{∗}}^{σ−1}/T (M̃_{1,(I_{P} (σ),J_{P} (σ))}×\overline{𝔐}_{0,ϱ}) ⊂ N_{ι_{σ−1,σ}}^{ide}

is a vector bundle for all ϱ∈A_{B}(σ; σ^{∗}) and

ι_{σ−1,σ}|_{M_{1,I_{P} (σ)⊔J_{P} (σ)}×𝔐_{0,ϱ}}^{∗}T \overline{𝔐}_{1,σ^{∗}}^{σ−1}/T (M_{1,(I_{P} (σ),J_{P} (σ))}×𝔐_{0,ϱ}) = ⨁_{i∈ℵ_{P} (ϱ)}π_{P}^{∗} L_{i}⊗π_{B}^{∗}π_{i}^{∗}L_{0}.

Thus, by the first equality in (4.6),

\begin{aligned}ι_{σ−1,σ}|_{M̃_{1,(I_{P} (σ),J_{P} (σ))}×\overline{𝔐}_{0,ϱ}}^{∗}T \overline{𝔐}_{1,σ^{∗}}^{σ−1}/T (M̃_{1,(I_{P} (σ),J_{P} (σ))}×\overline{𝔐}_{0,ϱ}) & = π_{P}^{∗} 𝕃⊗ ⨁_{i∈ℵ_{P} (ϱ)}π_{B}^{∗}π_{i}^{∗}L_{0} \\ & = π_{P}^{∗} 𝕃 ⊗ π_{B}^{∗}F_{ϱ;P} . \end{aligned}

It follows that

ι_{σ,σ}^{−1}(\overline{𝔐}_{1,σ^{∗}}^{σ}) = ⋃_{ϱ∈A_{B}(σ;σ^{∗})} ℙ(π_{P}^{∗} 𝕃 ⊗ π_{B}^{∗}F_{ϱ;P} )|_{M̃_{1,(I_{P} (σ),J_{P} (σ))}×\overline{𝔐}_{0,ϱ}}

= M̃_{1,(I_{P} (σ),J_{P} (σ))} × ( ⋃_{ϱ∈A_{B}(σ;σ^{∗})} ℙF_{ϱ;P} )

≡ M̃_{1,(I_{P} (σ),J_{P} (σ))} × ( ⋃_{ϱ∈A_{B}(σ;σ^{∗})} 𝔐̃_{0,ϱ}^{0}) = M̃_{1,(I_{P} (σ),J_{P} (σ))} × ( ⋃_{ϱ∈A_{B}(σ;σ^{∗})} 𝔐̃_{0,ϱ}^{ϱ_{σ}(σ)}),

as needed for the inductive assumption (I5) with σ−1 replaced by σ and σ′ =σ.

Furthermore, by (I12), N \overline{Z}_{σ}^{σ−1} is the closure of

\begin{aligned}N \overline{Z}_{σ}^{0}|_{Z_{σ}^{0}} & ≡ ker D_{0,σ}|_{M_{1,I_{P} (σ)⊔J_{P} (σ)}×𝔐_{0,(ℵ_{B} (σ),J_{B} (σ))}} \\ & = π_{P}^{∗} 𝕃 ⊗ π_{B}^{∗} ker D_{(ℵ_{B}(σ),J_{B}(σ))}|_{𝔐_{0,(ℵ_{B} (σ),J_{B} (σ))}} \end{aligned}

in π_{P}^{∗} 𝕃⊗π_{B}^{∗}F_{(ℵ_{B}(σ),J_{B}(σ))}, where D_{(ℵ_{B}(σ),J_{B}(σ))} is the bundle homomorphism described in Subsection 3.3. Thus, by the first statement of Lemma 4.5,

\overline{Z}_{σ}^{σ} ≡ ι_{σ,σ}^{−1}(\overline{𝔐}_{1,(0)}^{σ})

is the closure of

M_{1,(I_{P} (σ),J_{P} (σ))} × {b∈ℙF_{(ℵ_{B}(σ),J_{B}(σ))}|_{𝔐_{0,(ℵ_{B} (σ),J_{B} (σ))}} : D̃_{0}b=0}

in M̃_{1,(I_{P} (σ),J_{P} (σ))}×𝔐̃_{0,(ℵ_{B}(σ),J_{B}(σ))}^{ϱ_{σ}(σ)}. The inductive assumption (I7), with σ′ =σ and σ−1 replaced by σ, now follows from the first statement of Lemma 3.11.

We next verify that the inductive assumptions (I4)-(I7) hold for σ′ <σ, with σ−1 replaced by σ. If σ′ ⊀σ, then

ϱ_{σ′}(σ) = ϱ_{σ′}(σ−1)  and  \overline{𝔐}_{1,σ′}^{σ−1} ∩ \overline{𝔐}_{1,σ}^{σ−1} = ∅,

by definition and (I3), respectively. It then follows that

ι_{σ,σ′} = ι_{σ−1,σ′}, N_{ι_{σ,σ′}}^{ide} = N_{ι_{σ−1,σ′}}^{ide} ,
and \overline{𝔐}_{1,σ′}^{σ} ∩ \overline{𝔐}_{1,σ^{∗}}^{σ} = \overline{𝔐}_{1,σ′}^{σ−1} ∩ \overline{𝔐}_{1,σ^{∗}}^{σ−1} ∀ σ^{∗} ∈\{(0)\}⊔A_{1}(d, k).

Thus, the inductive assumptions (I4)-(I7), as stated above, imply the corresponding statements with σ−1 replaced by σ.

Suppose that σ′ ≺ σ. By (I4) and (1) of Lemma 3.3, the domain of the immersion ι_{σ,σ′} induced by ι_{σ−1,σ′} is the blowup of

M̃_{1,(I_{P} (σ′),J_{P} (σ′))} × 𝔐̃_{0,(ℵ_{B}(σ′),J_{B}(σ′))}^{ϱ_{σ′}(σ−1)}

along the preimage of \overline{𝔐}_{1,σ}^{σ−1} under ι_{σ−1,σ′} in

id×π_{ϱ_{σ′}(σ−1)} : M̃_{1,(I_{P} (σ′),J_{P} (σ′))}×𝔐̃_{0,(ℵ_{B}(σ′),J_{B}(σ′))}^{ϱ_{σ′}(σ−1)} ⟶ M̃_{1,(I_{P} (σ′),J_{P} (σ′))}×𝔐̃_{0,(ℵ_{B}(σ′),J_{B}(σ′))}^{0}.

By (I5), this preimage is

M̃_{1,(I_{P} (σ′),J_{P} (σ′))} × ( ⋃_{ϱ∈A_{B}(σ′;σ)} 𝔐̃_{0,ϱ}^{ϱ_{σ′}(σ−1)}).

By the inductive assumption (I5) in Subsection 3.4 and the second paragraph after Lemma 4.9,

𝔐̃_{0,ϱ_{1}}^{ϱ_{σ′}(σ−1)} ∩ 𝔐̃_{0,ϱ_{2}}^{ϱ_{σ′}(σ−1)} = ∅ ∀ ϱ_{1}, ϱ_{2} ∈A_{B}(σ′; σ), ϱ_{1} ≠ϱ_{2}.

Thus, by the construction of Subsection 3.4, the blowup of 𝔐̃_{0,(ℵ_{B}(σ′),J_{B}(σ′))}^{ϱ_{σ′}(σ−1)} along

⋃_{ϱ∈A_{B}(σ′;σ)} 𝔐̃_{0,ϱ}^{ϱ_{σ′}(σ−1)}

is 𝔐̃_{0,(ℵ_{B}(σ′),J_{B}(σ′))}^{ϱ_{σ′}(σ)}, as needed for the inductive statement (I4), with σ−1 replaced by σ. The inductive requirement (I5) is obtained by the same reasoning, using the last statement of Lemma 3.3.

Since \overline{𝔐}_{1,σ}^{σ−1} is not contained in \overline{𝔐}_{1,σ′}^{σ−1}, the bundle homomorphism

ι_{σ−1,σ′}^{∗}T \overline{𝔐}_{1,σ}^{σ−1} ⟶ N_{ι_{σ−1,σ′}}^{ide}

must be surjective on every fiber over ι_{σ−1,σ′}^{−1}(\overline{𝔐}_{1,σ}^{σ−1}) by (I14). Thus, the inductive assumption (I6), for σ′ < σ, continues to hold. Furthermore, by (I7) and the last statement of Lemma 3.3, \overline{Z}_{σ′}^{σ} is the closure of

\begin{matrix}M̃_{1,(I_{P} (σ′),J_{P} (σ′))} × (D̃_{ϱ_{σ′}(σ−1)}^{−1}(0) − ⋃_{ϱ_{σ′}(σ−1)<ϱ^{ϱ∈A_{B} (ℵ_{B} (σ′);d,J_{B} (σ′))}} 𝔐̃_{0,ϱ}^{ϱ_{σ′}(σ−1)}) \\ = M̃_{1,(I_{P} (σ′),J_{P} (σ′))} × (D̃_{ϱ_{σ′}(σ−1)}^{−1}(0) − ⋃_{ϱ∈A_{B}(σ′;σ)} 𝔐̃_{0,ϱ}^{ϱ} − ⋃_{ϱ_{σ′}(σ)<ϱ^{ϱ∈A_{B} (ℵ_{B} (σ′);d,J_{B} (σ′))}} 𝔐̃_{0,ϱ}^{ϱ_{σ′}(σ)}) \end{matrix}

in M̃_{1,(I_{P} (σ′),J_{P} (σ′))}×𝔐̃_{0,(ℵ_{B}(σ′),J_{B}(σ′))}^{ϱ_{σ′}(σ)}. By the construction of Subsection 3.4,

D̃_{ϱ_{σ′}(σ−1)}|_{𝔐̃_{0,(ℵ_{B} (σ′),J_{B} (σ′))}^{ϱ_{σ′} (σ)}−⋃_{ϱ∈A_{B} (σ′;σ)}𝔐̃_{0,ϱ}^{ϱ}} = D̃_{ϱ_{σ′}(σ)}|_{𝔐̃_{0,(ℵ_{B} (σ′),J_{B} (σ′))}^{ϱ_{σ′} (σ)}−⋃_{ϱ∈A_{B} (σ′;σ)}𝔐̃_{0,ϱ}^{ϱ}}.

Since D̃_{ϱ_{σ′}(σ)} is transverse to the zero set outside of the subvarieties 𝔐̃_{0,ϱ}^{ϱ_{σ′}(σ)} with ϱ > ϱ_{σ′}(σ) by the inductive requirement (I4) in Subsection 3.4, we conclude that the first part of the inductive assumption (I7), with σ − 1 replaced by σ, is satisfied. The second part follows from the last statement of Lemma 4.5.

It remains to verify the inductive assumption (I9)-(I12), with σ − 1 replaced by σ. Suppose σ^{∗} ∈A_{1}(d, k) is such that σ <σ^{∗}. If σ ⊀σ^{∗}, then

ρ_{σ^{∗}}(σ) = ρ_{σ^{∗}}(σ−1)  and  \overline{𝔐}_{1,σ^{∗}}^{σ−1} ∩ \overline{𝔐}_{1,σ}^{σ−1} = ∅,

by definition and (I3), respectively. It then follows that

ι_{σ,σ^{∗}} = ι_{σ−1,σ^{∗}}, N_{ι_{σ,σ^{∗}}}^{ide} = N_{ι_{σ−1,σ^{∗}}}^{ide} ,
and \overline{𝔐}_{1,σ^{∗}}^{σ} ∩ \overline{𝔐}_{1,σ′}^{σ} = \overline{𝔐}_{1,σ^{∗}}^{σ−1} ∩ \overline{𝔐}_{1,σ′}^{σ−1} ∀σ′ ∈\{(0)\}⊔A_{1}(d, k).

Thus, the inductive assumptions (I9)-(I12), as stated above, imply the corresponding statements with σ−1 replaced by σ.

Suppose that σ ≺ σ^{∗}. By (I9) and (1) of Lemma 3.3, the domain of the immersion ι_{σ,σ^{∗}} induced by ι_{σ−1,σ^{∗}} is the blowup of

\overline{M}_{1,(I_{P} (σ^{∗}),J_{P} (σ^{∗}))}^{ρ_{σ^{∗}}(σ−1)} × \overline{𝔐}_{0,(ℵ_{B}(σ^{∗}),J_{B}(σ^{∗}))}

along the preimage of \overline{𝔐}_{1,σ}^{σ−1} under ι_{σ−1,σ^{∗}} in

π_{ρ_{σ^{∗}}(σ−1)}×id: \overline{M}_{1,(I_{P} (σ^{∗}),J_{P} (σ^{∗}))}^{ρ_{σ^{∗}}(σ−1)}×\overline{𝔐}_{0,(ℵ_{B}(σ^{∗}),J_{B}(σ^{∗}))} ⟶ \overline{M}_{1,I_{P} (σ^{∗})⊔J_{P} (σ^{∗})}×\overline{𝔐}_{0,(ℵ_{B}(σ^{∗}),J_{B}(σ^{∗}))}

By (I10), this preimage is

( ⋃_{ρ∈A_{P} (σ^{∗};σ)} \overline{M}_{1,ρ}^{ρ_{σ^{∗}}(σ−1)}) × \overline{𝔐}_{0,(ℵ_{B}(σ^{∗}),J_{B}(σ^{∗}))}.

By Lemma 2.6 and the paragraph after Lemma 4.6,

\overline{M}_{1,ρ_{1}}^{ρ_{σ^{∗}}(σ−1)} ∩ \overline{M}_{1,ρ_{2}}^{ρ_{σ^{∗}}(σ−1)} = ∅ ∀ ρ_{1}, ρ_{2} ∈A_{P} (σ^{∗}; σ), ρ_{1} ≠ρ_{2}.

Thus, by the construction of Subsection 2.3, the blowup of \overline{M}_{1,(I_{P} (σ^{∗}),J_{P} (σ^{∗}))}^{ρ_{σ^{∗}}(σ−1)} along

⋃_{ρ∈A_{P} (σ^{∗};σ)} \overline{M}_{1,ρ}^{ρ_{σ^{∗}}(σ−1)}

is \overline{M}_{1,(I_{P} (σ^{∗}),J_{P} (σ^{∗}))}^{ρ_{σ^{∗}}(σ)}, as needed for the inductive statement (I9), with σ −1 replaced by σ. The inductive assumptions (I10) and (I11) are verified similarly, using the last statement of Lemma 3.3 and Lemma 4.3. The argument for (I11) is nearly identical to the verification of the inductive assumption (I11) in Subsection 3.4. Finally, the inductive requirement (I12), with σ−1 replaced by σ, follows from the last statement of Lemma 4.5, along with the assumptions (I1) and (I2).

We conclude this blowup construction after the σ_{max} step and put

𝔐̃_{1,k}^{0}(ℙ^{n}, d) = \overline{𝔐}_{1,(0)}^{σ_{max}},  π̃ = π_{σ_{max}} |\overline{𝔐}_{1,(0)}^{σmax} , and Z̃_{σ}(ℙ^{n}, d) = \overline{Z}_{σ}^{σ_{max}}.

The inductive assumptions (I1)-(I8) imply that

π̃: 𝔐̃_{1,k}^{0}(ℙ^{n}, d) ⟶ \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d)

is a desingularization as described in Subsection 1.2. By (I3), the final result of this blowup construction is independent of the choice of full ordering < extending the natural partial ordering ≺ on A_{1}(d, k).

5 Proof of Theorem 1.2

5.1 Pushforwards of Vector Bundles

In this section we prove Theorem 1.2 by lifting the construction of Section 4 from stable maps into ℙ^{n} to stable maps into (the total space of) the line bundle L associated to the locally free sheaf O_{ℙ^{n}}(a).

Let τ : L ⟶ ℙ^{n} be the bundle projection map. We denote by \overline{𝔐}_{1,k}(L, d) the moduli space of degree-d stable maps from genus-one curves with k marked points into L. The projection map τ induces a morphism,

p: \overline{𝔐}_{1,k}(L, d) ⟶ \overline{𝔐}_{1,k}(ℙ^{n}, d), [Σ, u] ⟶ [Σ, τ ∘ u].

Since no fiber of L contains the image of a non-constant holomorphic map, the ghost components of (Σ, τ ∘ u) are precisely the same as the ghost components of (Σ, u). We note that

p^{−1}([Σ, u]) = H^{0}(Σ; u^{∗}L)/Aut(Σ, u).

In particular, p is a bundle of vector spaces, but of two possible ranks: da and da+1. Let S_{L} denote the sheaf of (holomorphic) sections of

p|_{\overline{𝔐}_{1,k}^{0}(ℙ^{n},d)}: \overline{𝔐}_{1,k}(L, d)|_{\overline{𝔐}_{1,k}^{0}(ℙ^{n},d)} ⟶ \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d).

Similarly, denote by S̃_{L} the sheaf of sections of

p: π̃^{∗}\overline{𝔐}_{1,k}(L, d) ⟶ 𝔐̃_{1,k}^{0}(ℙ^{n}, d),

where π̃: 𝔐̃_{1,k}^{0}(ℙ^{n}, d)⟶\overline{𝔐}_{1,k}^{0}(ℙ^{n}, d) is the desingularization map of Theorem 1.1:

\overline{𝔐}_{1,k}(L, d)

\overline{𝔐}_{1,k}(ℙ^{n}, d)

p p

π^{∗}\overline{𝔐}_{1,k}(L, d)

𝔐̃_{1,k}^{0}(ℙ^{n}, d) π̃

π̃

Lemma 5.1 With notation as in Theorem 1.2 and above, (1) the sheaves S_{L} and π_{∗}ev_{∗}O_{ℙ^{n}}(a) over \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d) are isomorphic;

(2) the sheaves S̃_{L} and π_{∗}π̃^{∗}ev_{∗}O_{ℙ^{n}}(a) over 𝔐̃_{1,k}^{0}(ℙ^{n}, d) are isomorphic.

Let 𝔘_{L} be the universal curve over \overline{𝔐}_{1,k}(L, d)|_{\overline{𝔐}_{1,k}^{0}(ℙ^{n},d)}, with structure map π_{L} and evaluation map ev_{L}. The projection map τ induces a morphism p̃ on 𝔘_{L} so that the diagram

𝔘_{L} 𝔘 p̃

L ev_{L} ℙ^{n} ev τ

\overline{𝔐}_{1,k}(L, d)|_{\overline{𝔐}_{1,k}^{0}(ℙ^{n},d)} \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d) p

π_{L} π

commutes. Suppose W ⊂\overline{𝔐}_{1,k}^{0}(ℙ^{n}, d) is an open subset.

(i) An element

s ∈ {π_{∗}ev_{∗}O_{ℙ^{n}}(a)}(W) ≡ H^{0}(π^{−1}(W); ev^{∗}L)

induces a morphism s̃: π^{−1}(W)⟶L so that ev=τ ∘ s̃. In turn, s̃ induces morphisms f_{s} and f̃_{s} to \overline{𝔐}_{1,k}(L, d)|_{\overline{𝔐}_{1,k}^{0}(ℙ^{n},d)} and 𝔘_{L},

π^{−1}(W) 𝔘_{L} f̃_{s}

W \overline{𝔐}_{1,k}(L, d)|_{\overline{𝔐}_{1,k}^{0}(ℙ^{n},d)} f_{s}

π π_{L}

L ev_{L}

so that s̃=ev_{L}∘ f̃_{s}. Then,

ev ∘ p̃ ∘ f̃_{s} = τ ∘ ev_{L}∘ f̃_{s} = τ ∘ s̃ = ev: π^{−1}(W) ⟶ ℙ^{n}  ⟹  p ∘ f_{s} = id_{W} ,

since π ∘ p̃ ∘ f̃_{s} = p ∘ f_{s}∘ π. Thus, f_{s} ∈S_{L}(W). It is immediate that the map

{π_{∗}ev_{∗}O_{ℙ^{n}}(a)}(W) ⟶ S_{L}(W), s ⟶ f_{s},

induces a sheaf homomorphism. (ii) Conversely, let σ ∈S_{L}(W), i.e. σ: W ⟶\overline{𝔐}_{1,k}(L, d) is a morphism such that p ∘ σ = id_{W}. Since 𝔘_{L} =p^{∗}𝔘,

π^{−1}(W) ≡ 𝔘|_{W} = σ^{∗}𝔘_{L}.

Thus, σ lifts to a morphism

σ̃: π^{−1}(W) = σ^{∗}𝔘_{L} ⟶ 𝔘_{L}.

Let g_{σ} =ev_{L}∘ σ̃. Then,

τ ∘ g_{σ} = τ ∘ ev_{L}∘ σ̃ = ev ∘ p̃ ∘ σ̃ = ev,

i.e. g_{σ} ∈H^{0}(π^{−1}(W); ev^{∗}L). It is immediate that the map

S_{L}(W) ⟶ {π_{∗}ev_{∗}O_{ℙ^{n}}(a)}(W), σ ⟶ g_{σ},

induces a sheaf homomorphism. Furthermore,

g_{f_{s}} = s ∀ s ∈ {π_{∗}ev_{∗}O_{ℙ^{n}}(a)}(W)  and  f_{g_{σ}} = σ ∀ σ ∈ S_{L}(W).

These observations imply the first statement of Lemma 5.1. The second claim is proved similarly.

Let

\overline{𝔐}_{1,k}^{0}(L, d) ⊂ \overline{𝔐}_{1,k}(L, d)

be the closure of the locus of maps from smooth domains. We show in Subsection 5.4 that the proper transform 𝔐̃_{1,k}^{0}(L, d) of \overline{𝔐}_{1,k}^{0}(L, d) in

π̃^{∗}\overline{𝔐}_{1,k}(L, d) ⟶ 𝔐̃_{1,k}^{0}(ℙ^{n}, d)

is smooth. Similarly to the case of 𝔐̃_{1,k}^{0}(ℙ^{n}, d), the main stratum of 𝔐̃_{1,k}^{0}(L, d),

𝔐_{1,k}^{eff}(L, d) ≡ \overline{𝔐}_{1,k}(L, d)|_{𝔐_{1,k}^{eff}(ℙ^{n},d)} = 𝔐̃_{1,k}^{0}(L, d) − ⋃_{σ∈A_{1}(d,k)} p^{−1}(Im ι_{σ_{max},σ}),

is smooth. On the other hand, by the inductive assumption (I1) and the last paragraph of Subsection 5.4, for each σ ∈A_{1}(d, k)

𝔐̃_{1,k}^{0}(L, d) ∩ p^{−1}(Im ι_{σ_{max},σ})

is the image of a smooth variety under the bundle homomorphism j̄_{σ_{max},σ} lifting the embedding ῑ_{σ_{max},σ} of Subsection 4.3. Thus,

𝔐̃_{1,k}^{0}(L, d) ∩ p^{−1}(Im ι_{σ_{max},σ})

is a smooth subvariety of 𝔐̃_{1,k}^{0}(L, d). As its normal cone in 𝔐̃_{1,k}^{0}(L, d) is a line bundle by the inductive assumption (I1) of Subsection 5.4 for every σ ∈ A_{1}(d, k), we conclude that the entire space 𝔐̃_{1,k}^{0}(L, d) is smooth. Furthermore, the fibers of

p̃: 𝔐̃_{1,k}^{0}(L, d) ⟶ 𝔐̃_{1,k}^{0}(ℙ^{n}, d)

are vector spaces of the same rank and 𝔐̃_{1,k}^{0}(L, d) contains 𝔐̃_{1,k}^{0}(ℙ^{n}, d) as the zero section. Thus, p̃ is a vector bundle.

Lemma 5.1 and the previous paragraph imply (1) of Theorem 1.2. The second claim of this theorem is obtained in the last paragraph of Subsection 5.4. Finally, (3) of Theorem 1.2 follows from (4) of Theorem 1.1 and the following lemma.

Lemma 5.2 Suppose π̃: 𝔐̃ ⟶ \overline{𝔐} is a morphism between varieties, 𝔘 ⟶ \overline{𝔐} is a flat family of curves, L⟶𝔘 is a line bundle, and π: 𝔘̃⟶𝔐̃ and L̃⟶𝔘̃ are the pullbacks of 𝔘 and L via π̃:

𝔘

L

\overline{𝔐}

π π

𝔘̃=π^{∗}𝔘

L̃=π̃^{∗}L

𝔐̃ π̃

π̃

If the morphism π̃ is surjective and its fibers are compact and connected, then

π̃_{∗}π_{∗}L̃ = π_{∗}L.

Since L is locally trivial, Lemma 5.2 follows from

π̃_{∗}O𝔐̃ = O\overline{𝔐}.

In turn, this identity follows from the fact that every holomorphic function on a compact connected variety is constant. Thus, if W ⊂ \overline{𝔐} is any open subset and f̃ is a holomorphic function on π̃^{−1}(W) ⊂ 𝔐̃, then f̃ is constant on the fibers of π̃, i.e. f̃= π̃^{∗}f for some holomorphic function f on W.

5.2 Construction of Bundle Homomorphism

In this subsection we describe the surjective bundle homomorphism that appears in the second statement of Theorem 1.2; see Proposition 5.5. The construction of this homomorphism is similar to the construction of the homomorphism D̃_{(ℵ,J)} in Subsections 3.3 and 3.4.

Let L⟶ℙ^{n} be a line bundle as in Subsection 1.2. If J is a finite set, let

V_{0} = \overline{𝔐}_{0,\{0\}⊔J}(L, d) ⟶ \overline{𝔐}_{0,\{0\}⊔J}(ℙ^{n}, d)

be the corresponding cone. In particular, if [Σ, u]∈\overline{𝔐}_{0,\{0\}⊔J}(ℙ^{n}, d), then

V_{0}|_{[Σ,u]} = H^{0}(Σ; u^{∗}L)/Aut(Σ, u).

In this, genus-zero, case, this is a vector bundle of the expected rank. Let

∇^{u}: Γ(Σ; u^{∗}L) ⟶ Γ(Σ; T^{∗}Σ⊗u^{∗}L)

be the pullback of the standard Hermitian connection in L by u. We define

𝔇_{0} ∈ Γ(\overline{𝔐}_{0,\{0\}⊔J}(ℙ^{n}, d); Hom(L_{0}⊗V_{0}, ev_{0}^{∗}L))
= Γ(\overline{𝔐}_{0,\{0\}⊔J}(ℙ^{n}, d); Hom(L_{0}, Hom(V_{0}, ev_{0}^{∗}L)))
= Γ(\overline{𝔐}_{0,\{0\}⊔J}(ℙ^{n}, d); Hom(V_{0}, Hom(L_{0}, ev_{0}^{∗}L)))
by 𝔇_{0}ξ = ∇^{u}ξ|_{x_{0}(Σ,u)} ∀ ξ ∈H^{0}(Σ; u^{∗}L),

where x_{0}(Σ, u)∈Σ is the marked point labeled by 0 as before. We note that 𝔇_{0} vanishes identically on the subvarieties \overline{𝔐}_{0,σ}(ℙ^{n}, d) with σ ∈A_{0}(d, J) defined in Subsection 3.2.

If ℵ and J are finite sets, let

p_{(ℵ,J)} : V_{(ℵ,J)} ⟶ \overline{𝔐}_{0,(ℵ,J)}(ℙ^{n}, d)

be the vector bundle induced by L, where \overline{𝔐}_{0,(ℵ,J)}(ℙ^{n}, d) is as in Subsection 3.3. It is immediate that

V_{(ℵ,J)} = \{(ξ_{i})_{i∈ℵ} ∈⨁_{i∈ℵ} π_{i}^{∗}V_{0} : ev_{0}(ξ_{i})=ev_{0}(ξ_{i′}) ∀ i, i′ ∈ℵ\} = \overline{𝔐}_{0,(ℵ,J)}(L, d).

Note that for every σ =(m; J_{P} , J_{B})∈A_{0}(d, J),

ι_{σ}^{∗}V_{0} =π_{B}^{∗}V_{([m],J_{B})} ⟶ \overline{M}_{0,\{0\}⊔[m]⊔J_{B}} ×\overline{𝔐}_{0,([m],J)}(ℙ^{n}, d),

where ι_{σ} is as in Subsection 3.2.

Lemma 5.3 If d∈ℤ^{+}, J, L, and V_{0} are as above, the bundle homomorphism

𝔇_{0} ∈ Γ(\overline{𝔐}_{0,\{0\}⊔J}(ℙ^{n}, d); Hom(V_{0}, L_{0}^{∗}⊗ev_{0}^{∗}L))

is surjective on the complement of the subvarieties \overline{𝔐}_{0,σ}(ℙ^{n}, d) with σ ∈A_{0}(d, J). Furthermore, for every

σ ≡(m; J_{P} , J_{B}) ∈ A_{0}(d, J),

the differential of 𝔇_{0},

∇𝔇_{0} : N_{ι_{σ}} ⟶ ι_{σ}^{∗} Hom(V_{0}, L_{0}^{∗}⊗ev_{0}^{∗}L) = π_{P}^{∗} L_{0}^{∗}⊗π_{B}^{∗}Hom(V_{([m],J_{B})}, ev_{0}^{∗}L),

in the normal direction to the immersion ι_{σ} is given by

∇𝔇_{0}|_{π_{P}^{∗} L_{i}⊗π_{B}^{∗}π_{i}^{∗}L_{0}} = π_{P}^{∗} s_{i}⊗π_{B}^{∗}π_{i}^{∗}𝔇_{0} ∀ i ∈ [m],

where s_{i} is the homomorphism defined in Subsection 2.2.

Lemma 5.3 can viewed as the analogue of Lemma 3.8 for vector bundle sections. The first claim of Lemma 5.3 is an immediate consequence of the fact that

H^{1}(Σ; u^{∗}L⊗O(−2z)) = {0}

for every genus-zero stable map (Σ, u) and a smooth point z ∈ Σ such that the restriction of u to the irreducible component of Σ containing z is not constant. The second statement follows from [Z5, Lemma 4.2]. With notation as in Subsection 3.3, let

𝔇_{(ℵ,J)} ∈ Γ(\overline{𝔐}_{0,(ℵ,J)}(ℙ^{n}, d); Hom(V_{(ℵ,J)}, Hom(F_{(ℵ,J)}, ev_{0}^{∗}L)))
= Γ(\overline{𝔐}_{0,(ℵ,J)}(ℙ^{n}, d); Hom(F_{(ℵ,J)}, Hom(V_{(ℵ,J)}, ev_{0}^{∗}L)))
= Γ(\overline{𝔐}_{0,(ℵ,J)}(ℙ^{n}, d); Hom(F_{(ℵ,J)}⊗V_{(ℵ,J)}, ev_{0}^{∗}L))

be the homomorphism defined by

𝔇_{(ℵ,J)}|_{π_{i}^{∗}L_{0}⊗π_{j}^{∗}V_{0}} = \begin{cases}π_{i}^{∗}𝔇_{0}, & if j =i; \\ 0, & otherwise; \end{cases} ∀ i, j ∈ℵ.

It induces a section

𝔇̃_{0} ∈Γ(𝔐̃_{0,(ℵ,J)}^{0}(ℙ^{n}, d); Hom(γ_{(ℵ,J)}, π_{ℙF_{(ℵ,J)}}^{∗}Hom(V_{(ℵ,J)}, ev_{0}^{∗}L)))
= Γ(𝔐̃_{0,(ℵ,J)}^{0}(ℙ^{n}, d); Hom(π_{ℙF_{(ℵ,J)}}^{∗}V_{(ℵ,J)}, 𝔼_{0}^{∗}⊗π_{F_{(ℵ,J)}}^{∗}ev_{0}^{∗}L)).

This section vanishes identically on the subvarieties 𝔐̃_{0,ϱ}^{0}(ℙ^{n}, d) of 𝔐̃_{0,(ℵ,J)}^{0}(ℙ^{n}, d) with ϱ∈A_{0}(ℵ; d, J), defined in Subsection 3.3. Lemma 5.4 The bundle homomorphism

𝔇̃_{0} ∈ Γ(𝔐̃_{0,(ℵ,J)}^{0}(ℙ^{n}, d); Hom(π_{ℙF_{(ℵ,J)}}^{∗}V_{(ℵ,J)}, 𝔼_{0}^{∗}⊗π_{F_{(ℵ,J)}}^{∗}ev_{0}^{∗}L))

is surjective on the complement of the subvarieties 𝔐̃_{0,ϱ^{∗}}^{0}(ℙ^{n}, d) with ϱ^{∗} ∈A_{0}(ℵ; d, J). Furthermore, for every ϱ^{∗} ∈A_{0}(ℵ; d, J) as in Lemma 3.10, the differential of 𝔇̃_{0},

∇𝔇̃_{0} : N_{ι_{0,ϱ^{∗}}} ⟶ι_{0,ϱ^{∗}}^{∗}Hom(π_{ℙF_{(ℵ,J)}}^{∗}V_{(ℵ,J)}, 𝔼_{0}^{∗}⊗π_{F_{(ℵ,J)}}^{∗}ev_{0}^{∗}L)
= π_{P}^{∗} 𝔼_{0}^{∗}⊗π_{B}^{∗}Hom(V_{(ℵ_{B}(ϱ^{∗}),J_{B}(ϱ^{∗}))}, ev_{0}^{∗}L),

in the normal direction to the immersion ι_{0,ϱ^{∗}} is given by

∇𝔇̃_{0}|_{π_{P}^{∗} L_{0,(l,i)}⊗π_{B}^{∗}π_{(l,i)}^{∗}L_{0}} = π_{P}^{∗} s_{0,(l,i)}⊗π_{B}^{∗}π_{(l,i)}^{∗}𝔇_{0} ∀ i∈[m_{l}^{∗}], l ∈ℵ_{P} (ϱ^{∗}),

and ∇𝔇̃_{0}|_{N_{ι_{0,ϱ^{∗}}}^{⊤}} = π_{P}^{∗} id⊗π_{B}^{∗}𝔇_{(ℵ_{B}(ϱ^{∗}),J_{B}(ϱ^{∗}))},

where s_{0,(l,i)} is the homomorphism defined in Subsection 2.4. This lemma follows immediately from Lemma 5.3. Proposition 5.5 With notation as above, there exists a surjective bundle homomorphism

𝔇̃_{(ℵ,J)} ∈ Γ(𝔐̃_{0,(ℵ,J)}(ℙ^{n}, d); Hom(π_{0,(ℵ,J)}^{∗}π_{ℙF_{(ℵ,J)}}^{∗}V_{(ℵ,J)}, 𝔼̃^{∗}⊗π_{0,(ℵ,J)}^{∗}π_{ℙF_{(ℵ,J)}}^{∗}ev_{0}^{∗}L))

such that

𝔇̃_{(ℵ,J)}|_{ℙF_{(ℵ,J)}^{0}} = 𝔇̃_{0}|_{ℙF_{(ℵ,J)}^{0}}, where

ℙF_{(ℵ,J)}^{0} = ℙF_{(ℵ,J)} − ⋃_{ϱ∈A_{0}(ℵ;d,J)} 𝔐̃_{0,ϱ}^{0}(ℙ^{n}, d) ⊂ 𝔐̃_{0,(ℵ,J)}^{0}(ℙ^{n}, d), 𝔐̃_{0,(ℵ,J)}(ℙ^{n}, d).

In fact, in the notation of Subsection 3.4, for every ϱ ∈ {0} ⊔ A_{0}(ℵ; d, J) there exists a bundle homomorphism

𝔇̃_{ϱ} ∈ Γ(𝔐̃_{0,(ℵ,J)}^{ϱ}; Hom(π_{ϱ}^{∗}π_{ℙF_{(ℵ,J)}}^{∗}V_{(ℵ,J)}, 𝔼_{ϱ}^{∗}⊗π_{ϱ}^{∗}π_{ℙF_{(ℵ,J)}}^{∗}ev_{0}^{∗}L))

such that (i) the restrictions of 𝔇̃_{ϱ} and 𝔇̃_{0} to ℙF_{(ℵ,J)}^{0} agree;

(ii) 𝔇̃_{ϱ} is surjective outside of the subvarieties 𝔐̃_{0,ϱ^{∗}}^{ϱ} with ϱ^{∗} >ϱ;

(iii) 𝔇̃_{ϱ} vanishes identically on the subvarieties 𝔐̃_{0,ϱ^{∗}}^{ϱ} with ϱ^{∗} >ϱ;

(iv) for each ϱ^{∗} >ϱ, the differential of 𝔇̃_{ϱ} in the normal direction to the immersion ι_{ϱ,ϱ^{∗}}

is given as in the statement of Lemma 5.4, but with s_{0,(l,i)} replaced by s_{ρ_{ϱ^{∗}}(ϱ),(l,i)}. Similarly to the construction of the bundle sections D̃_{ϱ} in Subsection 3.4, we construct the bundle homomorphisms 𝔇̃_{ϱ} inductively starting with 𝔇̃_{0} and twisting by the exceptional divisor at each step. The inductive assumptions (i)-(iv) are analogous to (I3), (I4), and (I12) in Subsection 3.4 and are verified similarly. Of course, we take

𝔇̃_{(ℵ,J)} = 𝔇̃_{ϱ_{max}}.

5.3 Structure of the Cone V_{1,k}^{d}

In this subsection we describe the structure of the cone

p_{0} : \overline{𝔐}_{1,k}(L, d) ⟶ \overline{𝔐}_{1,k}(ℙ^{n}, d),

restating the primary structural result of [Z5].

For each element σ =(m; J_{P} , J_{B}) of A_{1}(d, k), let

V_{1,σ}^{0} ≡\overline{𝔐}_{1,σ}(L, d) = p_{0}^{−1}(\overline{𝔐}_{1,σ}^{0}) ⊂ V_{1,k}^{0} ≡\overline{𝔐}_{1,k}(L, d),

with \overline{𝔐}_{1,σ}^{0} as on page 50. The subvarieties \overline{𝔐}_{1,σ}(L, d) of \overline{𝔐}_{1,k}(L, d) can also be defined analogously to the subvarieties \overline{𝔐}_{1,σ}(ℙ^{n}, d) of \overline{𝔐}_{1,k}(ℙ^{n}, d); see the beginning of Subsection 1.2. Similarly to Subsection 4.2, let

j_{0,σ}: \overline{M}_{1,(I_{P} (σ),J_{P} (σ))}^{0} × V_{(ℵ_{B}(σ),J_{B}(σ))} ⟶ V_{1,σ}^{0} ⊂ V_{1,k}^{0}

be the natural node-identifying immersion so that the diagram

\overline{M}_{1,(I_{P} (σ),J_{P} (σ))}^{0} × V_{(ℵ_{B}(σ),J_{B}(σ))} V_{1,σ}^{0} ⊂ V_{1,k}^{0}

\overline{M}_{1,(I_{P} (σ),J_{P} (σ))}^{0} × \overline{𝔐}_{0,(ℵ_{B}(σ),J_{B}(σ))} \overline{𝔐}_{1,σ}^{0} ⊂ \overline{𝔐}_{1,k}^{0}

j_{0,σ}

ι_{0,σ}

id p_{0,σ} p_{0}

commutes.

Lemma 5.6 If d, n ∈ ℤ^{+} and k ∈ ℤ̄^{+}, the collection {j_{0,σ}}_{σ∈A_{1}(d,k)} of immersions is properly self-intersecting. For every σ ∈A_{1}(d, k),

N_{j_{0,σ}}^{ide} = {id×p_{0,σ}}^{∗}N_{ι_{0,σ}}^{ide}

is an idealized normal bundle for j_{0,σ}.

The differential dp_{0} of p_{0} induces a surjective linear map

Im dj_{0,σ} ⟶ Im dι_{0,σ}.

Since the fibers of p_{0} are vector spaces, it follows that dp_{0} induces an injection

j_{0,σ}^{∗}T CV_{1,k}^{0}/Im dj_{0,σ} ⟶ ι_{0,σ}^{∗}T C\overline{𝔐}_{1,k}^{0}/Im dι_{0,σ}.

Thus, Lemma 5.6 follows from Lemma 4.7.

We denote by V_{1,(0)}^{0} the main component \overline{𝔐}_{1,k}^{0}(L, d) of the moduli space \overline{𝔐}_{1,k}(L, d). If σ ∈A_{1}(d, k), we put

W_{σ}^{0} = j_{0,σ}^{−1}(V_{1,(0)}^{0}) ≡ j_{0,σ}^{−1}(V_{1,(0)}^{0}∩ V_{1,σ}^{0}).

Note that

\{id×p_{0,σ}\}(W_{σ}^{0}) = Z̄_{σ}^{0} ≡ ι_{0,σ}^{−1}(\overline{𝔐}_{1,(0)}^{0}).

Let N W_{σ}^{0} ⊂ N_{j_{0,σ}}^{ide} be the normal cone N_{j_{0,σ}|V_{1,(0)}^{0}} for j_{0,σ}|_{W_{σ}^{0}} in V_{1,(0)}^{0}. Its structure is described in Lemma 5.7 below. Let

𝔇_{0,σ} ∈ Γ(\overline{M}_{1,(I_{P} (σ),J_{P} (σ))}^{0}×\overline{𝔐}_{0,(ℵ_{B}(σ),J_{B}(σ))}; Hom(π_{B}^{∗}V_{(ℵ_{B}(σ),J_{B}(σ))}, Hom(N_{ι_{0,σ}}^{ide}, π_{P}^{∗} 𝔼_{0}^{∗}⊗π_{B}^{∗}ev_{0}^{∗}L)))

= Γ(\overline{M}_{1,(I_{P} (σ),J_{P} (σ))}^{0}×\overline{𝔐}_{0,(ℵ_{B}(σ),J_{B}(σ))}; Hom(N_{ι_{0,σ}}^{ide}, π_{P}^{∗} 𝔼_{0}^{∗}⊗π_{B}^{∗}Hom(V_{(ℵ_{B}(σ),J_{B}(σ))}, ev_{0}^{∗}L)))

be the section defined by

𝔇_{0,σ}|_{π_{P}^{∗} L_{i}⊗π_{B}^{∗}π_{i}^{∗}L_{0}} = π_{P}^{∗} s_{i}⊗π_{B}^{∗}π_{i}^{∗}𝔇_{0}, ∀ i∈[m],

where s_{i} and 𝔇_{0} are as in Subsections 2.2 and 5.2, respectively. If ξ ∈π_{B}^{∗}V_{(ℵ_{B}(σ),J_{B}(σ))}, we will view 𝔇_{0,σ}ξ as a homomorphism

𝔇_{0,σ}ξ: N_{j_{0,σ}}^{ide} |_{ξ} =N_{ι_{0,σ}}^{ide} |_{{id×p_{0,σ}}(ξ)} ⟶ π_{P}^{∗} 𝔼_{0}^{∗}⊗π_{B}^{∗}ev_{0}^{∗}L|_{{id×p_{0,σ}}(ξ)}.

Lemma 5.7 For all σ ∈A_{1}(d, k), V_{1,(0)}^{0} is a proper subvariety of V_{1,k}^{0} relative to the immersion j_{0,σ}. The homomorphism

N W_{σ}^{0} ⟶ {id×p_{0,σ}}^{∗}N Z̄_{σ}^{0}

induced by dp_{0} is injective. Furthermore,

W_{σ}^{0}|_{Z_{σ}^{0}} = {ξ ∈π_{B}^{∗}V_{(ℵ_{B}(σ),J_{B}(σ))}|_{Z_{σ}^{0}} : ker {𝔇_{0,σ}ξ}|_{N Z̄_{σ}^{0}|_{{id×p_{0,σ}}(ξ)}} ≠{0}}
and  N W_{σ}^{0}|_{ξ} = ker {𝔇_{0,σ}ξ}|_{N Z̄_{σ}^{0}|_{{id×p_{0,σ}}(ξ)}} ⊂ N_{j_{0,σ}}^{ide}  ∀ ξ ∈W_{σ}^{0}|_{Z_{σ}^{0}}.

Finally, W_{σ}^{0} is the closure of W_{σ}^{0}|_{Z_{σ}^{0}} in \overline{M}_{1,(I_{P} (σ),J_{P} (σ))}^{0}×V_{(ℵ_{B}(σ),J_{B}(σ))} and N W_{σ}^{0} is the closure of N W_{σ}^{0}|_{W_{σ}^{0}|_{Z^{0}_{σ}}} in N_{j_{0,σ}}^{ide}.

Since the fibers of p_{0} are vector spaces, the first two sentences of this lemma follow from Lemma 4.10. The middle claim of Lemma 5.7 is a restatement of [Z5, Lemma 3.4]. The remaining claims of the lemma follow from [Z5, Lemma 3.4] by dimension counting, similarly to the argument following Lemma 4.10.

Remark: It may appear that the statement of Lemma 5.7 depends on the choice of a hermitian connection (or metric) in the line bundle L⟶ℙ^{n}. As explained in detail in [Z5, Subsect. 3.3], the dependence is only on the holomorphic structure of L, as the case should be.

5.4 Desingularization Construction

In this subsection we lift the inductive blowup construction of Subsection 4.3 to the cone

p_{0} : V_{1,k}^{0} ⟶ \overline{𝔐}_{1,k}^{0}.

For each σ ∈A_{1}(d, k), let V_{1,k}^{σ} ≡π_{σ}^{∗}V_{1,k}^{0} be the pullback of V_{1,k}^{0} to \overline{𝔐}_{1,k}^{σ}:

V_{1,k}^{σ} ≡π_{σ}^{∗}V_{1,k}^{0} V_{1,k}^{0}

\overline{𝔐}_{1,k}^{σ} \overline{𝔐}_{1,k}^{0}

π_{σ}

π_{σ}

p_{σ} p_{0}

For each σ′ ∈A_{1}(d, k), let

V_{1,σ′}^{σ} = V_{1,k}^{σ}|\overline{𝔐}_{1,σ′}^{σ} = π_{σ}^{−1}(V_{1,k}^{0}|\overline{𝔐}{0 \atop 1,σ′} ).

The bundle homomorphisms j_{0,σ′} lift to bundle homomorphisms onto V_{1,σ′}^{σ} covering the immersion ι_{σ,σ′} of Subsection 4.3:

M̃_{1,(I_{P} (σ′),J_{P} (σ′))} × π_{ϱ_{σ′}(σ)}^{∗}π_{ℙF_{(ℵ,J)}}^{∗}V_{(ℵ_{B}(σ′),J_{B}(σ′))} V_{1,σ′}^{σ} ⊂ V_{1,k}^{σ}

M̃_{1,(I_{P} (σ′),J_{P} (σ′))} × 𝔐̃_{0,(ℵ_{B}(σ′),J_{B}(σ′))}^{ϱ_{σ′}(σ)} \overline{𝔐}_{1,σ′}^{σ} ⊂ \overline{𝔐}_{1,k}^{σ}

j_{σ,σ′}

ι_{σ,σ′}

id p_{σ,σ′} p_{σ} σ′ ≤σ:

\overline{M}_{1,(I_{P} (σ′),J_{P} (σ′))}^{ρ_{σ′}(σ)} × V_{(ℵ_{B}(σ′),J_{B}(σ′))} V_{1,σ′}^{σ} ⊂ V_{1,k}^{σ}

\overline{M}_{1,(I_{P} (σ′),J_{P} (σ′))}^{ρ_{σ′}(σ)} × \overline{𝔐}_{0,(ℵ_{B}(σ′),J_{B}(σ′))} \overline{𝔐}_{1,σ′}^{σ} ⊂ \overline{𝔐}_{1,k}^{σ}

j_{σ,σ′}

ι_{σ,σ′}

id p_{σ,σ′} p_{σ} σ′ >σ:

The collection {ι_{σ,σ′}}_{σ′∈A_{1}(d,k)} of immersions is properly self-intersecting by the inductive assumption (I14) of Subsection 4.3. Thus, by the same argument as in the paragraph following Lemma 5.6, so is the collection {j_{σ,σ′}}_{σ′∈A_{1}(d,k)}. Furthermore,

N_{j_{σ,σ′}}^{ide} = {id×p_{σ,σ′}}^{∗}N_{ι_{σ,σ′}}^{ide}  (5.1)

is an idealized normal bundle for j_{σ,σ′}. These two observations also follow from Lemma 5.6 by induction using Lemmas 4.2 and 4.3.

Lemma 5.8 If σ ∈A_{1}(d, k), V_{1,σ}^{σ−1} is a smooth subvariety of V_{1,k}^{σ−1} and

p_{σ}: V_{1,k}^{σ} ⟶ \overline{𝔐}_{1,k}^{σ}

is the idealized blowup of V_{1,k}^{σ−1} along V_{1,σ}^{σ−1}.

Recall from Subsection 4.3 that the immersion

ῑ_{σ−1,σ}: (M̃_{1,(I_{P} (σ),J_{P} (σ))}×\overline{𝔐}_{0,(ℵ_{B}(σ),J_{B}(σ))})/G_{σ} ⟶ \overline{𝔐}_{1,σ}^{σ−1} ⊂ \overline{𝔐}_{1,k}^{σ−1}

induced by ι_{σ−1,σ} is an embedding and

π̃_{σ}: \overline{𝔐}_{1,k}^{σ} ⟶ \overline{𝔐}_{1,k}^{σ−1}

is the idealized blowup along \overline{𝔐}_{1,σ}^{σ−1}. Thus, the immersion

j̄_{σ−1,σ}: (M̃_{1,(I_{P} (σ),J_{P} (σ))}×V_{(ℵ_{B}(σ),J_{B}(σ))})/G_{σ} ⟶ V_{1,σ}^{σ−1} ⊂ V_{1,k}^{σ−1}

induced by j_{σ−1,σ} is also an embedding and V_{1,σ}^{σ−1} is a smooth subvariety of V_{1,k}^{σ−1}. Let

π̃_{σ}: V ⟶ V_{1,k}^{σ−1}

be the idealized blowup along V_{1,k}^{σ−1}. Since

N_{j_{σ−1,σ}}^{ide} = {id×p_{σ−1,σ}}^{∗}N_{ι_{σ−1,σ}}^{ide}

and the linear map

j_{σ−1,σ}^{∗}T CV_{1,k}^{σ−1}/Im dj_{σ−1,σ} ⟶ ι_{σ−1,σ}^{∗}T C\overline{𝔐}_{1,k}^{σ−1}/Im dι_{σ−1,σ}

induced by dp_{σ−1} is injective, p_{σ−1} lifts to a map p over the blowdown maps π̃_{σ}:

V V^{σ−1}_{1,k}

𝔐^σ_{1,k} 𝔐^{σ−1}_{1,k}

π̃_σ

π̃_σ

p p_{σ−1}

Then p and the top arrow π̃_{σ} factor through a morphism f to π̃_{σ}^{∗}V_{1,k}^{σ−1}:

V

p

π̃_{σ} f

π̃_{σ}^{∗}V_{1,k}^{σ−1} V_{1,k}^{σ−1}

\overline{𝔐}_{1,k}^{σ} \overline{𝔐}_{1,k}^{σ−1}

π̃_{σ}
π̃_{σ}

p_{σ} p_{σ−1}

We show in the next paragraph that f is an isomorphism. Since π̃_{σ}^{∗}V_{1,k}^{σ−1} = V_{1,k}^{σ}, this implies the second statement of Lemma 5.8. By construction, the maps

π̃_{σ}: \overline{𝔐}_{1,k}^{σ} ⟶ \overline{𝔐}_{1,k}^{σ−1} and π̃_{σ}: V ⟶ V_{1,k}^{σ−1}

are isomorphisms on the complements of the idealized exceptional divisors

\overline{𝔐}_{1,σ}^{σ} ≡ E_{\overline{𝔐}_{1,σ}^{σ−1}}^{ide} ⊂ \overline{𝔐}_{1,k}^{σ}  and  V_{1,σ}^{σ} ≡ E_{V_{1,σ}^{σ−1}}^{ide} ⊂ V.

Thus, f: V ⟶π̃_{σ}^{∗}V_{1,k} is an isomorphism over the complement of \overline{𝔐}_{1,σ}^{σ} in \overline{𝔐}_{1,k}^{σ}. In particular, f is linear on all fibers of p. Furthermore,

π̃_{σ}^{∗}V_{1,k}^{σ−1}|\overline{𝔐}_{1,σ}^{σ} = \{(ℓ, v)∈ℙN_{\overline{𝔐}_{1,σ}^{σ−1}}^{ide} ×V_{1,k}^{σ−1} : π̃_{σ}(ℓ)=p_{σ−1}(v)\}.

On the other hand, since

N_{V_{1,σ}^{σ−1}}^{ide} = p_{σ−1}^{∗}N_{\overline{𝔐}_{1,σ}^{σ−1}}^{ide}

by (5.1), we have

V_{1,σ}^{σ} = p_{σ−1}^{∗}ℙN_{\overline{𝔐}_{1,σ}^{σ−1}}^{ide} = \{(v, ℓ)∈V_{1,k}^{σ−1}×ℙN_{\overline{𝔐}_{1,σ}^{σ−1}}^{ide} : p_{σ−1}(v)=π̃_{σ}(ℓ)\}.

Thus, the restriction of f to V_{1,σ}^{σ} must interchange v and ℓ, i.e. it is a vector bundle isomorphism over \overline{𝔐}_{1,k}^{σ}. Finally, \overline{𝔐}_{1,σ}^{σ−1} is a smooth subvariety of V_{1,k}^{σ−1} and

T (\overline{𝔐}_{1,k}^{σ−1}∩V_{1,σ}^{σ−1}) = T \overline{𝔐}_{1,σ}^{σ−1} = T C\overline{𝔐}_{1,k}^{σ−1} ∩ T V_{1,σ}^{σ−1} ⊂ T CV_{1,k}^{σ−1}.

Thus, similarly to (1) of Lemma 3.3, the proper transform of \overline{𝔐}_{1,k}^{σ−1} in V is the blowup of \overline{𝔐}_{1,k}^{σ−1} along

\overline{𝔐}_{1,k}^{σ−1}∩V_{1,σ}^{σ−1} = \overline{𝔐}_{1,σ}^{σ−1},

i.e. V contains \overline{𝔐}_{1,k}^{σ} as the zero section. The map f must be the identity on \overline{𝔐}_{1,k}^{σ}. Since f is a linear isomorphism on all fibers of p by the above, it then follows that f is an isomorphism everywhere.

Remark: If V_{1,k}^{σ−1} is a vector bundle over \overline{𝔐}_{1,k}^{σ−1}, the second statement of Lemma 5.8 applies to

standard blowups of \overline{𝔐}_{1,k}^{σ−1} and V_{1,k}^{σ−1} as well. However, the second statement does not generally apply to standard blowups in the setting of Lemma 5.8, as the analogue of the morphism f may not be surjective.

By the inductive assumption (I1) of Subsection 4.3, the projection map π_{σ} is an isomorphism outside of the subvarieties V_{1,σ′}^{σ} with σ′ ≤σ. We denote by

V_{1,(0)}^{σ} ⊂ V_{1,k}^{σ}

the proper transform of V_{1,(0)}^{0}. For each σ′ ∈A_{1}(d, k), let

W_{σ′}^{σ} = j_{σ,σ′}^{−1}(V_{1,(0)}^{σ}) = j_{σ,σ′}^{−1}(V_{1,(0)}^{σ}∩V_{1,σ′}^{σ}).

By the inductive assumption (I15) of Subsection 4.3, \overline{𝔐}_{1,(0)}^{σ} is a proper subvariety of \overline{𝔐}_{1,k}^{σ} with respect to the immersion ι_{σ,σ′}. Thus, by the same argument as in the paragraph following Lemma 5.7, the subvariety V_{1,(0)}^{σ} of V_{1,k}^{σ} is proper with respect to the immersion j_{σ,σ′}. Furthermore, if

N W_{σ′}^{σ} ≡ N W_{j_{σ,σ′}|V_{1,(0)}^{σ}} ⊂ N_{j_{σ′,σ}}^{ide}

denotes the normal cone for j_{σ′,σ}|_{W_{σ′}^{σ}} in V_{1,(0)}^{σ}, then the homomorphism

N W_{σ′}^{σ} ⟶ {id×p_{σ,σ′}}^{∗}N Z̄_{σ′}^{σ}

induced by dp_{σ} is injective. These two observations also follow from Lemma 5.7 by induction using Lemma 4.5.

If σ′ ∈A_{1}(d, k), let

Z̃_{σ′;B}^{0} = D̃_{0}^{−1}(0) ∩ ℙF_{(I_{P} (σ′),J_{P} (σ′))}|_{𝔐_{0,(ℵ_{B} (σ),J_{B} (σ))}}.

By the inductive assumptions (I7) in Subsection 4.3 and (I4) in Subsection 3.4,

\overline{Z}_{σ′}^{σ} ≡ ι_{σ,σ′}^{−1}(\overline{𝔐}_{1,(0)}^{σ})

is the closure of M̃_{1,(I_{P} (σ′),J_{P} (σ′))}×Z̃_{σ′;B}^{0} in

M̃_{1,(I_{P} (σ′),J_{P} (σ′))} × 𝔐̃_{0,(ℵ_{B}(σ′),J_{B}(σ′))}^{ϱ_{σ′}(σ)}

for all σ ∈A_{1}(k, d) such that σ′ ≤σ.

Suppose σ ∈{0}∪A_{1}(d, k) and σ′ ∈A_{1}(d, k). We claim that (I1) if σ′ ≤σ, then W_{σ′}^{σ} is the closure of

\begin{aligned}M̃_{1,(I_{P} (σ′),J_{P} (σ′))} × ker 𝔇̃_{0}|Z̃_{σ′;B}^{0} & ⊂ M̃_{1,(I_{P} (σ′),J_{P} (σ′))} × π_{ℙF_{(ℵ,J)}}^{∗}V_{(ℵ_{B}(σ′),J_{B}(σ′))}, \\ & M̃_{1,(I_{P} (σ′),J_{P} (σ′))} × π_{ϱ_{σ′}(σ)}^{∗}π_{ℙF_{(ℵ,J)}}^{∗}V_{(ℵ_{B}(σ′),J_{B}(σ′))} \end{aligned}

in M̃_{1,(I_{P} (σ′),J_{P} (σ′))} × π_{ϱ_{σ′}(σ)}^{∗}π_{ℙF_{(ℵ,J)}}^{∗}V_{(ℵ_{B}(σ′),J_{B}(σ′))} and

N W_{σ′}^{σ} = N_{j_{σ,σ′}}^{ide} |_{W_{σ′}^{σ}};

(I2) if σ′ >σ, then W_{σ′}^{σ} and N W_{σ′}^{σ} are the closures of

W_{σ′}^{0}|_{Z_{σ′}^{0}} ⊂ \overline{M}_{1,(I_{P} (σ′),J_{P} (σ′))}^{0}×V_{(ℵ_{B}(σ′),J_{B}(σ′))}, \overline{M}_{1,(I_{P} (σ′),J_{P} (σ′))}^{ρ_{σ′}(σ)}×V_{(ℵ_{B}(σ′),J_{B}(σ′))}

and N W_{σ′}^{0}|_{W_{σ′}^{0}|_{Z{0 \atop σ′}}} ⊂ N_{j_{0,σ′}}^{ide} |_{W_{σ′}^{0}|_{Z{0 \atop σ′}}} ⊂ N_{j_{0,σ′}}^{ide} , N_{j_{σ,σ′}}^{ide}

in \overline{M}_{1,(I_{P} (σ′),J_{P} (σ′))}^{ρ_{σ′}(σ)}×V_{(ℵ_{B}(σ′),J_{B}(σ′))} and in N_{j_{σ,σ′}}^{ide} , respectively.

If σ = 0, the assumption (I1) is trivially satisfied, while (I2) constitutes part of Lemma 5.7. Suppose σ ∈ A_{1}(d, k) and the two assumptions hold with σ replaced by σ−1. By Lemma 5.8, V_{1,k}^{σ} is the idealized blowup of V_{1,k}^{σ−1} along V_{1,σ}^{σ−1}. Thus, by the last statement of Lemma 4.5 both of the inductive assumptions continue to hold for σ′ ≠σ.

On the other hand, let

Z_{σ;B} = {b∈𝔐_{0,(ℵ_{B}(σ),J_{B}(σ))} : ker D_{(ℵ_{B}(σ),J_{B}(σ))} ≠0},
W_{σ;B}^{0} = {ξ ∈V_{(ℵ_{B}(σ),J_{B}(σ))}|_{Z_{σ;B}}: ker {𝔇_{(ℵ_{B}(σ),J_{B}(σ))}ξ}|_{ker D_{(ℵ_{B} (σ),J_{B} (σ))}|_{p_{σ−1,σ}(ξ)}} ≠{0}},  and
N W_{σ;B}^{0} = {(ξ, υ): ξ ∈W_{σ;B}^{0}, υ ∈ker{𝔇_{0,σ}ξ}|_{ker D_{(ℵ_{B} (σ),J_{B} (σ))}|_{p_{σ−1,σ}(ξ)}}} ⊂ p_{σ−1,σ}^{∗}F_{(ℵ_{B}(σ),J_{B}(σ))}.

By the inductive assumption (I12) in Subsection 4.3,

\overline{Z}_{σ}^{σ−1} ∩ (M̃_{1,(I_{P} (σ),J_{P} (σ))}×𝔐_{0,(ℵ_{B}(σ),J_{B}(σ))}) = M̃_{1,(I_{P} (σ),J_{P} (σ))}×Z_{σ;B} and
N Z̄_{σ}^{0}|_{M̃_{1,(I_{P} (σ),J_{P} (σ))}×Z_{σ;B}} = π_{P}^{∗} 𝕃 ⊗ π_{B}^{∗} ker D_{(ℵ_{B}(σ),J_{B}(σ))}.

By the inductive assumption (I2) above, Lemma 5.7, and the inductive assumption (I11) in Subsection 4.3, W_{σ}^{σ−1} and N W_{σ}^{σ−1} are the closures of

W_{σ}^{0}|_{Z_{σ}^{0}} = {ξ ∈π_{B}^{∗}V_{(ℵ_{B}(σ),J_{B}(σ))}|_{Z_{σ}^{0}} : ker {𝔇_{0,σ}ξ}|_{N Z̄_{σ}^{0}|_{{id×p_{0,σ}}(ξ)}} ≠{0}}
and N W_{σ}^{0}|_{W_{σ}^{0}|_{Z^{0}_{σ}}} = {(ξ, υ): ξ ∈W_{σ}^{0}|_{Z_{σ}^{0}}, υ ∈ker{𝔇_{0,σ}ξ}|_{N Z̄_{σ}^{0}|_{{id×p_{0,σ}}(ξ)}}} ⊂ N_{j_{0,σ}}^{ide}, N_{j_{σ−1,σ}}^{ide}

in M̃_{1,(I_{P} (σ),J_{P} (σ))}×V_{(ℵ_{B}(σ),J_{B}(σ))} and in

N_{j_{σ−1,σ}}^{ide} = π_{P}^{∗} 𝕃 ⊗ π_{B}^{∗}p_{σ−1,σ}^{∗}F_{(ℵ_{B}(σ),J_{B}(σ))}.

As before,

π_{P} , π_{B}: M̃_{1,(I_{P} (σ),J_{P} (σ))}×V_{(ℵ_{B}(σ),J_{B}(σ))} ⟶ M̃_{1,(I_{P} (σ),J_{P} (σ))}, V_{(ℵ_{B}(σ),J_{B}(σ))}

are the projections onto the principle and bubble components. The bundle homomorphisms s_{i} and s̃_{i} of Subsection 2.3 agree on

M_{1,(I_{P} (σ),J_{P} (σ))} ⊂ \overline{M}_{1,(I_{P} (σ),J_{P} (σ))}, M̃_{1,(I_{P} (σ),J_{P} (σ))}.

The homomorphism s̃_{i} is an isomorphism from L̃_{i} to 𝔼̃^{∗} over M̃_{1,(I_{P} (σ),J_{P} (σ))}, and both line bundles are isomorphic to 𝕃. It follows that W_{σ}^{σ−1} and N W_{σ}^{σ−1} are the closures of

M̃_{1,(I_{P} (σ),J_{P} (σ))} × W_{σ;B}^{0}  and  π_{P}^{∗} 𝕃 ⊗ π_{B}^{∗}N W_{σ;B}^{0}

in M̃_{1,(I_{P} (σ),J_{P} (σ))}×V_{(ℵ_{B}(σ),J_{B}(σ))} and in

N_{j_{σ−1,σ}}^{ide} = π_{P}^{∗} 𝕃 ⊗ π_{B}^{∗}p_{σ−1,σ}^{∗}F_{(ℵ_{B}(σ),J_{B}(σ))}.

Thus, by the first statement of Lemma 4.5,

W_{σ}^{σ} ≡ j_{σ,σ}^{−1}(V_{1,(0)}^{σ})

is the closure of

\begin{aligned}ℙ(π_{P}^{∗} 𝕃⊗π_{B}^{∗}N W_{σ;B}^{0}) & = M̃_{1,(I_{P} (σ),J_{P} (σ))} × ℙN W_{σ;B}^{0} = M̃_{1,(I_{P} (σ),J_{P} (σ))} × ker 𝔇̃_{0}|Z̃_{σ;B}^{0} \\ & ⊂ ℙN_{j_{σ−1,σ}}^{ide} = M̃_{1,(I_{P} (σ),J_{P} (σ))}×π_{ℙF_{(ℵ,J)}}^{∗}V_{(ℵ_{B}(σ),J_{B}(σ))}, \end{aligned}

i.e. the first part of the inductive assumption (I1) for σ′ =σ is satisfied. Furthermore, by the second part of (1) of Lemma 4.5,

N W_{σ}^{σ} = γ_{V_{1,σ}^{σ−1}} |_{W_{σ}^{σ}} = \{id×p_{σ,σ}\}^{∗}γ\overline{𝔐}{σ−1 \atop 1,σ} |_{W_{σ}^{σ}} = \{id×p_{σ,σ}\}^{∗}N_{ι_{σ,σ}}^{ide} |_{W_{σ}^{σ}} = N_{j_{σ,σ}}^{ide} |_{W_{σ}^{σ}}.

We have thus verified the second part of the inductive assumption (I1) for σ′ =σ.

Since the immersions ῑ_{σ,σ′} with σ′ ≤σ are embeddings by the inductive assumption (I8) in Subsection 4.3, so are the immersions

j̄_{σ,σ′}: (M̃_{1,(I_{P} (σ′),J_{P} (σ′))}×π_{ϱ_{σ′}(σ)}^{∗}π_{ℙF_{(ℵ,J)}}^{∗}V_{(ℵ_{B}(σ′),J_{B}(σ′))})/G_{σ′} ⟶ V_{1,σ′}^{σ} ⊂ V_{1,k}^{σ}

induced by j_{σ,σ′}. In particular, all of the morphisms

j̄_{σ_{max},σ′}: (M̃_{1,(I_{P} (σ′),J_{P} (σ′))}×π_{0,(ℵ_{B}(σ′),J_{B}(σ′))}^{∗}π_{ℙF_{(ℵ,J)}}^{∗}V_{(ℵ_{B}(σ′),J_{B}(σ′))})/G_{σ′}
⟶ V_{1,σ′}^{σ_{max}} ⊂ V_{1,k}^{σ_{max}} = π̃^{∗}\overline{𝔐}_{1,k}(L, d)

are embeddings. On the other hand, by the inductive assumption (I1),

W̃_{σ′} ≡ W_{σ′}^{σ_{max}} ≡ j_{σ_{max},σ′}^{−1}(𝔐̃_{1,k}^{0}(L, d)) ≡ j_{σ_{max},σ′}^{−1}(V_{1,(0)}^{σ_{max}})

is the closure of

M̃_{1,(I_{P} (σ′),J_{P} (σ′))} × ker 𝔇̃_{0}|Z̃_{σ′;B}^{0} ⊂ M̃_{1,(I_{P} (σ′),J_{P} (σ′))}×π_{0,(ℵ_{B}(σ′),J_{B}(σ′))}^{∗}π_{ℙF_{(ℵ,J)}}^{∗}V_{(ℵ_{B}(σ′),J_{B}(σ′))}.

By Proposition 5.5 and the inductive assumption (I8) in Subsection 4.3, this closure is

M̃_{1,(I_{P} (σ′),J_{P} (σ′))} × ker 𝔇̃_{(I_{P} (σ′),J_{P} (σ′))}|Z̃_{σ′;B}, where
Z̃_{σ′;B} = D̃_{(I_{P} (σ′),J_{P} (σ′))}^{−1}(0).

Since the bundle section D̃_{(I_{P} (σ′),J_{P} (σ′))} is transverse to the zero set, Z̃_{σ′;B} is a smooth subvariety of 𝔐̃_{0,(I_{P} (σ′),J_{P} (σ′))}(ℙ^{n}, d) and

W̃_{σ′} ⟶ M̃_{1,(I_{P} (σ′),J_{P} (σ′))} × Z̃_{σ′;B}

is a smooth vector bundle by Proposition 5.5. We conclude that

𝔐̃_{1,k}^{0}(L, d) ∩ V_{1,σ′}^{σ_{max}}

is a smooth subvariety of 𝔐̃_{1,k}^{0}(L, d) for all σ′ ∈A_{1,k}(k, d). Its normal cone is a line bundle by the inductive assumption (I1).

A Most Frequently Used Symbols and Notation

d degree of stable maps to ℙ^{n} k number of marked points for genus-one stable maps n dimension of projective space, ℙ^{n} ℂ, ℚ the sets of rational numbers, of complex numbers ℤ, ℤ^{+}, ℤ̄^{+} the sets of integers, of positive integers, of nonnegative integers

[k] {1, …, k} γ_{ϱ}, γ_{(N ,J)} tautological line bundles on ℙF_{ϱ}, ℙF_{(ℵ,J)} s_{i} natural pairing of L_{i} with 𝔼 on \overline{M}_{1,I} and with L_{0} on \overline{M}_{0,0⊔I}; p25 s_{ρ,i} pairing of L_{ρ,i} with 𝔼_{ρ} on \overline{M}_{1,(I,J)}^{ρ} induced by s_{i}, ρ∈A_{1}(I, J); p28 s̃_{i} nondegenerate pairing of L̃_{i} with 𝔼̃ on M̃_{1,(I,J)} induced by s_{i}; p28 s_{ρ,(l,i)} pairing of L_{ρ,(l,i)} with 𝔼_{ρ} on M̃_{0,ϱ}^{ρ} induced by π_{l}^{∗}s_{i}, ρ∈A_{0}(ϱ); Subs. 2.4 s̃_{(l,i)} nondegenerate pairing of L̃_{(l,i)} with 𝔼̃ on M̃_{0,ϱ} induced by s_{i}; p31

A_{g}(I), A_{g}(I, J) collections of subvarieties of \overline{M}_{g,I}, of \overline{M}_{g,I⊔J}; p24, p26 A_{0}(ϱ) collection of subvarieties of ℙF_{ϱ}, ϱ=(I_{l}, J_{l})_{l∈ℵ}; p30 A_{g}(d, k) collection of subvarieties of \overline{𝔐}_{g,k}(ℙ^{n}, d); p5, p34 A_{0}(ℵ; d, J) collection of subvarieties of ℙF_{(ℵ,J)}; p38 D_{0}, 𝔇_{0} “derivatives” of maps at the 0th marked point; p36, p67 D_{(ℵ,J)}, 𝔇_{(ℵ,J)} “sums” of various D_{0}, 𝔇_{0}; p41, p68 D̃_{ϱ}, 𝔇̃_{ϱ} bundle maps over 𝔐̃_{(ℵ,J)}^{ϱ} induced by D_{(ℵ,J)}, 𝔇_{(ℵ,J)}; p44, p69 𝔼 Hodge line bundle of holomorphic differentials on \overline{M}_{1,J} 𝔼_{ρ}, 𝔼̃ line bundles on \overline{M}_{1,(I,J)}^{ρ}, M̃_{1,(I,J)}, ρ∈A_{1}(I, J), or M̃_{0,ϱ}^{ρ}, M̃_{0,ϱ}, ρ∈A_{1}(ϱ), or

𝔐̃_{0,(ℵ,J)}^{ρ}, 𝔐̃_{0,(ℵ,J)}(ℙ^{n}, d), ρ∈A_{1}(ℵ; d, J), obtained from 𝔼, or γ_{ϱ}, or γ_{(ℵ,J)}; p28, p31, p44

F_{ϱ}, F_{(ℵ,J)} natural vector bundles over \overline{M}_{0,ϱ}, \overline{𝔐}_{0,(ℵ,J)}(ℙ^{n}, d); p29, p37 F_{ϱ;P}, F_{ϱ;S} subbundles of F_{(ℵ,J)}, ϱ∈A_{0}(ℵ; d, J); p39 G_{σ} symmetry group of the immersions ι_{σ}, ι_{σ^{∗},σ}; p9 Im^{s}ι singular locus of immersion ι; p32 L_{j} tangent line bundle for jth marked point over \overline{M}_{g,J} or \overline{𝔐}_{g,J}(ℙ^{n}, d), j ∈J L_{ρ,i}, L̃_{i} line bundles on \overline{M}_{1,(I,J)}^{ρ}, M̃_{1,(I,J)}, obtained from L_{i}; ρ∈A_{1}((I, J); p28 L_{ρ,(l,i)}, L̃_{(l,i)} line bundles on M̃_{0,ϱ}^{ρ}, M̃_{0,ϱ}, ρ∈A_{0}(ϱ), obtained from π_{l}^{∗}L_{i}; p31 𝕃 universal tangent line bundle on M̃_{1,(I,J)}, M̃_{0,ϱ}; p26, p30 L the line bundle associated to the sheaf O_{ℙ^{n}}(a), a>0

\overline{M}_{g,J} Deligne-Mumford moduli space of stable genus-g curves with marked

points indexed by the set J

\overline{M}_{g,ρ} smooth subvariety of \overline{M}_{g,J}, ρ∈A_{g}(J); p24 \overline{M}_{1,ρ}^{ρ},\overline{M}_{1,ρ^{∗}}^{ρ} exceptional divisor in \overline{M}_{1,(I,J)}^{ρ}, proper transform of \overline{M}_{1,ρ^{∗}}, ρ, ρ^{∗} ∈A_{1}(I, J),

ρ^{∗} >ρ; p28

\overline{M}_{1,(I,J)}^{ρ}, M̃_{1,(I,J)} ρ-stage and final blowups of \overline{M}_{1,I⊔J}, ρ∈A_{1}(I, J); Subs. 2.3 \overline{M}_{σ;P},M̃_{σ;P} the “curve part” of \overline{𝔐}_{1,σ}(ℙ^{n}, d) and its (final) blowup, σ ∈A_{1}(d, k); p8 \overline{M}_{0,ϱ} moduli space of tuples of genus-zero curves; p29 M̃_{0,ρ}^{0},M̃_{0,ρ}^{ρ} smooth subvariety of ℙF_{ϱ}, exceptional divisor in M̃_{0,ϱ}^{ρ}, ρ∈A_{0}(ϱ); p30 M̃_{0,ϱ}^{ρ}, M̃_{0,ϱ} ρ-stage and final blowups of ℙF_{ϱ}, ρ∈A_{0}(ϱ); Subs. 2.4

\overline{𝔐}_{g,J}(ℙ^{n}, d) moduli space of genus-g degree-d stable maps to ℙ^{n} with marked points

indexed by the set J

\overline{𝔐}_{g,σ}(ℙ^{n}, d) subvariety in \overline{𝔐}_{1,k}(ℙ^{n}, d), σ ∈A_{0}(d, k), or \overline{𝔐}_{0,J}(ℙ^{n}, d), σ ∈A_{1}(d, J); p6, p34 \overline{𝔐}_{σ;B}(ℙ^{n}, d) the “map part” of \overline{𝔐}_{g,σ}(ℙ^{n}, d); p8, p35 \overline{𝔐}_{0,(ℵ,J)}(ℙ^{n}, d) moduli space of ℵ-tuples of stable maps; p37 𝔐̃_{0,ϱ^{∗}}^{0}, 𝔐̃_{0,ϱ^{∗}}^{ϱ} subvariety in ℙF_{(ℵ,J)}, its proper transform in 𝔐̃_{(ℵ,J)}^{ϱ}, ϱ, ϱ^{∗} ∈A_{0}(ℵ; d, J),

ϱ^{∗} >ϱ; p39, p42

𝔐̃_{0,(ℵ,J)}^{ϱ}, ϱ-stage blowup of ℙF_{(ℵ,J)}, ϱ∈A_{0}(ℵ; J, d); p42 𝔐̃_{1,k}^{σ} σ-stage idealized blowup of \overline{𝔐}_{1,k}(ℙ^{n}, d), σ ∈\{0\} ⊔ A_{1}(d, k); p50, p56 \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d) main component of \overline{𝔐}_{1,k}(ℙ^{n}, d); p2 𝔐̃_{1,k}^{0}(ℙ^{n}, d) desingularization of \overline{𝔐}_{1,k}^{0}(ℙ^{n}, d) constructed in this paper; p7

N_{ι} normal cone for immersion ι; p32 N_{ι}^{ide} idealized normal bundle for for immersion ι; Defin. 4.1

Department of Mathematics, Stanford University, Stanford, CA 94305-2125 vakil@math.stanford.edu

Department of Mathematics, SUNY, Stony Brook, NY 11794-3651 azinger@math.sunysb.edu

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