THE ENUMERATIVE GEOMETRY OF RATIONAL AND ELLIPTIC CURVES IN PROJECTIVE SPACE

RAVI VAKIL

Abstract. We study the geometry of moduli spaces of genus 0 and 1 curves in ℙ^{n} with specified contact with a hyperplane H. We compute intersection numbers on these spaces that correspond to the number of degree d curves incident to various general linear spaces, and tangent to H with various multiplicities along various general linear subspaces of H. (The numbers of classical interest, the numbers of curves incident to various general linear spaces and no specified contact with H, are a special case.) In the genus 0 case, these numbers are candidates for relative Gromov-Witten invariants of the pair (ℙ^{n}, H), and in the genus 1 case they generalize the enumerative consequences of Kontsevich’s reconstruction theorem for ℙ^{n}. The intersection numbers are rcursively computed by degenerating conditions. As an example, the enumerative geometry of quartic elliptic space curves is worked out in detail.

The methods used may be of independent interest, especially i) the surprisingly intricate geometry of maps of pointed curves to ℙ^{1}, and ii) the study of the space of curves in ℙ^{n} via a smooth fibration (from an open set) to the space of maps of curves to ℙ^{1}. An unusual consequence of i) is an example of a map from a nodal curve to ℙ^{1} that can be smoothed in two different ways.

Contents

1. Introduction 1 2. Motivating Examples 3 3. Definitions and Strategy 7 4. Maps to ℙ^{1} 11 5. Maps to ℙ^{n} 20 6. Components of D_{H} 26 7. Recursive formulas 28 8. Examples 36 Appendix A. Background: The moduli space of stable maps 40 References 41

1. Introduction

In this paper, we study the geometry of moduli spaces of genus 0 and 1 curves in ℙ^{n} with specified contact with a hyperplane H. Recursions are given to compute the number of degree d curves (of genus 0 or 1) incident to various general linear spaces, and tangent to H with various multiplicities along various general linear subspaces of H; call these numbers enumerative invariants of ℙ^{n}. (Gathmann refers to them as degeneration invariants, [Ga1].) The number usually of interest, the number of curves incident to various general linear spaces and with no specified contact with H, is a special case; call these ordinary enumerative invariants.

Date: March 16, 1999.

Li and Ruan, and Ionel and Parker have proposed a definition of relative Gromov-Witten invariants ([LR], [R], [IPa]) in the symplectic category, which in the case of (ℙ^{n}, H) agrees with the genus 0 enumerative invariants defined here. Enumerative invariants of ℙ^{n} are thus potentially a good test-case for any candidate for an algebraic definition of relative Gromov-Witten invariants. Hence the recursions can be interpreted as a full reconstruction theorem for relative Gromov-Witten invariants of ℙ^{n} (and genus 1 enumerative invariants, cf. Getzler’s reconstruction theorem for genus 1 Gromov-Witten invariants of ℙ^{n} [G]).

The approach to the enumerative problem is classical: one of the general linear spaces is specialized to lie in H, and the resulting degenerations and multiplicities are analyzed. (Even if one is only interested in ordinary enumerative invariants, one is forced to deal with all enumerative invariants.) Thanks to the power of Kontsevich’s moduli space of maps, the overall strategy is simple, and most of the article is spent checking details. The reader may wish to see a few motivating examples to understand the issues that come up (Section 2; many phenomena are reminiscent of [CH]), and then read the basic definitions and strategy (Section 3).

As an example, many enumerative invariants of quartic elliptic space curves were computed (by hand, Section 8), including the fact that there are 52,832,040 such curves through 16 general lines in ℙ^{3}. This number was earlier computed by Avritzer and Vainsencher ([AVa], with the actual number corrected in [A]), and independently by Getzler (announced in [G], proof to appear in [GP]). Getzler’s method gives recursions for ordinary enumerative invariants of genus 1 curves in ℙ^{3}.

Gathmann has extended many of these results to the case where H is replaced by a hypersurface ([Ga1]). He has also used extended these ideas to give a different algebro-geometric proof of the number of rational curves of all degrees on the quintic threefold ([Ga2]).

As a surprising aside, a map to ℙ^{1} is given that has two distinct smoothings (Section 4.15).

1.1. Brief history. The enumerative geometry of space curves has been of interest since classical times (see [K2] for an excellent history; see also [KSX] and [PiZ]). Interest in such problems has been reinvigorated by recent ideas motivated by physics, and in particular Kontsevich’s introduction of the moduli space of stable maps. Recursions for ordinary enumerative invariants of rational curves in ℙ^{n} were one of the first applications of this space, via the First Reconstruction Theorem ([KoM], see also [RT]). Similar techniques have been brought to bear on (maps from) genus 1 curves (see [P], [I], [GP] for various enumerative results).

Caporaso and Harris used degeneration methods to give recursions for the enumerative geometry of plane curves (of arbitrary genus, [CH]). Although they use the Hilbert scheme, a reading of their paper from the perspective of stable maps is enlightening, and motivated this work. Such ideas can also be used to calculate genus g Gromov-Witten invariants of del Pezzo surfaces (or equivalently, count curves) and count curves on rational ruled surfaces ([V1]).

1.2. Acknowledgements. This article contains the majority of the author’s 1997 Harvard Ph.D. thesis (and the e-print math.AG/9709007), and was partially supported by an NSERC 1967 Fellowship and a Sloan Foundation Dissertation Fellowship. The author is extremely grateful to his advisor, J. Harris, for inspiration and advice. Conversations with A.J. de Jong have greatly improved the exposition and argumentation. The author also wishes to thank D. Abramovich, M. Thaddeus, R. Pandharipande, T. Graber, T. Pantev, A. Vistoli, M. Roth, L. Caporaso, E. Getzler, and A. Gathmann for many useful discussions.

A. Gathmann has written a program computing genus 0 enumerative invariants of ℙ^{n}, available upon request from the authour.

L_{4} L_{1} L_{2}  l L_{3} L_{3} L_{4} L_{1} L_{2} l

Figure 1. Possible positions of l after L_{1} and L_{2} have degenerated to H

2. Motivating Examples

To count the number of curves in ℙ^{n} incident to various general linear subspaces and tangent with various multiplicities to various general linear subspaces in H, we successively specialize the linear subspaces (not in H) to lie in H. By following through this idea in special cases, we get a preview of the behavior that will turn up in general.

2.1. Two lines through four fixed general lines in ℙ^{3}. Fix four general lines L_{1}, L_{2}, L_{3}, L_{4} in ℙ^{3}, and a hyperplane H. There are a finite number of lines in ℙ^{3} intersecting L_{1}, L_{2}, L_{3}, L_{4}. Call one of them l. We will specialize the lines L_{1}, L_{2}, L_{3}, and L_{4} to lie in H one at a time and see what happens to l. First, specialize the line L_{1} to (a general line in) H, and then do the same with L_{2} (see Figure 1; H is represented by a parallelogram). If l doesn’t pass through the intersection of L_{1} and L_{2}, it must still intersect both L_{1} and L_{2}, and thus lie in H. Then l is uniquely determined: it is the line through L_{3} ∩ H and L_{4} ∩ H. Otherwise, if l passes through the point L_{1} ∩ L_{2}, it is once again uniquely determined (as only one line in ℙ^{3} can pass through two general lines and one point — this can also be seen through further degeneration). This argument can be tightened to rigorously show the classical fact that there are two lines in ℙ^{3} intersecting four general lines.

2.2. 92 conics through eight fixed general lines in ℙ^{3}. The example of conics in ℙ^{3} is a simple extension of that of lines in ℙ^{3}, and gives a hint as to why stable maps are the correct way to think about these degenerations. Consider the question: How many conics pass through 8 general lines L_{1}, …, L_{8}? (For another discussion of this classical problem, see [H] p. 26.) We introduce a pictorial shorthand that will allow us to easily follow the degenerations (see Figure 2).

We start with the set of conics through 8 general lines (the top row of the diagram — the label 92 indicates the number of such conics, which we will calculate last) and specialize one of the lines L_{1} to H to get row 7. (The line L_{1} in H is indicated by the dotted line in the figure.) When we specialize another line L_{2}, one of two things can happen: the conic can intersect H at the point L_{1} ∩ L_{2} and one other (general) point, or it can intersect H once on L_{1} and once on L_{2} (at general points). (The requirement that the conic must pass through a fixed point in the first case is indicated by the thick dot in the figure.)

In this second case (the picture on the right in row 6), if we specialize another line L_{3}, one of three things can happen.

1. The conic can stay smooth, and not lie in H, in which case it must intersect H at {L_{1} ∩ L_{3}, L_{2}} or {L_{1}, L_{2} ∩ L_{3}} (hence the “×2” in the figure).

2. The conic could lie in H. In this case, there are eight conics through five fixed points L_{4} ∩ H, …, L_{8} ∩ H with marked points on the lines L_{1}, L_{2}, and L_{3}.

876543 # general lines  ^{92}^{92}^{18}  ^{74}^{×2}  ^{×8}  ^{30}^{1}^{18}  ^{×4}^{×2}^{10}  ^{4}  ^{1}^{2}  ^{1}

Figure 2. Counting 92 conics in ℙ^{3} through 8 general lines

3. The conic can degenerate into the union of two intersecting lines, one (l_{0}) in H and one (l_{1}) not. These lines must intersect L_{4}, …, L_{8}. (The line l_{0} already intersects L_{1}, L_{2}, L_{3}, so these conditions are automatically satisfied.) Either three or four of {L_{4}, …, L_{8}} intersect l_{1}. In the first case, there are \binom{5}{3} choices of the three lines, and two configurations (l_{0}, l_{1}) once the three lines are chosen (2 choices for l_{1} from Section 2.1). In the second case there are a total of \binom{5}{4} × 2 configurations by similar reasoning. Thus the total number of such configurations is 30.

We fill out the rest of the diagram in the same way. Then, using the enumerative geometry of lines in ℙ^{3} and conics in ℙ^{2} we can work our way up the table, attaching numbers to each picture, finally deducing that there are 92 conics through 8 general lines in ℙ^{3}. To make this argument rigorous, precise dimension counts and multiplicity calculations are needed.

The algorithm described in this article is slightly different: we parametrize rational curves with various conditions and marked intersections with H. In the case of conics through 8 lines, for example, we would count 184 conics through 8 lines with 2 marked points on H, and then divide by 2. The notation will then be cleaner. The resulting pictorial table is almost identical to Figure 2; the only difference is in the first two lines (see Figure 3).

2.3. Twisted cubics through 12 fixed general lines. The situation for curves in ℙ^{n} in general is not much more complicated in principle than our calculations for conics in ℙ^{3}. Two additional twists come up, which are illustrated in the case of the 80, 160 twisted cubics through 12 general lines in ℙ^{3}, indicated pictorially in Figure 4. The third figure in row 8 represents a nodal (rational) cubic in H. There are 12 nodal cubics through 8 general points in ℙ^{2}. (The algorithm described in this paper will actually calculate 80, 160 × 3! cubics with marked points on H through 12 general lines.)

184^{92} 87 # general lines  ^{×2}

Figure 3. Counting 184 conics with two marked points on H through 8 general lines

12 10 987 # general lines 65  ^{80160}^{80160}^{70296}  ^{9864}^{4968}^{2552}  ^{364}  ^{12}  ^{2380} ^{12}  ^{74}  ^{40}^{110} ^{12}^{50}^{58} ^{18}  ^{126}  ^{12}  ^{610}^{210}^{504}^{522}  ^{920}^{92}^{1312}^{7632}^{2808}  ^{420}^{190} ^{×16}  ^{×2}^{×3}^{×2}  ^{×8}  ^{×27}  ^{×2} ^{×9}^{×4}^{×2}^{×2}  ^{×2}  ^{×3}^{45600}  ^{9864}  ^{×2} 11  ^{×2}  ^{×81}  ^{×3}  ^{1220}^{×3}

Figure 4. Counting 80, 160 cubics in ℙ^{3} through 12 general lines

On the left side of row 8 we see a new degeneration (from twisted cubics through nine general lines intersecting H along three fixed general lines in H): a conic tangent to H, intersecting a line in H. (The tangency of the conic is indicated pictorially by drawing its lower horizontal tangent inside the parallelogram representing H.) We also have an unexpected multiplicity of 2 here.

The appearance of these new degenerations indicate why, in order to enumerate rational curves through general linear spaces by these degeneration methods, we must expand the set of curves under consideration to include those required to intersect H with given multiplicity, along linear subspaces.

12 11 10 987 # general lines  ^{1500}^{1500}^{1350}  ^{150}^{1}  ^{×2}^{150}^{×27} 6  ^{1023}^{18}  ^{588}  ^{127}  ^{1}^{1}^{11}^{90}  ^{5} ^{1}  ^{8} * * *^{×3}  ^{×3}  ^{×9}^{×3}  ^{×2}  ^{×3}^{×3}^{14}

Figure 5. Counting 1500 elliptic cubics through 12 general lines in ℙ^{3}

2.4. Genus 1 cubics through 12 fixed general lines in space. The example of smooth elliptic cubics in ℙ^{3} illustrates some of the degenerations we will see, and shows a new complication in genus 1. There are 1500 smooth elliptic cubics in ℙ^{3} through 12 general lines, and we can use the same degeneration ideas to calculate this number. Figure 5 is a pictorial table of the degenerations; a smooth elliptic curve is represented by a squiggle with an open circle on the end.

The degenerations marked with an asterisk have a new twist. For example, consider the cubics through 9 general lines L_{1}, …, L_{9} and 3 lines L_{10}, L_{11}, L_{12} in H (the middle figure in row 9) and specialize L_{9} to lie in H. The limit cubic could be a smooth plane curve in H (the left-most picture of row 8 in the figure). In this case, it must pass through the eight points L_{1} ∩ H, …, L_{8} ∩ H. But there is an additional restriction. The cubics (before specialization) intersected L_{10}, L_{11}, L_{12} in three points p_{10}, p_{11}, p_{12} (p_{i} ∈ L_{i}), and as elliptic cubics are planar, these three points must have been collinear. Thus the possible limits are those curves in H through L_{1} ∩ H, …, L_{8} ∩ H and passing through collinear points p_{10}, p_{11}, p_{12} (with p_{i} ∈ L_{i}). (There is also a choice of a marked point of the curve on L_{9}, which will give a multiplicity of 3.) This collinearity condition can be written as π^{∗}(O(1)) ≅ O(p_{10} + p_{11} + p_{12}) in the Picard group of the curve.

We will have to count elliptic curves with such a divisorial condition involving the marked points; this locus forms a divisor on a family of stable maps. Fortunately, we can express this divisor in terms of divisors we understand well (Section 7.7). As a side benefit, we get enumerative data about elliptic curves in ℙ^{n} with a divisorial condition as well.

3. Definitions and Strategy

3.1. Conventions. We work over ℂ. By scheme, we mean scheme of finite type over ℂ. By variety, we mean a separated integral scheme. Curves are assumed to be complete and reduced. All morphisms of schemes are assumed to be defined over ℂ, and fibre products are over ℂ unless otherwise specified. If f : C → X is a morphism of schemes and Y is a closed subscheme of X, then define f^{−1}(Y) as C ×_{X} Y; f^{−1}Y is a closed subscheme of C. Sing(f) is the set of singular points of f (in C). Similar definitions apply for stacks, which are assumed to be of Deligne-Mumford type. A brief summary of basic facts about the moduli stack of stable maps \overline{M}_{g,m}(ℙ^{n}, d) is included in Appendix A. If H is a hyperplane in ℙ^{n}, and q is a marked point, we will occasionally let {π(q) ∈ H} denote the Cartier divisor ev_{q}^{∗}H in order to be geometrically suggestive. If X is a zero-dimensional scheme (or stack), let #X be the number of points of X. (One should really count each point with multiplicity 1/G, where G is the cardinality of the isotopy group of the point, but this will be 1 in these applications.)

If Δ is a collection (of subspaces of ℙ^{n}, for example), indexed by a set S(Δ), let |Δ| be the cardinality of S(Δ). We use set notation for collections.

Throughout this paper, H is a hyperplane of ℙ^{n}, and A is a hyperplane of H.

3.2. The stacks X and W. Fix positive integers n and d. Suppose Δ = {Δ^{α}}_{α∈S(Δ)} is a collection of general linear spaces of ℙ^{n} indexed by a set S(Δ). (We use a collection rather than a set as we will need to consider the case when all of the Δ^{α} are dimension n, i.e. ℙ^{n}.) For each positive integer m, suppose Γ_{m} = {Γ_{m}^{α}}_{α∈S(Γ_{m})} is a collection of general linear spaces of H indexed by a set S(Γ_{m}), where ∑_{m} m|Γ_{m}| = d. Let Γ = {Γ_{m}}_{m>0}. Then define X_{n}(d, Γ, Δ) to be the (stack-theoretic) closure in \overline{M}_{0,∑ |Γ_{m}|+|Δ|}(ℙ^{n}, d) of points corresponding to stable maps π : C → ℙ^{n} of genus 0 curves with marked points q^{α} (α ∈ S(Δ)) and p_{m}^{α} (m > 0, α ∈ S(Γ_{m})), such that

(i) π(q^{α}) ∈ Δ^{α} for all α ∈ S(Δ), and π(p_{m}^{α}) ∈ Γ_{m}^{α} for all m > 0, α ∈ S(Γ_{m}). (ii) π^{∗}H = ∑_{m,α∈S(Γ_{m})} mp_{m}^{α}.

(As a consequence, no component of C is mapped to H.) Informally, this stack parametrizes rational curves incident (at a marked point) to the linear spaces {Δ^{α}}, and m-fold tangent to H (at a marked point) along the space Γ_{m}^{α} (for all m > 0, α ∈ S(Γ_{m})).

Define W_{n}(d, Γ, Δ) to be the closure in \overline{M}_{1,∑ |Γ_{m}|+|Δ|}(ℙ^{n}, d) of points corresponding to stable maps π : C → ℙ^{n} of genus 1 curves with marked points q^{α} (α ∈ S(Δ)) and p_{m}^{α} (for all m > 0, α ∈ S(Γ_{m})), satisfying (i) and (ii) above and also

(iii) No connected union of components of C of arithmetic genus 1 is contracted.

Informally, this stack parametrizes genus 1 curves incident to the linear spaces {Δ_{α}}, and m-fold tangent to H along the space Γ_{m}^{α} (for all m > 0, α ∈ S(Γ_{m})).

The subscript n will often be suppressed to keep the notation from becoming too complicated.

3.3. The Strategy. We calculate #W_{n}(d, Γ, Δ) (or #X_{n}(d, Γ, Δ)) by degenerating one of the linear spaces Δ^{β} (general in ℙ^{n}) to a general linear space in H, and observing how the maps corresponding to points of W_{n}(d, Γ, Δ) degenerate. We can express this degeneration method as follows. Let Δ′ be the same as Δ except Δ′^{β} is dimension one greater than Δ^{β}. Let D_{H} be the Cartier divisor ev_{β}^{∗}H (i.e. π(q^{β}) ∈ H), and D_{H′} the Cartier divisor ev_{β}^{∗}H′, where H′ is a general hyperplane in ℙ^{n}. Then W_{n}(d, Γ, Δ) = W_{n}(d, Γ, Δ′) ∩ D_{H′} (Proposition 5.3(c)). The degeneration corresponds to counting the points of W_{n}(d, Γ, Δ′) ∩ D_{H}, with multiplicity. As D_{H} is linearly equivalent to D_{H′}, these numbers are the same.

In short, we must understand the irreducible components (and the corresponding multiplicities) of the divisor D_{H} = ∑ m_{i}D_{i} on W_{n}(d, Γ, Δ). This is the key problem addressed in this paper. We make three reductions to simplify the problem.

Reduction A. The condition of requiring a marked point to lie on a fixed general hyperplane imposes one transverse condition on any irreducible substack of \overline{M}_{g,m}(ℙ^{n}, d) (by Kleiman-Bertini 5.1), so requiring a marked point to lie on a codimension k fixed general linear space imposes k transverse conditions. Hence it suffices to know the components and multiplicities of D_{H} when each Γ_{m}^{α} is H and each Δ^{α} is ℙ^{n} (as one can then reduce to the case when the Γ_{m}^{α} and Δ^{α} are smaller by intersecting the D_{i}’s with the appropriate transverse conditions).

Reduction B. Now that there are no conditions on the marked points q^{α}, it suffices to know the components and multiplicities of D_{H} in the case when S(Δ) = {β} (as we can then reduce to the case where |Δ| > 1 by adding marked points).

Reduction C. Projection from the hyperplane A of H gives a rational map ℙ^{n} −−→ ℙ^{1} (sending H to a point ∞ ∈ ℙ^{1}) that induces a rational map ρ_{A} : \overline{M}_{g,m}(ℙ^{n}, d) −−→ \overline{M}_{g,m}(ℙ^{1}, d) that is a morphism on an open set corresponding to maps π : C → ℙ^{n} where π^{−1}A = ∅. It extends to a morphism on the open set V where π^{−1}A is a union of reduced smooth points of C, and ρ_{A} is smooth on V (Proposition 5.5). Each component of D_{H} meets V. Hence if we can solve the problem for n = 1, we can solve the problem in general: the divisor D_{H} (restricted to V) is the pullback of the corresponding divisor D_{∞} on W_{1}(d, Γ′, Δ′ = {ℙ^{1}}), and the components of D_{H} (restricted to V) are pullbacks of the analogous components of D_{∞} (and also with W replaced by X). As ρ_{A} is smooth on V, the multiplicities are the same.

For this reason, we first turn to the space of maps from curves to ℙ^{1} with specified ramification (at marked points) over a point ∞ ∈ ℙ^{1}, with one other marked point q^{β}, and find the components and multiplicities of D_{∞} = ev_{q^{β}}^{∗}∞ = {π(q^{β}) = ∞} (Section 4). This is technically the most intricate part of the argument.

3.4. Summary. In short, the proof of the recursion for the number of genus 0 and 1 curves with prescribed incidences and tangencies is as follows. We first study stacks of maps to ℙ^{1}, with prescribed ramification over ∞ (at marked points), and one other marked point q^{β}, and find the components and multiplicities of ev_{β}^{∗}∞ (Section 4). Then, pulling back by the smooth morphism ρ_{A} (Reduction C), we have a result on stacks of maps to ℙ^{n} with prescribed intersection with a hyperplane H and one other marked point q^{β}, giving components and multiplicities of ev_{β}^{∗}H. By adding additional marked points (Reduction B) and requiring them to lie on various numbers of general hyperplanes (Reduction A), we have a linear equivalence of divisors on stacks of maps to ℙ^{n} with various incidence and tangency conditions (Section 6). If the stack is one-dimensional, we get an expression giving each enumerative invariant in terms of “simpler” enumerative invariants, i.e. recursions (Section 7.1).

3.5. Definitions: Components of D_{H}. Suppose A is a stack of m-pointed, arithmetic genus g, degree d stable maps to ℙ^{n}, and let D_{i} = ev_{i}^{∗}H (1 ≤ i ≤ m). If ∏_{i=1}^{m} D_{i}^{n_{i}}[A] = 0 for all m-tuples (n_{1}, …, n_{m}) adding to dim A, we say A is enumeratively irrelevant; otherwise it is enumeratively relevant. The components of D_{H} that are enumeratively irrelevant will not contribute to enumerative calculations, and may be discarded.

If A^{(j)} := A ×_{\overline{M}_{g,m}(ℙ^{n},d)} \overline{M}_{g,m+j}(ℙ^{n}, d) is enumeratively irrelevant for all j ≥ 0, we say A is stably enumeratively irrelevant. (The stack A^{(1)} is the universal curve over A, and A^{(j+1)} is the universal curve over A^{(j)}.) Informally speaking, if the “image of A in the Chow variety” is of dimension less than that of A, then A is stably enumeratively irrelevant. (See [V2] 2.1 for a similar definition.) Enumerative irrelevance can arise because of moduli in a contracted component, see Proposition 5.2.

3.6. The components of the divisor D_{H} on stacks of form X or W that are enumeratively relevant will turn out to be stacks of the form defined below. The enumerative geometry of these will be obviously related to the enumerative geometry of stacks of the form X and W.

Fix n, d, Γ, Δ, and a non-negative integer l. Let ∑_{k=0}^{l} d(k) be a partition of d. Let the points {p_{m}^{α}}_{m,α} be partitioned into l + 1 subsets {p_{m}^{α}(k)}_{m,α} for k = 0, …, l. This induces a partition of each Γ_{m} into ∐_{k=0}^{l} Γ_{m}(k). Let the points {q^{α}}_{α} be partitioned into l + 1 subsets {q^{α}(k)}_{α} for k = 0, …, l. This induces a partition of Δ into ∐_{k=0}^{l} Δ(k). For k > 0, define m^{k} := d(k) − ∑_{m} m|Γ_{m}(k)|, and assume m^{k} > 0 for all k = 1, …, l. In Definitions 3.7, 3.8, 3.10, and 3.11 below, the marked points on \overline{M}_{g,∑ |Γ_{m}|+|Δ|}(ℙ^{n}, d) are labelled \{p_{m}^{α}\}_{α∈S(Γ)} and {q^{α}}_{α∈S(Δ)}

Substacks of the following form will appear as components of D_{H} on stacks of the form X (d, Γ, Δ).

3.7. Definition. The stack

Y_{n}(d(0), Γ(0), Δ(0); …; d(l), Γ(l), Δ(l))

is the (stack-theoretic) closure of the locally closed substack of \overline{M}_{0,∑ |Γ_{m}|+|Δ|}(ℙ^{n}, d) representing stable maps (C, {p_{m}^{α}}, {q^{α}}, π) satisfying the following conditions Y1. The curve C consists of l + 1 irreducible components C(0), …, C(l) with all components

meeting C(0). The map π has degree d(k) on curve C(k) (0 ≤ k ≤ l). Y2. The points {p_{m}^{α}(k)}_{m,α} and {q^{α}(k)}_{α} lie on C(k), and π(p_{m}^{α}(k)) ∈ Γ_{m}^{α}(k), π(q^{α}(k)) ∈

Δ^{α}(k). Y3. As sets, π^{−1}H = C(0) ∪ {p_{m}^{α}}_{m,α}, and for k > 0,

(π |_{C(k)})^{∗}H = ∑_{m,α} mp_{m}^{α}(k) + m^{k}(C(0) ∩ C(k)).

Pictorial representations of such maps are given in the Figures of Section 2. Note that d(k) > 0 for all positive k by the last condition.

Substacks of the following forms will appear as components of D_{H} on stacks of the form W(d, Γ, Δ).

3.8. Definition. The stack

Y_{n}^{a}(d(0), Γ(0), Δ(0); …; d(l), Γ(l), Δ(l))

is the closure of the locally closed substack of \overline{M}_{1,∑ |Γ_{m}|+|Δ|}(ℙ^{n}, d) representing stable maps (C, {p_{m}^{α}}, {q^{α}}, π) satisfying conditions Y1–Y3 above, and Y^{a}4. The curve C(1) has genus 1 (and the other components are genus 0).

3.9. Remark. For given d, Γ, Δ, m, and choices α ∈ Γ_{m}, β ∈ Γ, let Γ′ be the same as Γ except Γ′_{m}^{α} = Γ_{m}^{α} ∩ Δ^{β}. Then there is a natural isomorphism

ϕ : X_{n}(d, Γ′, Δ \ {Δ^{β}}) → Y_{n}(0, {Γ_{m}^{α}}, {Δ^{β}}; d, Γ \ {Γ_{m}^{α}}, Δ \ {Δ^{β}}).

The map from the left to the right involves gluing a contracted ℙ^{1} (with marked points p_{m}^{α} and q^{β}) to the point p′_{m}^{α}.

Similarly, there is a natural isomorphism (which we sloppily denote ϕ as well)

ϕ : W_{n}(d, Γ′, Δ \ {Δ^{β}}) → Y_{n}^{a}(0, {Γ_{m}^{α}}, {Δ^{β}}; d, Γ \ {Γ_{m}^{α}}, Δ \ {Δ^{β}}).

3.10. Definition. The stack

Y_{n}^{b}(d(0), Γ(0), Δ(0); …; d(l), Γ(l), Δ(l))

is the closure of the locally closed substack of \overline{M}_{1,∑ |Γ_{m}|+|Δ|}(ℙ^{n}, d) representing stable maps (C, {p_{m}^{α}}, {q^{α}}, π) satisfying conditions Y1–Y2 above, and Y^{b}3. As sets, π^{−1}H = C(0) ∪ {p_{m}^{α}}_{m,α}, and for k > 1,

(π |_{C(k)})^{∗}H = ∑_{m,α} mp_{m}^{α}(k) + m^{k}(C(0) ∩ C(k)).

Y^{b}4. All components of C are rational. The curves C(0) and C(1) intersect at two distinct points {a_{1}, a_{2}}. (These points are not marked; monodromy may exchange them.) Also,

(π |_{C(1)})^{∗}H = ∑_{m,α} mp_{m}^{α}(k) + m_{1}^{1}a_{1} + m_{2}^{1}a_{2}

where m_{1}^{1} + m_{2}^{1} = m^{1}. Thus Y^{b}(d(0), …, Δ(l)) is naturally the union of [m^{1}/2] (possibly reducible) stacks (where

[·] is the greatest-integer function), indexed by m_{1}^{1}. For convenience, label these stacks

{Y^{b}(d(0), …, Δ(l))_{m_{1}^{1}}}_{1≤m_{1}^{1}<m^{1}},

so Y^{b}(d(0), …, Δ(l))_{m_{1}^{1}} = Y^{b}(d(0), …, Δ(l))_{m^{1}−m_{1}^{1}}.

3.11. Definition. The stack

Y_{n}^{c}(d(0), Γ(0), Δ(0); …; d(l), Γ(l), Δ(l))

is the closure of the locally closed substack of \overline{M}_{1,∑ |Γ_{m}|+|Δ|}(ℙ^{n}, d) representing stable maps (C, {p_{m}^{α}}, {q^{α}}, π) satisfying conditions Y1–Y3 above, and Y^{c}4. The curve C(0) has genus 1 (and the other components are genus 0). The morphism π

has positive degree on every component. Y^{c}5. In Pic(C(0)),

π^{∗}(O_{ℙ^{n}}(1)) ⊗ O_{C(0)} (∑_{k=1}^{l} m^{k}(C(0) ∩ C(k))) ≅ O_{C(0)} ( ∑_{m,α∈Γ_{m}(0)} mp_{m}^{α}(0)) .

3.12. Remark. This Picard condition Y^{c}5 was actually present in Y, Y^{a} and Y^{b}, but as C(0) was rational in each of these cases, the requirement reduced to

d(0) + ∑_{k=1}^{l} m^{k} = ∑_{m} m|Γ_{m}(0)|

which was always true.

The six types of stacks W, X, Y, Y^{a}, Y^{b}, Y^{c} are illustrated in Figure 6. In the figure, the dual graph of the curve corresponding to a general point of the stack is given. Vertices corresponding to components mapped to H are labelled with an H, and vertices corresponding to genus 1 components are open circles.

Because of the divisorial condition Y^{c}5 in the definition of Y^{c}, we will also be interested in the variety parametrizing smooth degree d genus 1 curves in ℙ^{n} (n ≥ 2) with a condition in the Picard group of the curve involving the marked points and π^{∗}(O_{ℙ^{n}}(1)). Let Δ = {Δ^{α}}_{α∈S(Δ)} be a set of linear spaces in ℙ^{n}. Let D be a linear combination of the formal variables {q^{α}}_{α∈S(Δ)} with integral coefficients summing to d.

W X Y^a H

Y^b Y^c

H H

Y

H

Figure 6. Six types of stacks

3.13. Definition. The stack Z_{n}(d, Δ)_{D} is the (stack-theoretic) closure of the locally closed subset of \overline{M}_{1,|Δ|}(ℙ^{n}, d) (where the points are labelled \{q^{α}\}_{α}) representing stable maps (C, {q^{α}}, π) where C is smooth, π(q^{α}) ∈ Δ^{α} for all α, and π^{∗}(O_{ℙ^{n}}(1)) ≅ O_{C}(D) in Pic(C).

For example,

Z_{2}(d = 4, Δ = {11 general points})_{q^{1}+q^{2}+q^{3}+q^{4}}.

parametrizes the finite number of two-nodal quartic plane curves C through 11 fixed general points {q^{j}}_{1≤j≤11} such that q^{1} + q^{2} + q^{3} + q^{4} is linearly equivalent to the hyperplane section on the normalization of the curve C. (There are 62 such curves, see the example at the end of Section 7.7 and Table 2.)

When Γ and Δ are general, all of the varieties W, X, Y, Y^{a}, Y^{b}, Y^{c}, Z defined above will be seen to be of the expected dimension (Proposition 5.7).

4. Maps to ℙ^{1}

4.1. Quasi-stable maps. For the purposes of this Section, define a quasi-stable map in the same way as a stable map, except the source curve is not required to be connected. Then the entire theory of stable maps carries through for quasi-stable maps, and for smooth projective X there is a fine moduli stack, which we’ll denote \overline{M}_{g,m}(X, β)^{Q}, of finite type, and \overline{M}_{g,m}(X, β) is an open and closed substack of \overline{M}_{g,m}(X, β)^{Q}. For more details on this (essentially trivial) variation on stable maps, see [V1] Section 2.5.

4.2. Deformations of maps from curves to curves. Suppose π : C → ℙ^{1} is a stable degree d map with marked points {p_{i}}. Call étale neighborhoods of connected components A of Sing(π) ∪ {p_{i}} ⊂ C (containing only one copy of A and no other points of Sing(π) ∪ {p_{i}}) special loci of π. Special loci are étale neighborhoods of ramification points of C, nodes of C, marked points, or unions of contracted components of π. The map π is stable, so the functor parametrizing deformations of π is pro-representable by the formal neighborhood Def of the corresponding point in the moduli stack of stable maps.

If A_{1}, …, A_{n} are the special loci of π (or more precisely, if étale neighborhoods of A_{1}, …, A_{n} are the special loci), let Def_{A_{1}}, …, Def_{A_{n}} be the deformation spaces (i.e. hulls) of the special loci. (Implicit here is the fact that Def_{A_{j}} is independent of the neighborhood of A_{j} chosen; this will follow from the proof of the following proposition.)

4.3. Proposition. — The natural map Def → Def_{A_{1}} × ⋯ × Def_{A_{n}} is an isomorphism.

In the analytic category, this proposition is clear. Proof. The deformation theory of π is controlled by Ext^{·}(\underline{Ω}_{π}, O_{C}) (see Appendix A). Let e_{j} : C_{j} → C be an étale neighborhood of A_{j} as described above, so Def_{A_{j}} is constructed using Ext(e_{j}^{∗}\underline{Ω}_{π}, e_{j}^{∗}O_{C}) (as e_{j}^{∗}\underline{Ω}_{π} = \underline{Ω}_{π∘e_{j}}).

Let K = Ker(\underline{Ω}_{π}) and Q = Coker(\underline{Ω}_{π}), so K is supported on the special loci and hence splits canonically into ⊕_{j}K_{j}, with K_{j} supported on A_{j} (and similarly for Q = ⊕_{j}Q_{j}).

There are exact sequences

0 → K[1] → \underline{Ω}_{π} → Q → 0 (1)

and (for each j)

0 → K_{j}[1] → e_{j}^{∗}\underline{Ω}_{π} → Q_{j} → 0 (1_{j})

with morphisms (1) → (1_{j}) induced by e_{j}. By considering (1) → ⊕_{j} (1_{j}) and taking the associated long exact Ext sequence, we have

… → Ext^{i}(Q, O_{C}) →  Ext^{i}(\underline{Ω}_{π}, O_{C})  → Ext^{i−1}(K, O_{C}) → …
||  ↓  ||
… → ⊕_{j} Ext^{i}(Q_{j}, O_{C_{j}}) → ⊕_{j} Ext^{i}(e_{j}^{∗}\underline{Ω}_{π}, O_{C_{j}}) → ⊕_{j} Ext^{i−1}(K_{j}, O_{C_{j}}) → ….

(2)

By the five lemma, the vertical arrow is an isomorphism for all i, and by the construction of Def from Ext^{i}(\underline{Ω}_{π}, O_{C}) (and Def_{A_{j}} from Ext^{i}(\underline{Ω}_{π∘e_{j}}, O_{C_{j}})), the result follows.

4.4. Substacks of \overline{M}_{g,m}(ℙ^{1}, d). Fix a positive integer d and a point ∞ on ℙ^{1}, and let h⃗ = (h_{1}, h_{2}, …) represent a partition of d with h_{1} 1’s, h_{2} 2’s, etc., so ∑_{m} mh_{m} = d. Let V = V^{d,g}(h⃗) be the closure in \overline{M}_{g,∑ h_{m}+1}(ℙ^{1}, d)^{Q} of points representing quasi-stable maps (C, \{p_{m}^{α}\}, q, π) where C is a smooth curve with ∑ h_{m} + 1 (distinct) marked points Γ_{m} = {p_{m}^{α}}_{1≤α≤h_{m}} and q, and π^{∗}(∞) = ∑_{m,α} mp_{m}^{α}. (For example, X_{1}(d, Γ, Δ) is an open and closed substack of V^{d,0}(h⃗), where Γ_{m} consists of h_{m} copies of ∞, and Δ = {ℙ^{1}}. Similarly, W_{1}(d, Γ, Δ) is an open and closed subset of V^{d,1}(h⃗).)

For the map corresponding to a general point in V, each special locus A_{j} is either a marked ramification above the point ∞, a simple unmarked ramification (of which there are d + 2g − 2 + ∑ h_{m} by Riemann-Hurwitz), or the point q (at which π is smooth). In these three cases, the formal deformation space of A_{j} inside V is 0, Spf ℂ[[t]], and Spf ℂ[[t]] respectively. Thus

dim V = d + 2g − 1 + ∑ h_{m} (3)

Let D_{∞} be the divisor ev_{q}^{∗}(∞) = {π(q) = ∞}. There are three natural questions to ask:

1. What are the components of the divisor D_{∞}?

2. With what multiplicity do they appear?

3. What is the local structure of V near these components?

We partially answer these three questions.

Fix a component Y of the divisor D_{∞} and a map (C, {p_{m}^{α}}, q, π) corresponding to the general element of Y. Notice that π collapses a component of C to ∞, as otherwise π^{−1}(∞) is a union of points, and

d = deg π^{∗}(∞) ≥ deg_{q} π^{∗}(∞) + ∑_{m,α} deg_{p_{m}^{α}} π^{∗}(∞) ≥ 1 + ∑_{m} mh_{m} = d + 1.

Let C(0) be the connected component of π^{−1}(∞) containing q, and let C̃ be the closure of C \C(0) in C (see Figure 7; C(0) is the union of those curves contained in the dotted rectangle, and C̃ is the rest of C).

Let Γ_{m}(0) = {p_{m}^{α}(0)} be the points of Γ_{m} on C(0), and Γ̃_{m} = {p̃_{m}^{α}} = Γ_{m} \ Γ_{m}(0) be the points on C̃ (with h_{m}(0) = |Γ_{m}(0)|, h̃_{m} = |Γ̃_{m}|). Let s be the number of intersections of C(0) and C̃, and label these points r^{1}, …, r^{s}. Thus g = p_{a}(C(0)) + p_{a}(C̃) + s − 1. Let m^{k} be the multiplicity of (π|C̃)^{∗}(∞) at r^{k}. The data (m^{1}, …, m^{s}) is constant for any choice of (C, {p_{m}^{α}}, q, π) in an open subset of Y.

π^{p_{1}^{1}} ^{p_{2}^{1}}^{p_{1}^{3}} ∞  ℙ^{1}^{q}^{p_{1}^{2}}
Figure 7. The map (C, {p_{m}^{α}}, q, π}) ∈ Y

4.5. Proposition. — The map (C̃, {p_{m}^{α}(0)}, {r^{k}}, π) has no collapsed components, and only simple ramification away from π^{−1}(∞). The curve C̃ is smooth.

The map (C̃, {p_{m}^{α}(0)}, {r^{k}}, π) will turn out to correspond to a general element in V^{d,g′}(h⃗′) for some g′, h⃗′. Proof. Let A_{1}, …, A_{l} be the special loci of π, and say q ∈ A_{1}.

The map (C, {p_{m}^{α}}, q, π) lies in V and hence can be deformed to a curve where each special locus is either a marked ramification above ∞, a simple unmarked ramification, or the point q. If A_{k} (k > 1) is not one of these three forms then by Proposition 4.3 there is a deformation of the map (C, {p_{m}^{α}}, q, π) preserving π at A_{i} (i ≠ k) but changing A_{k} into a combination of special loci of these three forms. Such a deformation (in which A_{1} is preserved and thus still smoothable) is actually a deformation in the divisor D_{∞} = ev_{q}^{∗}∞ = {π(q) = ∞}, contradicting the generality of (C, {p_{m}^{α}}, q, π) in Y.

4.6. Thus the map (C̃, {p̃_{m}^{α}}, {r^{k}}, π) must lie in V^{d,p_{a}(C̃)}(h⃗′) where h⃗′ is the partition corresponding to (π|C̃)^{∗}(∞). By (3), C̃ moves in a family of dimension at most

d + 2p_{a}(C̃) − 2 + (∑ h̃_{m} + s) ,

and the curve C(0) (as a nodal curve with marked points {p_{m}^{α}(0)}_{m,α∈Γ_{m}(0)}, {r^{k}}_{1≤k≤s}, and q) moves in a family of dimension at most

3p_{a}(C(0)) − 3 + ∑ h_{m}(0) + s + 1,

so Y is contained in a family of dimension

(d + 2p_{a}(C̃) − 2 + ∑ h̃_{m} + s) + (3p_{a}(C(0)) − 3 + ∑ h_{m}(0) + s + 1)
= d + 2g − 1 + ∑ h_{m} − 1 + p_{a}(C(0))
(4)  = dim V − 1 + p_{a}(C(0))

by (3).

4.7. Components Y of D_{∞} satisfying p_{a}(C(0)) = 0. For each choice of a partition Γ_{m} = Γ_{m}(0) ∐ Γ̃_{m} (inducing a partition of h_{m} into h_{m}(0) + h̃_{m}), a positive integer s, and

q^{q}^{p_{1}^{1}}  ^{p_{1}^{1}}  ^{p_{1}^{1}}  ^{p_{1}^{2}}^{p_{1}^{1}}  V^{2,0}(h_{1} = 2)^{p_{1}^{2}}^{p_{1}^{2}}  ^{p_{1}^{2}}^{q}  ^{p_{1}^{2}}^{p_{1}^{1}}  ^{q}^{q}

Figure 8. The four possible components of D_{∞} on V^{2,0}(h_{1} = 2)

(m^{1}, …, m^{s}) satisfying ∑ m^{k} + ∑ mh̃_{m} = d, consider the closure in \overline{M}_{g,∑ h_{m}+1}(ℙ^{1}, d)^{Q} of points corresponding to maps (C(0) ∪ C̃, {p_{m}^{α}}, q, π) where

• The curve C(0) ∪ C̃ is nodal, and the curves C(0) and C̃ intersect at the points {r^{k}}.

• The curve C(0) is isomorphic to ℙ^{1}, has labelled points {p_{m}^{α}(0)} and q, and π(C(0)) = ∞.

• The curve C̃ is smooth of arithmetic genus g − s + 1 with labelled points {p̃_{m}^{α}}. The map π is degree d on C̃, and

(π|C̃)^{∗}(∞) = ∑ mp̃_{m}^{α} + ∑ m^{k}r^{k}.

Let U be the union of these substacks (over all choices of partitions of Γ_{m}); dim U = dim V − 1 by (4).

An irreducible component Y of the divisor D_{∞} satisfying p_{a}(C(0)) = 0 has dimension dim V − 1 and is a closed substack of U, which also has dimension dim V − 1. Hence Y must be a component of U and the stable map corresponding to a general point of Y satisfies the three properties listed in the previous paragraph. (We don’t yet know that all such Y are subsets of V, but this will follow from Proposition 4.8 below.)

For example, if d = 2, g = 0, and h_{1} = 2, there are four components of U (see Figure 8; π^{−1}(∞) is indicated by a dashed line). The components (from left to right) are a subset of the following.

1. The curve C̃ is irreducible and maps with degree 2 to ℙ^{1}, ramifying over two general points of ℙ^{1}. The marked points q and p_{1}^{1} lie on C(0), and p_{1}^{2} lies on C̃. The curve C(0) is attached to C̃ at the point

(π|C̃)^{−1}(∞) \ {p_{1}^{2}}.

2. This case is the same as the previous one with p_{1}^{1} and p_{1}^{2} switched.

3. The curve C̃ is the disjoint union of two ℙ^{1}’s, each mapping to ℙ^{1} with degree 1. Both intersect C(0), which contains all the marked points.

4. The curve C̃ is irreducible and maps with degree 2 to ℙ^{1}, and one of its branch points is ∞. All of the marked points lie on C(0).

Given a component Y of U, we can determine the multiplicity of the divisor D_{∞} along Y. As this multiplicity will turn out to be positive, Y is a subset of V, so as sets, U ⊂ D_{∞}.

4.8. Proposition. — Fix such a component Y with p_{a}(C(0)) = 0. The multiplicity of D_{∞} along Y is ∏_{k=1}^{s} m^{k}.

C′(0) p_{m}^{α} ∞  0

Figure 9. Step 2: It suffices to study deformations of a simpler map

For example, in Figure 8, the first 3 components appear with multiplicity 1, and the fourth with multiplicity 2. Proof. We make a series of reductions to simplify the proof.

Step 1: Deformations of A_{1}. The multiplicity may be computed on a formal neighborhood of a general point of Y. By Proposition 4.3, this space is naturally a product of the hulls of the special loci. If A_{1} is the special locus containing q, then D_{∞} is the pullback of a Cartier divisor on Def_{A_{1}}, and Y is the pullback of a Weil divisor on Def_{A_{1}}. Thus we need only consider deformations of the special locus A_{1}.

Step 2: Simpler maps. Fix a point 0 ∈ ℙ^{1} (distinct from ∞). Let V′ be the closure in \overline{M}_{0,|Γ(0)|+1+s}(ℙ^{1}, ∑_{k=1}^{s} m^{k}) of points representing maps (C, \{p_{m}^{α}(0)\}, q, \{y^{k}\}_{k=1}^{s}, π) where π^{∗}(∞) = ∑ mp_{m}^{α}(0) and π^{∗}(0) = ∑ m^{k}y^{k}. Let Y′ be the closure of points representing maps from a nodal curve C′(0) ∪ C(1) ∪ ⋯ ∪ C(s), where

• C′(0) is glued to C(k) at a point (call it r^{k}) (1 ≤ k ≤ s)

• The marked curve (C′(0), {p_{m}^{α}(0)}, q, {r^{k}}) is isomorphic to the marked curve (C(0), {p_{m}^{α}(0)}, q, {r^{k}}), and is collapsed to ∞ by π.

• C(k) maps to ℙ^{1} with degree m^{k}, and is totally ramified over 0 (at y^{k}) and ∞ (at r^{k}).

(See Figure 9.) (An étale neighborhood of) the special locus C′(0) of a general map in Y′ is isomorphic to (an étale neighborhood of) A_{1}. As the only other special loci of such a map are the points y^{k}, the formal deformations in V′ of a general map in Y′ are given by Def_{A_{1}}. As it suffices to consider the case when Γ = Γ(0) (all the marked points mapping to ∞ are on C(0)), we now assume that this is the case.

Step 3: Fixing the marked curve. There is a morphism of stacks α : V′ → \overline{M}_{0,∑ h_{m}+1+s} that sends each map to the stable model of the underlying pointed nodal curve. Given any smooth marked curve (C(0), \{p_{m}^{α}\}, q, \{y^{k}\}) in \overline{M}_{0,∑ h_{m}+1+s}, the stable map (C, \{p_{m}^{α}\}, q, \{y^{k}\}, π) defined in Step 2 corresponds to a point in α^{−1}(C(0), {p_{m}^{α}}, q, {y^{k}}), so α|_{Y′} is surjective. Let F_{α} be a general fiber of α. By Sard’s theorem, α|_{Y′} is regular in a Zariski-open subset of Y′, so [Y′] ∩ [F_{α}] = [Y′ ∩ F_{α}] in the Chow group of [V′].

In order to determine the multiplicity of D_{∞}|_{V′} along Y′, it suffices to determine the multiplicity of the Cartier divisor D_{∞}|_{F_{α}} along Y′ ∩ F_{α} (in the Chow group of F_{α}). (Proof: As D_{∞} is a Cartier divisor, [D_{∞}|_{F_{α}}] = D_{∞} · [F_{α}]. Thus if [D_{∞}|_{V′}] = m[Y′] in A^{1}V′ then, intersecting with [F_{α}], [D_{∞}|_{F_{α}}] = D_{∞} · [F_{α}] = m[Y′] · [F_{α}] = m[Y′ ∩ F_{α}] in A^{1}F_{α}.)

With this in mind, fix a general (C, \{p_{m}^{α}\}, q, \{y^{k}\}) in \overline{M}_{0,∑ h_{m}+1+s} and let V′′_{o} be the points of \overline{M}_{0,∑ h_{m}+1+s}(ℙ^{1}, d)^{Q} representing stable maps (C, \{p_{m}^{α}\}, q, \{y^{k}\}, π) where π^{∗}(∞) = ∑ mp_{m}^{α} and π^{∗}(0) = ∑ m^{k}y^{k}. Let f and g be sections of O_{C}(d) with associated divisors

(f) = ∑ mp_{m}^{α}, (g) = ∑ m^{k}y^{k}.

Then the maps in V′′_{o} are those of the form [βf, γg] where

[β, γ] ∈ ℙ^{1} \ {[0, 1], [1, 0]}

where ∞ = [0, 1] and 0 = [1, 0].

Let V′′ = V′ ∩ F_{α} be the closure of V′′_{o}, and define Y′′ = Y ∩ F_{α} similarly. Let V′′ be the course moduli scheme of V′′, and Y′′ the course moduli scheme of Y′′.

Step 4: V′′ is isomorphic to ℙ^{1}. The variety V′′ is proper, and V′′_{o} ≅ ℙ^{1} \ {[0, 1], [1, 0]}, so the normalization of the variety V′′ is ℙ^{1}.

The evaluation map gives a morphism from the curve V′′ to ℙ^{1}:

[(C, {p_{m}^{α}}, q, {y^{k}}, π)] → π(q)

and this map is an isomorphism from V′′_{o} to ℙ^{1} \ {[0, 1], [1, 0]}, so it must be an isomorphism from V′′ to ℙ^{1}.

Step 5: Calculating the multiplicity. Let w be a general point of the target ℙ^{1}. Then the divisor {π(q) = w} is linearly equivalent to D_{∞}|_{V′′} = ev_{q}^{∗}∞|_{V′′} = {π(q) = ∞}|_{V′′}, and is O_{V′′}(1).

Thus, on V′′, D_{∞}|_{V′′} = [1, 0] = Y′′. But the limit map has automorphism group

ℤ/m^{1}ℤ ⊕ ⋯ ⊕ ℤ/m^{s}ℤ

(as Aut(C(k), π|_{C(k)}) = m^{k}) so as stacks [D_{∞}|_{V′′}] = (∏ m^{k}) [Y′′]. Therefore [D_{∞}] = ∏ m^{k}[Y].

In order to extend these results to components for which p_{a}(C(0)) = 1, we will need the following result.

4.9. Proposition. — Let Y be a component of D_{∞}, with (C, {p_{m}^{α}}, q, π) the map corresponding to a general point of Y, C(0) ∩ C̃ = {r^{1}, ..., r^{s}}, and m^{k} the multiplicity of π^{∗}(∞) on C̃ at r^{k}. Then

O_{C(0)} (∑_{m,α} mp_{m}^{α}(0)) ≅ O_{C(0)} (∑_{k=1}^{s} m^{k}r^{k})

where Γ(0) ⊂ Γ are the marked points whose limits lie in C(0). Proof. For a map (C, {p_{m}^{α}}, q, π) corresponding to a general point in V, we have the following relation in the Picard group of C:

π^{∗}(O_{ℙ^{1}}(1)) ≅ O_{C}(∑_{m,α} mp_{m}^{α}).

Thus for the curve corresponding to a general point of Y the invertible sheaf O_{C}(∑_{m,α} mp_{m}^{α}) must be a possible limit of π^{∗}(O_{ℙ^{1}}(1)). The statement of the lemma depends only on an étale neighborhood of C(0), so we may assume (as in Step 2 of the proof of Proposition 4.8) that Γ = Γ(0), and C̃ consists of k rational tails C(1), …, C(k) each totally ramified where they intersect C(0). As the dual graph of C is a tree, C is of compact type (i.e. Pic^{0} C is compact). One possible limit of π^{∗}(O_{ℙ^{1}}(1)) is the line bundle that is trivial on C(0) and degree m^{k} on C(k). If a curve C′ is the central fiber of a one-dimensional family of curves, and C′ = C_{1} ∪ C_{2}, and a line bundle L is the limit of a family of line bundles, then the line bundle L′ whose restriction to C_{i} is L|_{C_{i}}((−1)^{i}C_{1} ∩ C_{2}) is another possible limit. Thus the line bundle that is trivial on C̃ and O_{C}(∑ m^{k}r^{k}) on C(0) is a possible limit of π^{∗}(O_{ℙ^{1}}(1)).

If two line bundles on a curve C of compact type are possible limits of the same family of line bundles, and they agree on all components but one of C, then they must agree on the remaining component. But O_{C}(∑ mp_{m}^{α}) is another limit of π^{∗}(O_{ℙ^{1}}(1)) that is trivial on C̃, so the result follows.

4.10. We now determine all components Y of D_{∞} satisfying p_{a}(C(0)) = 1. For each choice of a partition of Γ_{m} into Γ_{m}(0) ∐ Γ̃_{m}, a positive integer s, and (m^{1}, …, m^{s}) satisfying ∑ m^{k} + ∑ mh̃_{m} = d, consider the closure in \overline{M}_{g,∑ h_{m}+1}(ℙ^{1}, d) of points corresponding to maps

(C(0) ∪ C̃, {p_{m}^{α}}, q, π)

where

1. The curve C(0) ∪ C̃ is nodal, and C(0) and C̃ intersect at the points {r^{k}}.

2. The curve C(0) is a smooth genus 1 curve with labelled points {p_{m}^{α}(0)} and q, where O_{C(0)}(∑ mp_{m}^{j}(0)) ≅ O_{C(0)}(∑ m^{k}r^{k}), and π(C(0)) = ∞.

3. The curve C̃ is smooth of arithmetic genus g − s with labelled points {p̃_{m}^{α}}. The map π is degree d on C̃, and

(π|C̃)^{∗}(∞) = ∑ mp̃_{m}^{α} + ∑ m^{k}r^{k}.

Let U be the union of all such substacks (over all choices of s, partitions of Γ, etc.). The divisorial condition O_{C(0)}(∑ mp_{m}^{α}(0)) ≅ O_{C(0)}(∑ m^{k}r^{k}) defines a substack M′ of pure codimension 1 in M_{1,∑ h_{m}+1+s}: for any

(C, {p_{m}^{α}}, q, {r^{k}}_{k>1}) ∈ M_{1,∑ h_{m}+1+(s−1)}

the subscheme of points r^{1} ∈ C satisfying

O_{C}(m^{1}r^{1}) ≅ O_{C} (∑ mp_{m}^{α}(0) − ∑_{k>1} m^{k}r^{k})

is reduced of degree (m^{1})^{2}. Thus the stack M′ is a degree (m^{1})^{2} étale cover of M_{1,∑ h_{m}+1+(s−1)}. By this observation and (4), U has pure dimension dim V − 1.

An irreducible component Y of the divisor D_{∞} satisfying p_{a}(C(0)) = 1 has dimension dim V − 1 and is a substack of U, which also has dimension dim V − 1. Hence Y must be a component of U and the stable map corresponding to a general point of Y satisfies properties 1–3 above.

The determination of multiplicity and local structure is identical to the genus 0 case.

4.11. Proposition. — Fix such a component Y with p_{a}(C(0)) = 1. If m^{1}, …, m^{s} are the multiplicities of π^{∗}(∞) along C̃ at the s points C(0) ∩ C̃, then this divisor appears with multiplicity ∏_{k} m^{k}. Proof. The proof is identical to that of Proposition 4.8. We summarize the steps here.

Step 1. If A_{1} is the special locus of π containing q, then it suffices to analyze Def_{A_{1}}.

Step 2. We may consider instead deformations of the map consisting of C(0), with s rational tails ramifying completely over ∞ (at points r^{k}) and over another point 0 (at points y^{k}). In particular, we assume Γ = Γ(0).

Step 3. Let \overline{M}′_{1,∑ h_{m}+1+s} be the substack of \overline{M}_{1,∑ h_{m}+1+s} that is the closure of the set of points representing smooth stable curves where O(∑ mp_{m}^{α}) ≅ O(∑ m^{k}y^{k}). If α is defined by

α : V′ → \overline{M}′_{1,∑ h_{m}+1+s},

then α|_{Y} is dominant, so we may consider a fixed general stable curve

(C, \{p_{m}^{α}\}, q, \{y^{k}\}) ∈ \overline{M}′_{1,∑ h_{m}+1+s}.

Steps 4 and 5. The variety V′′ is ℙ^{1}, and the multiplicity calculation is identical.

4.12. We have now found all components of D_{∞} on X_{1}(d, Γ, Δ) and W_{1}(d, Γ, Δ) where Δ = {ℙ^{1}} (as in these cases p_{a}(C(0)) ≤ 1), and the multiplicity of D_{∞} along each component. We summarize this in two theorems which will be invoked later.

4.13. Theorem (Genus 0 maps to ℙ^{1}). — The components of D_{∞} on X_{1}(d, Γ, Δ) are of the form

Y_{1}(0, Γ(0), Δ; d(1), Γ(1), ∅; …; d(l), Γ(l), ∅)

for some positive integer l and partitions d = ∑_{k=1}^{l} d(k), Γ = ∐_{k=0}^{l} Γ(k). If m^{k} = d(k) − ∑_{m} m|Γ_{m}(k)| (as in Section 3.6), this component appears with multiplicity ∏_{k} m^{k}.

4.14. Theorem (Genus 1 maps to ℙ^{1}). — Let Y be a component of D_{∞} on W_{1}(d, Γ, Δ). Fix a positive integer l and partitions d = ∑_{k=1}^{l} d(k) and Γ_{m} = ∐_{k=0}^{l} Γ_{m}(k). Then Y is a component of

Y_{1}^{a}(0, Γ(0), Δ; d(1), Γ(1), ∅; …; d(l), Γ(l), ∅),
Y_{1}^{b}(0, Γ(0), Δ; d(1), Γ(1), ∅; …; d(l), Γ(l), ∅)_{m_{1}^{1}} (for some m_{1}^{1}), or
Y_{1}^{c}(0, Γ(0), Δ; d(1), Γ(1), ∅; …; d(l), Γ(l), ∅).

If m^{k} = d(k)−∑_{m} m|Γ_{m}(k)|, the components of the first and third types appear with multiplicity ∏_{k=1}^{l} m^{k} and those of the second type appear with multiplicity m_{1}^{1}(m^{1} − m_{1}^{1}) ∏_{k=2}^{l} m^{k}.

In all cases, the multiplicity is the product of the “new ramifications” of the components not mapped to ∞.

For general g, the above argument identifies some of the components of D_{∞}, but further work is required to determine what happens when p_{a}(C(0)) > 1.

4.15. Aside: local structure near D_{∞}, and pathological behavior of \overline{M}_{g}(ℙ^{1}, d). The proofs of Propositions 4.8 and 4.11 can be refined to determine the local structure of V

near Y in both cases. As these results will not be needed, the proof is omitted.

4.16. Corollary. — Let Y be the same component as in Propositions 4.8 or 4.11. A formal neighborhood of a general point of Y in the stack V is isomorphic to

Spf ℂ[[a, b_{1}, …, b_{s}, c_{1}, …, c_{dim V−1}]]/(a = b_{1}^{m^{1}} = ⋯ = b_{s}^{m^{s}})

with D_{∞} given by (a = 0), and Y given set-theoretically by the same equation.

In particular, if gcd(m^{i}, m^{j}) > 1 for some i and j, V fails to be unibranch at a general point of Y. Similar phenomena occur in other situations ([CH] Proposition 4.8, [V3] Section 2.5, etc.), although the proofs seem unrelated.

4.17. Let \overline{M}_{g}(ℙ^{1}, d)^{o} be the closure of points corresponding to maps from irreducible genus g curves. One might hope that \overline{M}_{g}(ℙ^{1}, d)^{o} is smooth for general g and d. This is not the case. The phenomenon of Proposition 4.16 suggested the following example of a map to ℙ^{1} that can be smoothed in 2 different ways.

The dimension of \overline{M}_{4}(ℙ^{1}, 4)^{o} is 14. Consider the family Y of stable maps whose general element parametrizes a smooth genus 3 curve C(0) meeting a rational tail C(1) at two general points. The curve C(0) maps with degree 0 to ℙ^{1}, and the rational tail maps with degree 4 to ℙ^{1}, ramifying at both points of intersection with C(0).

The substack Y has dimension 13: 8 for the two-pointed genus 3 curve C(0), 1 for the image of C(0) in ℙ^{1}, and 4 for the other ramification points of C(1). Thus if Y is contained in \overline{M}_{4}(ℙ^{1}, 4)^{o}, it is a Weil divisor.

4.18. Proposition. — \overline{M}_{4}(ℙ^{1}, 4)^{o} has two smooth branches along Y, intersecting transversely.

By a similar argument, we can find a codimension 1 unibranch singularity of \overline{M}_{5}(ℙ^{1}, 5)^{o}, and singularities of \overline{M}_{8}(ℙ^{1}, 7)^{o} with several codimension 1 singular branches.

4.19. Maps of genus 1 curves to ℙ^{1}. We conclude with necessary results unrelated to the earlier part of this section.

4.20. Lemma. — Suppose π : C → ℙ^{1} is a map from a connected nodal curve of arithmetic genus 1.

(a) If π contracts no component of arithmetic genus 1, then

h^{1}(C, π^{∗}(O_{ℙ^{1}}(1))) = h^{1}(C, π^{∗}(O_{ℙ^{1}}(2))) = 0.

(b) If C has a contracted component E of arithmetic genus 1, where E intersects the rest of the components R at two points p and q (and possibly others) with π|_{R} étale at p, then h^{1}(C, π^{∗}(O_{ℙ^{1}}(2))) = 1.

By “contracting no component of C of arithmetic genus 1” we mean that all connected unions of contracted irreducible components of C have arithmetic genus 0. Proof. (a) By Serre duality, it suffices to show that

H^{0}(C, K_{C} ⊗ π^{∗}(O(−1))) = 0.

Assume otherwise that such (C, π) exists, and choose one with the fewest components, and choose a nonzero global section s of K_{C} ⊗ π^{∗}(O(−1)). If C = C′ ∪ R where R is a rational tail (intersecting C′ at one point), then s = 0 on R as

deg_{R}(K_{C} ⊗ π^{∗}(O(−1))) = −1 − deg_{π} R < 0.

Then s|_{C′} is a section of (K_{C}⊗π^{∗}(O(−1)))|_{C′} that vanishes on C′∩R. But K_{C′} = K_{C}(−C′∩R)|_{C′}, so this induces a non-zero section of K_{C′} ⊗ (π|_{C′})^{∗}(O(−1)), contradicting the minimality of the number of components. Thus C has no rational tails, and C is either an irreducible genus 1 curve or a cycle of rational curves. If C is an irreducible genus 1 curve, then C isn’t contracted by hypothesis, so K_{C} ⊗ π^{∗}(O(−1)) is negative on C as desired. If C is a cycle C_{1} ∪ ⋯ ∪ C_{s} of ℙ^{1}’s, then

deg_{C_{i}}(K_{C} ⊗ π^{∗}(O(−1))) = − deg C_{i} ≤ 0.

As one of the curves has positive degree, there are no global sections of K_{C} ⊗ π^{∗}(O(−1)). (b) The proof is essentially the same, and is omitted.

4.21. Proposition. — Suppose (C, \{p_{i}\}, π) is a stable map in \overline{M}_{1,m}(ℙ^{1}, d) satisfying

(a) C has no contracted component of arithmetic genus 1, or

(b) C has a contracted component E of arithmetic genus 1, where E intersects the rest of the components R at two points p and q (and possibly others) with π|_{R} étale at p.

Then \overline{M}_{1}(ℙ^{1}, d) is smooth of dimension 2d + m at (C, \{p_{i}\}, π).

The result is almost certainly true even without the étale condition in (b).

Proof. (a) As H^{1}(C, π^{∗}T_{ℙ^{1}}) = 0 by the previous lemma, \overline{M}_{1,m}(ℙ^{1}, d) is smooth of dimension deg π^{∗}T_{ℙ^{1}} + m = 2d + m. The argument is well-known, but for completeness we give it here.

From the exact sequence for infinitesimal deformations of stable maps (see Appendix A), we have

(5)  0 ⟶ Aut(C, {p_{i}}) ⟶ H^{0}(C, π^{∗}T_{ℙ^{1}})
⟶ Def(C, {p_{i}}, π) ⟶ Def(C, {p_{i}}) ⟶ H^{1}(C, π^{∗}T_{ℙ^{1}})
⟶ Ob(C, {p_{i}}, π) ⟶ 0

where Aut(C, {p_{i}}) (resp. Def(C, {p_{i}})) are the infinitesimal automorphisms (resp. infinitesimal deformations) of the marked curve, and Def(C, {p_{i}}, π) (resp. Ob(C, {p_{i}}, π)) are the infinitesimal deformations (resp. obstructions) of the stable map. As H^{1}(C, π^{∗}T_{ℙ^{1}}) = 0, Ob(C, {p_{i}}, π) = 0 from (5). Thus the deformations of (C, {p_{i}}, π) are unobstructed, and the dimension follows from:

dim Def(C, {p_{i}}, π) − dim Ob(C, {p_{i}}, π) = (dim Def(C, {p_{i}}) − dim Aut(C, {p_{i}}))
+(h^{0}(C, π^{∗}T_{ℙ^{1}}) − h^{1}(C, π^{∗}T_{ℙ^{1}}))
= m + 2d.

(b) For convenience (and without loss of generality) assume m = 0. As h^{1}(C, π^{∗}T_{ℙ^{1}}) = 1 (previous Lemma), our proof of (a) will not carry through. However, Def(C, {p_{i}}, π) does not surject onto Def(C, {p_{i}}) in long exact sequence (5), as it is not possible to smooth the nodes independently: one cannot smooth the node at p while preserving the other nodes even to first order. (This is well-known; one argument, due to M. Thaddeus, is to consider a stable map (C, π) in \overline{M}_{1}(ℙ^{1}, 1) and express the obstruction space 𝔼xt^{2}(\underline{Ω}_{π}, O_{C}) as the dual of H^{0}(C, F) for a certain sheaf F, [Th].) Thus the map Def(C) → H^{1}(C, π^{∗}T_{ℙ^{1}}) is not the zero map, so Def(C) surjects onto H^{1}(C, π^{∗}T_{ℙ^{1}}). Therefore Ob(C, π) = 0, so the deformations are unobstructed.

The rest of the proof is identical to that of (a)

5. Maps to ℙ^{n}

We begin with generalities about curves in projective space justifying Reductions A–C in Section 3.3.

The following proposition is a straightforward consequence of the Kleiman-Bertini theorem for stacks. (Kleiman’s original proof [K1] carries through completely in the category of Deligne-Mumford stacks.)

5.1. The Kleiman-Bertini Theorem. — Let A be a reduced irreducible substack of \overline{M}_{g,m}(ℙ^{n}, d), and let p be one of the labelled points. Then there is a Zariski-open subset U of the dual projective space (ℙ^{n})^{∗} such that for all [H′] ∈ U the intersection A ∩ ev_{p}^{∗}H′, if nonempty, is reduced of dimension dim A − 1.

Let B be a proper closed substack of A. Then there is a Zariski-open subset U′ of the dual projective space (ℙ^{n})^{∗} such that for all [H′] ∈ U′, each component of B ∩ ev_{p}^{∗}H′ is a proper closed substack of a component of A ∩ ev_{p}^{∗}H′.

5.2. Proposition. — Let A be an irreducible family of stable maps where the source curve is of constant topological type and the components are distinguished. Suppose E is a contracted genus g component with s special points (i.e. marked points or branches of nodes). If the induced map i : A → \overline{M}_{g,s} is non-constant then A is stably enumeratively irrelevant (3.5).

Proof. Each component of the universal curve over A also satisfies the hypotheses of the Proposition. Hence by replacing A with A^{(j)} for suitable j, it suffices to prove that A is enumeratively irrelevant. Suppose otherwise that A were enumeratively relevant.

If D = ev_{p_{j}}^{∗}H′ (p_{j} one of the marked points, H′ a general hyperplane), then the hypothesis also holds for each component of D. (Reason: if p is a point of \overline{M}_{g,s}, then each component of i^{∗}p is a proper substack. By Kleiman-Bertini 5.1, D contains no component of i^{∗}p, so i^{∗}p ∩ D is of codimension at least 2 in A. Thus i(D) is non-constant.)

As A is enumeratively relevant, some component of ev_{j}^{∗}H (for some j) is enumeratively relevant (and can be represented by a family also satisfying the hypotheses of the Proposition). By repeating this process dim A times, we are left with an irreducible dimension 0 stack also satisfying the hypotheses of the Proposition. This is impossible, as a map from a point to \overline{M}_{g,m} must be constant.

We next justify the Reduction steps described in Section 3.3, A and B in the next Proposition, and C in Proposition 5.5.

5.3. Proposition. — Fix d, Γ, and Δ. Let H′ be a general hyperplane in ℙ^{n}. Then:

(a) X (d, Γ, Δ ∪ {ℙ^{n}}) (resp. W(d, Γ, Δ ∪ {ℙ^{n}})) is the universal curve over X (d, Γ, Δ) (resp. W(d, Γ, Δ)).

(b) Fix m and α ∈ Γ_{m}. Suppose Γ′ is the same as Γ except Γ′_{m}^{α} = Γ_{m}^{α} ∩ H′. Then the Cartier divisor ev_{p_{m}^{α}}^{∗}H′ on X (d, Γ, Δ) (resp. W(d, Γ, Δ)) is X (d, Γ′, Δ) (resp. W(d, Γ′, Δ)).

(c) Fix α ∈ Δ. Suppose Δ′ is the same as Δ except Δ′^{α} = Δ^{α} ∩ H′. Then the Cartier divisor ev_{q^{α}}^{∗}H′ on X (d, Γ, Δ) (resp. W(d, Γ, Δ), Z(d, Δ)) is X (d, Γ, Δ′) (resp. W(d, Γ, Δ′), Z(d, Δ′)).

Proof. (a) follows from

X (d, Γ, Δ ∪ \{ℙ^{n}\}) = X (d, Γ, Δ) ×_{\overline{M}_{0,∑ |Γm|+|Δ|}(ℙ^{n},d)} \overline{M}_{0,∑ |Γ_{m}|+|Δ|+1}(ℙ^{n}, d)

(and the analogous statement for W).

(b) Clearly

X (d, Γ′, Δ) ⊂ ev_{p_{m}^{α}}^{∗}H′;

each component of X (d, Γ′, Δ) appears with multiplicity one by Kleiman-Bertini 5.1. The only other possible components of ev_{p_{m}^{α}}^{∗}H′ are those whose general point represents a map where π^{−1}H is not a union of points (i.e. contains a component of C). But such maps form a union of proper subvarieties of components of X (d, Γ, Δ), and by Kleiman-Bertini 5.1 such maps cannot form a component of ev_{p_{m}^{α}}^{∗}H′ ∩ X (d, Γ, Δ).

Replacing p_{m}^{α} with q^{α} in the previous paragraph gives a proof of (c) for X. The same arguments hold with X replaced by W, and for (c), Z.

5.4. Next, we justify Reduction C.

Let A be a general (n − 2)-plane in H. The projection p_{A} from A induces a rational map ρ_{A} : \overline{M}_{g,m}(ℙ^{n}, d) −−→ \overline{M}_{g,m}(ℙ^{1}, d), that is a morphism (of stacks) at points representing maps (C, \{p_{i}\}, π) whose image π(C) does not meet A. Let V ⊂ \overline{M}_{g,m}(ℙ^{n}, d) be the open substack corresponding to maps where π^{−1}A is a union of reduced points distinct from the m marked points {p_{i}}.

5.5. Proposition. — (a) The morphism ρ_{A} can be extended to V, where the image of a map (C, {p_{i}}, π) ∈ V is the

stable map (C̃, {p_{i}}, π_{ℙ^{1}}), with

C̃ = C ∪ C_{1} ∪ ⋯ ∪ C_{#π^{−1}A},

where C_{1}, …, C_{#π^{−1}A} are rational tails attached to C at the points of π^{−1}A,

π_{ℙ^{1}} |_{{C\π^{−1}A}}= (p_{A} ∘ π) |_{{C\π^{−1}A}}

(which extends to a morphism from all of C) and π_{ℙ^{1}} |_{C_{k}} is a degree 1 map to ℙ^{1} (1 ≤ k ≤ #π^{−1}A).

(b) If g = 0, then ρ_{A} is a smooth morphism of stacks (on V) of relative dimension (n−1)(d+1).

(c) If g = 1 and π doesn’t collapse any component of arithmetic genus 1, then ρ_{A} is a smooth morphism of stacks (at (C, {p_{i}}, π)) of relative dimension (n − 1)d.

5.6. To show that a morphism of stacks A → B is smooth at a point a ∈ A, where B is smooth and A is equidimensional, it suffices to show that the fiber is smooth at a, or equivalently that the Zariski tangent space to the fiber at a is of dimension dim A − dim B. Proof. Let ν : Bl → ℙ^{n} be the blow-up of ℙ^{n} along A, with exceptional divisor E. If [L] ∈ H_{2}(ℙ^{n}) is the class of a line, let β = dν^{∗}[L] ∈ H_{2}(Bl). Let V′ be the open substack of \overline{M}_{g,m}(Bl, β) corresponding to maps (C̃, \{p_{i}\}, π_{Bl}), where s := #π^{−1}A,

C̃ = C ∪ C_{1} ∪ ⋯ ∪ C_{s};

C_{i} ≅ ℙ^{1} (1 ≤ i ≤ s) and is mapped to E, isomorphically to fibers of ν; (π_{Bl}|_{C})^{∗}E is a union of s reduced points, disjoint from {p_{i}}; and the C_{i} are glued to C at those s points.

Let U be the universal curve over \overline{M}_{g,m}(ℙ^{n}, d) (with natural map U → ℙ^{n}). Define U′ := U ×_{ℙ^{n}} Bl; as A is a complete intersection, U′ is the blow-up of U along the pullback of A. If p ∈ V ⊂ \overline{M}_{g,m}(ℙ^{n}, d) is a closed point, U′_{p} is a curve C̃ as described in the previous paragraph. If U′|_{V} is defined as U′ ×_{\overline{M}_{g,m}(ℙ^{n},d)} V, then

\begin{matrix}U′|_{V} & → Bl \\ ↓ \\ V \end{matrix}

is a family of stable maps to Bl. (Proof: All that needs to be verified is that U′|_{V} → V is flat, which can be checked at closed points. Consider a point of π^{−1}A on U mapping to V, where the restriction of π^{−1}A to the fiber is a closed point. Let (S, 𝔫) be the local ring of this point, and (R, 𝔪) be the local ring of the image in V. Then π^{−1}A defines an ideal of S generated by 2 elements (say x, y, so (𝔪S, x, y) = 𝔫), and we can choose x and y so that neither x nor y vanishes to order 2 on the fiber (i.e. 𝔫 = (𝔪S, x) = (𝔪S, y)). Then the flatness of the blow-up of (x, y) can be checked by looking at patches. Where y ≠ 0, the patch is ψ : Spec(S[t]/(y − tx)) → Spec R. As y isn’t a 0-divisor on S/𝔪S, y − tx isn’t a 0-divisor on S[t]/𝔪S[t]. Hence by the local flatness criterion [Ma] Cor. to 22.5 p. 177, ψ is flat. The same argument holds with x and y interchanged.)

By the universal property of the moduli stack of stable maps, this induces a morphism V → \overline{M}_{g,m}(Bl, β); the image lies in V′. On the other hand, the morphism ν : Bl → ℙ^{n} induces a morphism

\overline{M}_{g,m}(Bl, β) → \overline{M}_{g,m}(ℙ^{n}, d),

and the image of V′ lies in V. These morphisms clearly commute on the level of closed points, so V ≅ V′.

Finally, the morphism m : Bl → ℙ^{1} (corresponding to projection from A) induces a morphism σ : V′ → \overline{M}_{g,m}(ℙ^{1}, d), proving (a). Let (C̃, \{p_{i}\}, π_{ℙ^{1}}) be the image of (C, \{p_{i}\}, π) as in the statement of (a).

In short, we have diagrams as follows:

\begin{aligned}V & ⟶^{∼} & V′ \\ & ρ_{A} ↘ & ↓ σ \\ & & \overline{M}_{g,m}(ℙ^{1}, d) \end{aligned}  \begin{aligned}(C, \{p_{i}\}, π) & → & (C̃, \{p_{i}\}, π_{Bl}) \\ & ρ_{A} ↘ & ↓ σ \\ & & (C̃, \{p_{i}\}, π_{ℙ^{1}}). \end{aligned}

In order to use Criterion 5.6, we compute the Zariski tangent space to the fiber of σ at a point of V′. From the induced morphism of long exact sequences for infinitesimal deformations of maps (see Appendix A), we have

Aut(C̃, {p_{i}}) → H^{0}(C, π_{Bl}^{∗}T_{Bl}) → Def(C̃, {p_{i}}, π_{Bl}) → Def(C̃, {p_{i}})
|| ↓ ↓ ||
Aut(C̃, {p_{i}}) → H^{0}(C, π_{ℙ^{1}}^{∗}T_{ℙ^{1}}) → Def(C̃, {p_{i}}, π_{ℙ^{1}}) → Def(C̃, {p_{i}})

Hence the Zariski tangent space to the fiber ker(Def(C̃, {p_{i}}, π_{Bl}) → Def(C̃, {p_{i}}, π_{ℙ^{1}})) is isomorphic to ker(H^{0}(C, π_{Bl}^{∗}T_{Bl}) → H^{0}(C, π_{ℙ^{1}}^{∗}T_{ℙ^{1}})). These are first-order deformations of the map π_{Bl} preserving the image map π_{ℙ^{1}} (keeping the source curve C̃ fixed).

As the image of the map π_{Bl}|_{C_{i}} is a fiber of ν, it may move in E, but may not move out of E (even to first order, as the degree of π_{Bl}^{∗}E on C_{i} is -1). Thus deformations of π_{Bl} in ker(H^{0}(C, π_{Bl}^{∗}T_{Bl}) → H^{0}(C, π_{ℙ^{1}}^{∗}T_{ℙ^{1}})) are naturally deformations of π_{Bl}|_{C} preserving π_{ℙ^{1}}|_{C}, with the s points of (π_{Bl}|_{C})^{∗}E required to stay on E.

These are naturally identified with the deformations of π (preserving π_{ℙ^{1}}|_{C}) where the points of π^{−1}A may not move from A. If π : C → ℙ^{n} is given by sections (s_{0}, …, s_{n}) of π^{∗}O_{ℙ^{n}}(1) (and A is given by the vanishing of the first two co-ordinates), then these are deformations of (s_{0}, …, s_{n}) keeping the sections s_{0}, s_{1} constant. Thus the Zariski tangent space to the fiber of σ is isomorphic to H^{0}(C, π^{∗}O_{ℙ^{n}}(1))^{n−1}.

The last paragraph can also be rephrased as follows. If T_{Bl}[−E] is the locally free sheaf whose sections are sections of T_{Bl}, and whose sections on E are required to have no normal component, then

0 → (ν^{∗}O_{ℙ^{n}}(1))^{n−1} → T_{Bl}[−E] → m^{∗}T_{ℙ^{1}} → 0

is easily seen to be exact. Then pull back to C and take global sections.

(b) Recall that \overline{M}_{0,m}(ℙ^{n}, d) is a smooth stack of dimension (n + 1)(d + 1) + m − 4 (see Appendix A). The Zariski tangent space to the fiber of σ has dimension

h^{0}(C, π^{∗}O_{ℙ^{n}}(1)^{n−1}) = (n − 1)h^{0}(C, π^{∗}O_{ℙ^{n}}(1))
= (n − 1)(d + 1)
= dim \overline{M}_{0,m}(ℙ^{n}, d) − dim \overline{M}_{0,m}(ℙ^{1}, d)

as desired.

(c) If no components of C of arithmetic genus 1 are collapsed by p_{A} ∘ π, then ρ_{A}(C, π) is a smooth point of \overline{M}_{1,m}(ℙ^{1}, d) by Lemma 4.21. If a component B of C of arithmetic genus 1 is collapsed by p_{A} ∘ π, then the degree of π|_{B} is at least 2 (as C has no genus 1 component contracted by π), so B contains at least 2 (reduced) points of π^{−1}A. In this case ρ_{A}(C, {p_{i}}, π) consists of a curve with a contracted elliptic component, and this elliptic component has at least two rational tails that map to ℙ^{1} with degree 1 (corresponding to the points of π^{−1}A on B). Thus by Lemma 4.21, ρ_{A}(C, π) is a smooth point of \overline{M}_{1,m}(ℙ^{1}, d) as well.

By Lemma 4.20, h^{1}(C, π^{∗}(O(1))) = 0, so h^{0}(C, π^{∗}(O(1))) = d by Riemann-Roch. The proof is then identical to the rest of (b).

5.7. Proposition. — Fix n, d, Γ = ∐_{k=0}^{l} Γ(k), Δ = ∐_{k=0}^{l} Δ(k), and D.

(a) X_{n}(d, Γ, Δ) has pure dimension

(n + 1)d + (n − 3) − ∑_{m,α∈S(Γ_{m})}(n + m − dim Γ_{m}^{α} − 2) − ∑_{α∈S(Δ)}(n − 1 − dim Δ^{α}).

The general element of each component is (a map from) a smooth curve. Also, Y_{n}(d(0), …, Δ(l)) has pure dimension dim X_{n}(d, Γ, Δ) − 1.

(b) W_{n}(d, Γ, Δ) has pure dimension

(n + 1)d − ∑_{m,α∈S(Γ_{m})}(n + m − dim Γ_{m}^{α} − 2) − ∑_{α∈S(Δ)}(n − 1 − dim Δ^{α}).

The general element of each component is (a map from) a smooth curve. Also, Y^{a} = Y_{n}^{a}(d(0), …, Δ(l)) (respectively Y^{b}, Y^{c}) has pure dimension dim W_{n}(d, Γ, Δ) − 1.

(c) Z_{n}(d, Δ)_{D} has pure dimension

(n + 1)d − ∑_{α∈S(Δ)}(n − 1 − dim Δ^{α}) − 1.

These are the dimensions one would naively expect. Proof. The proof for (a) is simpler than the proof for (b), and will be omitted for brevity.

(b) We will prove the result about dim W(d, Γ, Δ) in the special case Δ = ∅ and Γ consists of copies of H. Then the result holds in general by Reductions A and B (Proposition 5.3). In this special case, we must prove that each component of W(d, Γ, ∅) is (reduced) of dimension

(n + 1)d − ∑_{m}(m − 1)|Γ_{m}|.

Consider any point (C, {p_{m}^{α}}, π) on W(d, Γ, ∅) where no component maps to H and π collapses no component of arithmetic genus 1. The natural map W_{n}(d, Γ, ∅) −−→ W_{1}(d, Γ̃, ∅) induced by ρ_{A} : \overline{M}_{1,∑ |Γ_{m}|}(ℙ^{n}, d) −−→ \overline{M}_{1,∑ |Γ_{m}|}(ℙ^{1}, d) is smooth of relative dimension (n − 1)d at the point (C, {p_{m}^{α}}, π) by Proposition 5.5. The stack W_{1}(d, Γ̃, ∅) is reduced of dimension 2d + 1 − ∑(m − 1)|Γ_{m}| by Section 4, so W(d, Γ, ∅) has dimension

(n − 1)d + dim(W_{1}(d, Γ̃, ∅)) = (n + 1)d − ∑_{m}(m − 1)|Γ_{m}|

as desired. As the general element of W_{1}(d, Γ̃, ∅) is (a map from) a smooth curve, the same is true of W(d, Γ, ∅).

The same argument works for Y^{a}, Y^{b}, and Y^{c}, as in Section 4, it was shown that Y_{1}^{a}(d(0), …, Δ(l)), Y_{1}^{b}(d(0), …, Δ(l)), and Y_{1}^{c}(d(0), …, Δ(l)) are Weil divisors of W_{1}(d, Γ, ∅).

(c) By Proposition 5.3 (c), we may assume that Δ consists only of copies of ℙ^{n}.

It suffices to prove the result for the generically degree d! cover Z′_{n}(d, Δ)_{D} obtained by marking the points of intersection with a fixed general hyperplane H. If Γ_{1} consists of d copies of H, and Γ_{m} is empty for m > 1, then Z′_{n}(d, Δ)_{D} is a substack of W(d, Γ, Δ), and as

dim W_{n}(d, Γ, Δ) = (n + 1)d − ∑_{α}(n − 1 − dim Δ^{α}),

we wish to show that Z′_{n}(d, Δ)_{D} is a Weil divisor (in fact reduced) of the variety W_{n}(d, Γ, Δ).

Assume q^{α_{0}} appears in D with non-zero coefficient a (so D−aq^{α_{0}} is a sum of integer multiples of q^{α}, α ∈ S(Δ) \ {α_{0}}). Let W(d, Γ, Δ \ {Δ^{α_{0}}})^{o} be the open subset of W(d, Γ, Δ \ {Δ^{α_{0}}}) representing maps from smooth elliptic curves. On the universal curve over W(d, Γ, Δ\{Δ^{α_{0}}})^{o} there is a reduced divisor Z corresponding to points q such that

aq = (D − aq^{α}) − π^{∗}(O(1))

in the Picard group of the fiber. The universal curve over the stack W(d, Γ, Δ \ {Δ^{α_{0}}}) is W(d, Γ, Δ) by Proposition 5.3 (a), so by definition the closure of Z in W(d, Γ, Δ) is Z(d, Δ)_{D}.

The following proposition is completely irrelevant to the rest of the argument. It is included to ensure that we are actually counting what we might want to.

5.8. Proposition. — If n ≥ 3, and (C, {p_{m}^{α}}, {q^{α}}, π) is the stable map corresponding to a general point of a component of X (d, Γ, Δ) or W(d, Γ, Δ), then C is smooth and π is a closed immersion. Proof. By Reductions A and B (Proposition 5.3), we can assume Δ = ∅ and Γ_{m} consists of copies of H. By the previous proposition, the curve C is smooth. We need only check that π is a closed immersion. The line bundle π^{∗}O_{ℙ^{n}}(d) is very ample, so a given non-zero section s_{0} and three general sections t_{1}, t_{2}, t_{3} will separate points and tangent vectors. If π = (s_{0}, s_{1}, s_{2}, s_{3}, s_{4}, …, s_{n}) then the infinitesimal deformation (s_{0}, s_{1} + εt_{1}, s_{2} + εt_{2}, s_{3} + εt_{3}, s_{4}, …, s_{n}) will separate points and tangent vectors and still lie in X (d, Γ, Δ) (or W(d, Γ, Δ)). As (C, {p_{m}^{α}}, {q^{α}}, π) corresponds to a general point in X (d, Γ, Δ) (or W(d, Γ, Δ)), the map π must be a closed immersion at this point.

We will need to avoid the locus on W(d, Γ, Δ) where an elliptic component is contracted. The following lemma identifies maps which could lie in this locus.

5.9. Lemma. — Let C be a complete reduced nodal curve of arithmetic genus 1, and let π : C → ℙ^{n}. Assume (C, π) can be smoothed. If B is a connected union of contracted components of C of arithmetic genus 1, intersecting \overline{C \ B} in k points, and T_{1}, …, T_{k} are the tangent vectors to \overline{C \ B} at those points, then the vectors \{π(T_{i})\}_{i=1}^{k} in T_{π(B)}ℙ^{n} are linearly dependent.

More generally, this result will hold whenever π is a map to an n-dimensional variety X. Proof. Let Δ be a smooth curve parametrizing maps (C_{t}, π) (with total family (C, π)) to ℙ^{n}, with (C_{0}, π) = (C, π) and general member a map from a smooth curve. Thus the following diagram commutes.

C ⟶^{π} ℙ^{n} × Δ
↘  ↙

Δ

There is an open neighborhood U of B ⊂ C such that π |_{U\B} is an immersion. Thus π factors through a family C′ that is the same as C except B is contracted. Let π′ be the contraction π′ : C → C′. The family C′ is also flat, and its general fiber has genus 1. The central fiber is a union of rational curves, at most nodal away from the image of B. If the images of T_{1}, …, T_{k} in C′_{0} are independent, the reduced fiber above 0 would have arithmetic genus 0, so the central fiber (reduced away from the image of B) would have arithmetic genus at most zero, contradicting the constancy of arithmetic genus in flat families. Thus the images of T_{1}, …, T_{k} in T_{π′(B)}C′_{0} must be dependent, and hence their images in T_{π(B)}ℙ^{n} must be dependent as well.

It is likely that every stable map of the form described in the above lemma can be smoothed, which would suggest (via a dimension estimate) that when k ≤ n+1 those maps with a collapsed elliptic component intersecting k noncontracted components (with linearly dependent images of tangent vectors) form a Weil divisor of W(d, Γ, Δ). Because of the moduli of \overline{M}_{1,k}, none of these divisors would be enumeratively relevant. Thus the following result is not surprising.

5.10. Proposition. — If W′ is an irreducible substack of W(d, Γ, Δ) whose general map (C, {p_{m}^{α}}, {q^{α}}, π) has a contracted elliptic component E (or more generally a contracted connected union of components of arithmetic genus 1) and W′ is of codimension 1, then W′ is enumeratively irrelevant. Proof. We show more generally that W′ is stably enumeratively irrelevant (3.5), as this property behaves well with respect to Reductions A and B (Proposition 5.3). By these reductions, we may assume Δ = ∅ and Γ consists of copies of H. Restrict to the open substack of W where the source curve has constant topological type. Take an étale cover to distinguish the components. Let (C, {p_{m}^{α}}, π) be a general point of W′. Suppose E has s special points (markings or intersections with noncontracted components) including k intersections with noncontracted components.

If E (with the s special points) has moduli (i.e. the induced map W′ → \overline{M}_{1,s} is nonconstant), the family is stably enumeratively irrelevant by Proposition 5.2. Assume that this is not the case.

If s ≥ 3, replace E by a rational R = ℙ^{1}, with s fixed special points (i.e. constant in \overline{M}_{0,s}), to obtain a new stable map (C′, {p_{m}^{α}}, π′) ∈ X (d, Γ, ∅). The family of such (C′, π′) forms an étale cover of a subtack X′ of X (d, Γ, ∅), and X′ is contained in X′′ where in the latter we don’t impose the dependence of tangent vectors required by the previous lemma.

Then X′′ has codimension at least (s − 3) + 1 in X (d, Γ, ∅) (s − 3 from the moduli of \overline{M}_{0,s}, 1 from smoothing of the curve). The previous lemma imposes an additional max(n + 1 − k, 0) conditions, which are independent as the rational curves intersecting R can move freely under automorphisms of ℙ^{n} preserving H. Thus the codimension of X′ in X (d, Γ, ∅) is at least n−1+(s−k) ≥ n−1. But dim X (d, Γ, ∅)−dim W(d, Γ, ∅) = n−3, so dim W′ ≤ dim W(d, Γ, ∅)− 2 = dim W′ − 1, giving a contradiction.

The s = 2 case is the same, except E is replaced by nothing. (This is perhaps most quickly said by replacing E by a rational R and then stabilizing.) The same argument carries through.

Otherwise, k = s = 1. Then X′′ can be identified with the substack of X (d, Γ, {ℙ^{n}}) where the corresponding map π : (C, {p_{m}^{α}}, q) → ℙ^{n} is singular at q. As the singularity requirement imposes n conditions,

dim W′ ≤ dim X (d, Γ, {ℙ^{n}}) − n
= dim X (d, Γ, ∅) + 1 − n
= (dim W(d, Γ, ∅) + n − 3) + 1 − n
= dim W(d, Γ, ∅) − 2
= dim W′ − 1

giving a contradiction once again. Hence W′ is enumeratively irrelevant.

6. Components of D_{H}

Fix d and general linear spaces Γ and Δ (as in the definition of X (d, Γ, Δ) and W(d, Γ, Δ)). Let q be the marked point corresponding to one of the linear spaces Δ^{β} in Δ.

Let D_{H} = ev_{q}^{∗}H = {π(q) ∈ H} be the Cartier divisor on X (d, Γ, Δ) or W(d, Γ, Δ) that corresponds to requiring q to lie on H. The components of D_{H} on X (d, Γ, Δ) are given by the following result.

6.1. Theorem. — If Γ and Δ are general, each component of D_{H} (as a divisor on X (d, Γ, Δ)) is a component of

Y(d(0), Γ(0), Δ(0); …; d(l), Γ(l), Δ(l))

for some l, d(0), …, Δ(l), with d = ∑_{k=0}^{l} d(k), Γ = ∐_{k=0}^{l} Γ(k), Δ = ∐_{k=0}^{l} Δ(k), Δ^{β} ∈ Δ(0). This component appears with multiplicity ∏ m^{k} (with m^{k} as defined in Section 3.6). Proof. By Reductions A and B (Proposition 5.3), we may assume that Γ_{m} consists only of H’s for all m, and that Δ = {ℙ^{n}} (with Δ^{β} = ℙ^{n}). With these assumptions, the result becomes much simpler. The stack X (d, Γ, {ℙ^{n}}) is the universal curve over X (d, Γ, ∅), and we are asking which points of the universal curve lie in π^{−1}H.

Let (C, {p_{m}^{α}}, q, π) be the stable map corresponding to a general point of a component of D_{H}. Choose a general (n − 2)-plane A in H. By Kleiman-Bertini 5.1, the set π^{−1}A is a union of reduced points on C, so by Proposition 5.5 ρ_{A} is smooth (as a morphism of stacks) at the point representing (C, {p_{m}^{α}}, q, π). As a set, D_{H} contains the entire fiber of ρ_{A} above ρ_{A}(C, {p_{m}^{α}}, q, π), so ρ_{A}(D_{H}) is a Weil divisor on X_{1}(d, Γ̃, {ℙ^{1}}) that is a component of ev_{q}^{∗}∞ = {π(q) = ∞} where ∞ = p_{A}(H). By Theorem 4.13, the curve C is a union of irreducible components C(0)∪⋯∪C(l′) with ρ_{A} ∘ π(C(0)) = ∞ (i.e. π(C(0)) ⊂ H), C(0) meets C(k), and the marked points split up among the components: Γ̃ = ∐ Γ̃(k). If d(0) = deg π|_{C(0)}, then d(0) of the curves C(1), …, C(l′) are rational tails that are collapsed to the d(0) points of C(0)∩A; they contain no marked points. Let l = l′ − d(0). Also, Δ(k) = ∅ for k > 0, as the only incidence condition in Δ was q ∈ Δ^{β}, and q ∈ C(0).

Therefore this component of D_{H} is contained in

Y = Y(d(0), Γ(0), Δ(0); …; d(l), Γ(l), Δ(l)).

But dim Y = dim X (d, Γ, Δ) − 1 (by Proposition 5.7), so the set-theoretic result follows. For the multiplicity result, pull back the multiplicity statement of Theorem 4.13 along the smooth morphism ρ_{A}.

For enumerative calculations, we need only consider enumeratively relevant components. With this in mind, we restate Theorem 6.1 in language reminiscent of [CH] and [V1]. Let ϕ be the isomorphism of Remark 3.9. The following statement will be more convenient for computation.

6.2. Theorem. — If Γ and Δ are general, then each enumeratively relevant component of D_{H} (as a divisor on X (d, Γ, Δ)) is one of the following. (I) A component of ϕ(X (d, Γ′, Δ′)), where, for some m_{0} and Δ^{α_{0}} ∈ Γ_{m_{0}},

• Δ′ = Δ \ {Δ^{β}}, and

• Γ′ is the same as Γ except Γ_{m_{0}}^{α_{0}} is replaced by Γ_{m_{0}}^{α_{0}} ∩ Δ^{β}.

The component appears with multiplicity m_{0}. (See Remark 3.9 for the definition of ϕ.) (II) A component of Y(d(0), Γ(0), Δ(0); …; d(l), Γ(l), Δ(l)) for some l, d(0), …, Δ(l), with

d = ∑_{k=0}^{l} d(k), Γ = ∐_{k=0}^{l} Γ(k), Δ = ∐_{k=0}^{l} Δ(k), Δ^{β} ∈ Γ(0), and d(0) > 0. The component appears with multiplicity ∏ m^{k} (with m^{k} as defined in Section 3.6).

Call these components Type I components and Type II components respectively. Proof. Consider a component Y of D_{H} that is not a Type II component (so d(0) = 0). Let (C(0) ∪ ⋯ ∪ C(l), {p_{m}^{α}}, {q}, π) be the stable map corresponding to a general point of Y. The curve C(0) has at least 3 special points: q, one of {p_{m}^{α}} (call it p_{m_{0}}^{α_{0}}), and C(0) ∩ C(1). If C(0) had more than 3 special points, then the component would not be enumeratively relevant, by Proposition 5.2. Thus l = 1, and Y is a Type I component.

An analogous result is true in genus 1.

6.3. Theorem. — If Γ and Δ are general, each enumeratively relevant component of D_{H} (as a divisor on W(d, Γ, Δ)) is one of the following.

(I) A component of ϕ(W(d, Γ′, Δ′)), where, for some m_{0} and Δ^{α_{0}} ∈ Γ_{m_{0}}, • Δ′ = Δ \ {Δ^{β}}, and

• Γ′ is the same as Γ except Γ_{m_{0}}^{α_{0}} is replaced by Γ_{m_{0}}^{α_{0}} ∩ Δ^{β}.

The component appears with multiplicity m_{0}. (II) A component of Y^{a}(d(0), …, Δ(l)), Y^{b}(d(0), …, Δ(l))_{m_{1}^{1}}, or Y^{c}(d(0), …, Δ(l)) for some

l, d(0), …, Δ(l), with d = ∑_{k=0}^{l} d(k), Γ = ∐_{k=0}^{l} Γ(k), Δ = ∐_{k=0}^{l} Δ(k), Δ^{β} ∈ Γ(0), and d(0) > 0. The component appears with multiplicity ∏ m^{k} in cases Y^{a} and Y^{c}, and m_{1}^{1}(m^{1} − m_{1}^{1}) ∏_{k>1} m^{k} in case Y^{b}.

Call the components of (I) Type I components, and call the three types of components of (II) Type IIa, IIb, and IIc components respectively. Proof. As in the proof of Theorem 6.1, we assume that Γ consists of copies of H, and Δ = {ℙ^{n}} (and Δ^{β} = ℙ^{n}). We consider the map ρ_{A} : W_{n}(d, Γ, {ℙ^{n}}) −−→ W_{1}(d, Γ̃, {ℙ^{1}}) near a general point (C, {p_{m}^{α}}, {q}, π) of a component of D_{H}. As a set, D_{H} contains the entire fiber of ρ_{A} above ρ_{A}(C, π), so ρ_{A}(D_{H}) is a Weil divisor on W_{1}(d, Γ̃, {ℙ^{1}}) that is a component of ev_{q}^{∗}∞ = {π(q) = ∞}. By Theorem 4.14, the curve C is a union of irreducible components C(0)∪⋯∪C(l′) with ρ_{A} ∘π(C(0)) = ∞ (i.e. C(0) ⊂ π^{−1}H), C(0)∩C(k) ≠ ϕ, and the marked points split up among the components: Γ̃ = ∐ Γ̃(k). If d(0) = deg π|_{C(0)}, then d(0) of the curves C(1), …, C(l′) are rational tails that are collapsed to the d(0) points of C(0) ∩ A; they contain no marked points. Let l = l′ − d(0). Also, Δ(k) = ∅ for k > 0, as the only incidence condition was q ∈ Δ^{β}, and q ∈ C(0).

Case d(0) > 0. By Theorem 4.14, the component Y is contained in

Y^{a}(d(0), …, Δ(l)), Y^{b}(d(0), …, Δ(l)), or Y^{c}(d(0), …, Δ(l)).

As the dimensions of each of these three is dim W(d, Γ, Δ) − 1 = dim Y, Y must be a Type II component as described in the statement of the theorem.

Case d(0) = 0. By the same argument as in the proof of Theorem 6.2, Y is a Type I component.

The multiplicity proof is the same as in Theorem 6.2; pull back the analogous statement in Theorem 4.14 along the smooth morphism ρ_{A}.

7. Recursive formulas

7.1. The enumerative geometry of X and W. Theorems 6.2 and 6.3 express the enumerative geometry of stacks of the form X (d(0), …, Δ(l)) or W(d(0), …, Δ(l)) in terms of the enumerative geometry of stacks of dimension 1 less, of the form X, W, Y, Y^{a}, Y^{b}, or Y^{c}.

7.2. The enumerative geometry of Y from that of X, and Y^{a} and Y^{c} from W and X.

L_{0} H  L_{1}

Figure 10. How many (L_{0}, L_{1}) satisfy the desired conditions?

Counting maps in Y(d(0), …, Δ(l)) (or Y^{a}(d(0), …, Δ(l)) or Y^{c}(d(0), …, Δ(l))) corresponds to counting maps of pointed curves C(0), …, C(l) where the images of certain points are required to coincide. This is done by the classical method of “splitting the diagonal”.

The method is best understood through an example. Fix a hyperplane H ⊂ ℙ^{4}. In ℙ^{4} the number of ordered pairs of lines (L_{0}, L_{1}) consisting of lines L_{0} ⊂ H and L_{1} ⊂ ℙ^{4}, with L_{0} intersecting 3 fixed general lines a_{1}, a_{2}, a_{3} in H, L_{1} intersecting 5 fixed general 2-planes b_{1}, …, b_{5} in ℙ^{4}, and L_{0} intersecting L_{1} (see Figure 10), can be determined as follows.

There is a one-parameter family of lines L_{0} in H intersecting the general lines a_{1}, a_{2}, a_{3}. This family sweeps out a surface S ⊂ H of some degree d_{0}. The degree d_{0} is the number of lines intersecting the lines a_{1}, a_{2}, and a_{3} and another general line in H, so this is

#X_{3}(d = 1, Γ_{1} = {plane}, Δ = {a_{1}, a_{2}, a_{3}, another line}).

There is also a one-parameter family of lines L_{1} intersecting the general 2-planes b_{1}, …, b_{5}, and the intersection point of such L_{1} with H sweeps out a curve C ⊂ H of some degree d_{1}. The degree d_{1} is the number of lines intersecting the 2-planes b_{1}, …, b_{5} in ℙ^{4} and another general 2-plane in H. Thus

d_{1} = #X_{4}(d = 1, Γ = {plane in H}, Δ = {b_{1}, b_{2}, b_{3}, b_{4}, b_{5}}).

The answer we seek is #(C ∩ S) = d_{0}d_{1}.

We count #Y_{n}(d(0), …, Δ(l)) in general as follows. Choose general linear spaces γ(1), …, γ(l) in H of appropriate dimensions such that X_{n}(d(k), Γ′(k), Δ(k)) is finite, where Γ′(k) is Γ(k) with the additional condition of γ(k) in Γ_{m^{k}}(k). Let Γ′_{1}(0) consist of d(0) copies of a hyperplane H′ in ℙ^{n−1}, and Γ′_{m}(0) = ∅ for m > 1. Choose general linear spaces δ(1), …, δ(l) in H of dimension dim δ(k) = (n−1)−dim γ(k) (“dual to γ(k)”). Let Δ′(0) = Δ(0) ∐{δ(k)}_{k=1}^{l} ∐ Γ(0). Then X_{n−1}(d(0), Γ′(0), Δ′(0)) is finite as well. With these definitions,

#Y_{n}(d(0), …, Δ(l)) = \frac{#X_{n−1}(d(0), Γ′(0), Δ′(0))}{d(0)!} ∏_{k=1}^{l} #X_{n}(d(k), Γ′(k), Δ(k)).

The d(0)! is included to account for the possible labellings of the intersection points of a degree d(0) curve C(0) in H with a fixed general hyperplane H′ of H.

7.3. We sketch an argument to show that this gives a correct count. If Δ′′(0) is the same as Δ(0) ∐ Γ(0) but with an extra l copies of H, there is a natural map

ψ : X_{n−1}(d(0), Γ′_{1}(0), Δ′′(0)) → H^{l}

that is the product of the evaluation maps on the “extra l points”. If Γ′′(k) (k > 0) is the same as Γ(k) with an additional copy of H in Γ′′_{m^{k}}(k), there is a natural map

ψ_{k} : X_{n}(d(k), Γ′′(k), Δ(k)) → H

that is the evaluation map on the “additional point”. As the linear spaces in Γ and Δ are chosen generally, by Kleiman-Bertini 5.1 (with the subgroup of Aut(ℙ^{n}) preserving H, acting on ψ_{k}), the image of ψ meets the product of the images of ψ_{k} transversely. The transversality checks for the cases Y^{a}, Y^{b}, Y^{c} below are similar, and omitted.

7.4. Similar formulas (with the same justification) hold for Y^{a} and Y^{c}.

#Y_{n}^{c}(d(0), …, Δ(l)) = #Z_{n−1}(d(0), Δ′(0))_{∑_{m,α} mp_{m}^{α}(0)−∑_{k=1}^{l} m^{k}r^{k}} ∏_{k=1}^{l} #X_{n}(d(k), Γ′(k), Δ(k))

where Γ′ and Δ′ are defined in the same way as for Y.

# Y_{n}^{a}(d(0), …, Δ(l)) =

\frac{#X_{n−1}(d(0), Γ′(0), Δ′(0))}{d(0)!}#W_{n}(d(1), Γ′(1), Δ(1)) ∏_{k=2}^{l} #X_{n}(d(k), Γ′(k), Δ(k))

where Γ′ and Δ′ is defined in the same way, except γ(1) is chosen so that W_{n}(d(1), Γ′(1), Δ(1)) is finite.

7.5. The enumerative geometry of Y^{b} from that of X. Counting points of Y^{b} is similar. For convenience, let Ỹ^{b} correspond to the slightly different problem, where we mark one of the points of C(0) ∩ C(1) where C(1) intersects H with multiplicity m_{1}^{1}; this will avoid a factor of 1/2 that would otherwise arise if m_{1}^{1} = m_{2}^{1}. The (mild) additional complexity comes from requiring the curves C(0) and C(1) to intersect twice (at marked points of each curve, on H), so a natural object of study is the blow-up of H × H along the diagonal Δ, Bl_{Δ} H × H.

But when n = 2, the situation is simpler. The curve C(0) is H, and C(1) will always intersect it. In this case, for C(0) ∪ ⋯ ∪ C(l) to be determined by the incidence conditions (up to a finite number of possibilities), each of C(1), …, C(l) must also be determined (up to a finite number). The analogous formula to those of the previous subsection is

#Ỹ_{2}^{b}(d(0); …; Δ(l))_{m_{1}^{1}} = ∏_{k=2}^{l} #X_{2}(d(k), Γ′(k), Δ(k))

where for k > 1, Γ′(k) is the same as Γ(k) with the additional condition {point} in Γ_{m^{k}}(k), and Γ′(1) is the same as Γ(1) with additional conditions {point} in Γ_{m_{1}^{1}}(1) and Γ_{m_{2}^{1}}(1). This formula agrees with Theorem 1.3 of [CH].

We now calculate #Ỹ^{b} when n = 3; the same method works for n > 3. As an illustration of the method, consider the following enumerative problem.

Fix seven general lines L_{1}, …, L_{7} in ℙ^{3} and a point p on a hyperplane H. How many pairs of curves (C(0), C(1)) are there with C(0) a line in H through p and C(1) a conic intersecting L_{1}, …, L_{7} and intersecting C(0) at two distinct points, where the intersections are labelled a_{1} and a_{2}?

The answer to this enumerative problem is by definition

#Ỹ^{b} = #Ỹ_{3}^{b}(1, Γ(0) = {p}, ∅; 2, ∅, Δ = {L_{1}, …, L_{7}}).

The space of lines in H passing through p is one-dimensional, and thus defines a threedimensional locus in Bl_{Δ} H × H. The space of conics in ℙ^{3} passing through 7 general lines is one-dimensional, and thus defines a one-dimensional locus in Bl_{Δ} H × H parametrizing the points of intersection of the conic with H. Then #Ỹ^{b} is the intersection of these two classes.

Let h_{i} be the class (in the Chow group) of the hyperplane on the i^{th} factor of Bl_{Δ} H × H (i = 1, 2), and let e be the class of the exceptional divisor. Then the Chow ring of Bl_{Δ} H × H is generated (as a ℤ-module) by the classes listed below with the relations

h_{1}^{3} = 0, h_{2}^{3} = 0, e^{2} = 3h_{1}e − h_{1}^{2} − h_{1}h_{2} − h_{2}^{2}.
Codimension  Classes
0  1
1  h_{1}, h_{2}, e
2  h_{1}^{2}, h_{1}h_{2}, h_{2}^{2}, h_{1}e = h_{2}e
3  h_{1}^{2}h_{2}, h_{1}h_{2}^{2}, h_{1}^{2}e = h_{1}h_{2}e = h_{2}^{2}e
4  h_{1}^{2}h_{2}^{2}

Let the image of possible pairs of points on C(0) be the class C(0) = α(h_{1} + h_{2}) + βe in A^{3}(Bl_{Δ} H × H). Then C(0) · h_{1}^{2}h_{2} = α and C(0) · eh_{1}^{2} = −β. But C(0) · h_{1}^{2}h_{2} is the number of lines in H passing through p (class C(0)) and another fixed point (class h_{1}^{2}) with a marked point on a fixed general line (class h_{2}), so α = 1. Also, C(0) · eh_{1}^{2} is the number of lines in the plane through p (class C(0)) and another fixed point q (class h_{1}^{2}) with a marked point mapping to q, so β = −1. Thus C(0) = h_{1} + h_{2} − e.

Let the image of possible pairs of points on C(1) ∩ H be the class C(1) = γ(h_{1}^{2}h_{2} + h_{1}h_{2}^{2}) + δh_{1}h_{2}e. Then γ = C(1) · h_{1} is the number of conics in ℙ^{3} through 7 general lines in ℙ^{3} and a general line in H, which is 92 from Section 2.2. Also, −δ = C(1) · e counts the number of conics in ℙ^{3} through 7 general lines in ℙ^{3} and tangent to H, which is 116 (which can be inductively computed using the methods of this article). Thus C(1) = 92(h_{1}^{2}h_{2} + h_{1}h_{2}^{2}) − 116eh_{1}h_{2}.

Finally, the answer to the enumerative problem is

#Ỹ^{b} = (h_{1} + h_{2} − e) (92(h_{1}^{2}h_{2} + h_{1}h_{2}^{2}) − 116eh_{1}h_{2}) = 92 + 92 − 116 = 68.

When n = 3 in general, there are three cases to consider. Let C = C(0) ∪ C(1) ∪ ⋯ ∪ C(l) as usual.

Case i). If the conditions Γ(1) and Δ(1) specify C(1) up to a finite number of possibilities, then #Ỹ^{b}(d(0), …, Δ(l)) is:

#X (d(1), Γ′(1), Δ(1)) · #Y (d(0), Γ′(0), Δ(0); d(2), Γ(2), Δ(2); …; d(l), Γ(l), Δ(l))

where

• Γ′(1) is the same as Γ(1) with the additional conditions {H} in Γ_{m_{1}^{1}}(1) and Γ_{m_{2}^{1}}(1) (the curve C(1) intersects H at two points a_{1} and a_{2} with multiplicity m_{1}^{1} and m_{2}^{1} respectively; these will be the intersections with C(0)), and

• Γ′(0) is the same as Γ(0) with the additional two conditions {point} and {another point} in Γ′_{1}(0) (the curve C(0) must pass through two points of intersection a_{1} and a_{2} of C(1) with H).

Case ii). If the conditions Γ(1) and Δ(1) specify C(1) up to a one-parameter family, we are in the same situation as in the enumerative problem above. Then #Ỹ^{b}(d(0), …; Δ(l)) is

(d(0) (#X (d(1), Γ′, Δ(1)) + d(0)#X (d(1), Γ′′, Δ(1)))

−#X (d(1), Γ′′′, Δ(1))) · #Y(d(0), Γ′(0), Δ(0); d(2), …, Δ(l))

where

#  # #  #  # # −^{6}  ^{6} ×  1 58 74 # #  #  #  1 × =  ×^{8}^{8}^{7}^{6} # #  184  184  1 68 16  +^{7}  ^{7}  ^{7} − 92  92  116

Figure 11. Calculating #Ỹ^{b}: A pictorial example

• Γ′ is the same as Γ(1) with additional conditions {line} in Γ′_{m_{1}^{1}} and {H} in Γ′_{m_{2}^{1}} (the curve C(1) intersects H with multiplicity m_{1}^{1} at a_{1} along a fixed general line and with multiplicity m_{2}^{1} at a_{2} at another point of H),

• Γ′′ is the same as Γ(1) with additional conditions {H} in Γ′′_{m_{1}^{1}} and {line} in Γ′′_{m_{2}^{1}} (the curve C(1) intersects H with multiplicity m_{2}^{1} at a_{2} along a fixed general line and with multiplicity m_{1}^{1} at a_{1} at another point of H),

• Γ′′′ is the same as Γ(1) with the additional condition {H} in Γ′′′_{m^{1}} (the points a_{1} and a_{2} on the curve C(1) coincide, and C(1) is required to intersect H at this point with multiplicity m^{1} = m_{1}^{1} + m_{2}^{1}),

• Γ′(0) is the same as Γ(0) with the additional condition {point} in Γ′_{1}(0) (the curve C(0) is additionally required to pass through a fixed point in H).

Case iii). If the incidence conditions on C(0) ∪ C(2) ∪ ⋯ ∪ C(l) specify the union of these curves up to a finite number of possibilities (and the incidence conditions on C(1) specify C(1) up to a two-parameter family), a similar argument shows that #Ỹ^{b}(d(0), …, Δ(l)) is

(d(0)#X (d(1), Γ′(1), Δ(1)) − #X (d(1), Γ′′(1), Δ(1))) #Y (d(0), Γ(0), Δ(0); d(2), …, Δ(l))

where

• Γ′(1) is the same as Γ(1) with the additional conditions {line} in Γ′_{m_{1}^{1}}(1) and {another line} in Γ′_{m_{2}^{1}}(1)

• Γ′′(1) is the same as Γ(1) with the additional condition {line} in Γ′′_{m^{1}}(1).

These three cases are illustrated pictorially in Figure 11 for the special case of conics in ℙ^{3} intersecting a line in H at two points, with the entire configuration required to intersect 8 general lines in ℙ^{3}. One of the intersection points of the conic with H is marked with an “×” to remind the reader of the marking a_{1}. The distribution of the line conditions (e.g. the number of line conditions on the conic) is indicated by a small number. The bigger number beside each picture is the actual solution to the enumerative problem corresponding to the picture. For example, there are 116 conics in ℙ^{3} tangent to a general hyperplane H intersecting 7 general lines.

7.6. In ℙ^{n} (n > 3), the procedure is the same, but the Chow ring of Bl_{Δ}(H × H) is more complicated, so there are more cases.

7.7. The enumerative geometry of Z from that of X and W. For the purposes of this section, ^{d}Γ is defined by: ^{d}Γ_{1} contains of d copies of H, and ^{d}Γ_{k} = ∅ for k > 1.

We will use intersection theory on elliptic fibrations over a curve; the Chow ring modulo algebraic or numerical equivalence will suffice. Let F be an elliptic fibration over a smooth curve with smooth total space, whose fibers are smooth elliptic curves, except for a finite number of fibers which are irreducible nodal elliptic curves. Let F be the class of a fiber and K_{F} the (class of the) relative dualizing sheaf. The self-intersection of a section is independent of the choice of section. (Proof: K_{F} restricted to the generic fiber is trivial, so K_{F} is a sum of fibers. Let S_{1}, S_{2} be two sections. Using adjunction, S_{1}^{2} + K_{F} · S_{1} = (K_{F} + S_{1}) · S_{1} = 0, so S_{1}^{2} = −K_{F} · S_{1} = −K_{F} · S_{2} = S_{2}^{2}.)

For convenience, call the self-intersection of a section S^{2}. The parenthetical proof above shows that K_{F} = −S^{2}F.

7.8. Proposition. — Let S be a section, and C a class on F such that S = C on the general fiber. Then S = C + (\frac{S^{2}−C^{2}}{2} )F. Proof. As all fibers are irreducible, S = C + kF for some k. By adjunction,

0 = S · (K_{F} + S) = (C + kF)(C + (k − S^{2})F) = C^{2} + 2k − S^{2}.

Hence k = (S^{2} − C^{2})/2.

If the dimension of Z_{n}(d, Δ)_{∑ m^{β}q^{β}} is 0, then consider the universal family over the curve parametrizing maps to ℙ^{n} with only the incidence conditions of Δ (i.e. no divisorial condition). This is W_{n}(d,^{d}Γ, Δ) modulo the symmetric group S_{d} (to forget the d markings of intersection with H). The general point of W_{n}(d,^{d}Γ, Δ) represents a smooth elliptic curve. The remaining points represent curves that are either irreducible and rational, or elliptic with rational tails. (This can be proved by simple dimension counts on W_{1}(d,^{d}Γ, Δ).) Normalize the base (which will normalize the family), and blow down (-1)-curves in fibers.

Call the resulting family F. It is straightforward to check that the total space of F is smooth. (To check that the surface is smooth at the nodes of the family, show that the firstorder deformations of such maps surject onto the deformation space of the node, using the long exact deformation sequence of Appendix A and the fact that H^{1}(C, π^{∗}T_{X}) = 0. By appropriate application of Kleiman-Bertini 5.1, one checks that the point remains smooth once one requires marked points to lie on various general linear spaces.)

The curves blown down come from maps from nodal curves C(0) ∪ C(1), where C(0) is rational and C(1) is elliptic. Let H be the pullback of a hyperplane to F, and let Q^{β} be the section given by q^{β}.

7.9. Theorem. — Let D = H − ∑ m^{β}Q^{β}. Then #Z_{n}(d, Δ)_{∑ m^{β}q^{β}} = S^{2} − D^{2}/2. Proof. Let Q be any section. Let S be the section given by Q + H − ∑ m^{β}q^{β} in the Picard group of the generic fiber. Then

#Z_{n}(d, Δ)_{∑ m^{β}q^{β}} = S · Q.

(The sections S and Q intersect transversely from the proof of Proposition 5.7(c).) By the previous proposition, as S^{2} = Q^{2},

S = Q + D + (\frac{S^{2} − (Q + D)^{2}}{2} ) F
so S · Q = (Q + D + (\frac{S^{2} − (Q + D)^{2}}{2} ) F ) · Q
= Q^{2} + D · Q + \frac{S^{2} − Q^{2} − D^{2}}{2} − D · Q
= S^{2} − D^{2}/2.

To calculate

#Z_{n}(d, Δ)_{∑ m^{β}q^{β}} = S^{2} − (H − ∑ m^{β}Q^{β})^{2}/2,

we need to calculate H^{2}, H · Q^{β}, and Q^{β} · Q^{β′}, and these correspond to simpler enumerative problems.

If β ≠ β′, Q^{β} could intersect Q^{β′} in two ways. If dim Δ^{β} + dim Δ^{β′} ≥ n, the elliptic curve could pass through Δ^{β} ∩ Δ^{β′}, which will happen

#W(d,^{d}Γ, Δ \ {Δ^{β}, Δ^{β′}} ∐{Δ^{β} ∩ Δ^{β′}})/d! (6)

times. Or the curve could break into two intersecting components, one rational containing Q^{β} and Q^{β′} (which will be blown down in the construction of F, with images of Q^{β} and Q^{β′} meeting transversely), and the other smooth elliptic. This will happen

(7)  ∑_{\begin{matrix}d(0)+d(1)=d \\ Δ(0) ∐ Δ(1)=Δ \\ Δ^{β},Δ^{β′} \end{matrix}}(d(0)d(1))^{δ_{n,2}} (\frac{#X (d(0),^{d(0)}Γ, Δ(0))}{d(0)!}) (\frac{#W(d(1),^{d(1)}Γ, Δ(1))}{d(1)!})

times. The factor of (d(0)d(1))^{δ_{n,2}} corresponds to the fact when n = 2, π(C) is a plane curve, and the node of C could map to any node of the plane curve π(C). Transversality in both cases is simple to check, and both possibilities are of the right dimension. Thus Q^{β} · Q^{β′} is the sum of (6) and (7).

To determine H · Q^{β} on F, fix a general hyperplane h in ℙ^{n}, and let H be its pullback to the fibration F. Then H is a multisection of the elliptic fibration. The cycle H could intersect Q^{β} in two ways. Either π(q^{β}) ∈ h ∩ Δ^{β} — which will happen

#W (d,^{d}Γ, Δ \ {Δ^{β}} ∐{h ∩ Δ^{β}})/d! (8)

times — or the curve breaks into two pieces, one rational containing a point of h and q^{β}, which will happen

(9) ∑_{\begin{matrix}d(0)+d(1)=d \\ Δ(0) ∐ Δ(1)=Δ \\ Δ^{β}∈Δ(0) \end{matrix}}(d(0)d(1))^{δ_{n,2}}d(0) (\frac{#X (d(0),^{d(0)}Γ, Δ(0))}{d(0)!}) (\frac{#W(d(1),^{d(1)}Γ, Δ(1))}{d(1)!})

times. (The second d(0) in the formula comes from the choice of point of h on the degree d(0) rational component.) Thus H · Q^{β} is the sum of (8) and (9).

To determine H^{2}, fix a second general hyperplane h′ in ℙ^{n}, and let H′ be its pullback to F. Once again, H could intersect H′ in two ways depending on if the curve passes through h ∩ h′, or if the curve breaks into two pieces. The first case happens

\frac{#W(d,^{d}Γ, Δ ∐\{h ∩ h′\})}{d!} (10)

times, and the second happens

∑_{\begin{matrix}d(0)+d(1)=d \\ Δ(0) ∐ Δ(1)=Δ \end{matrix}}(d(0)d(1))^{δ_{n,2}}d(0)^{2} (\frac{#X (d(0),^{d(0)}Γ, Δ(0))}{d(0)!}) (\frac{#W(d(1),^{d(1)}Γ, Δ(1))}{d(1)!}) (11)

times. Hence H^{2} is the sum of (10) and (11).

The self-intersection of a section S^{2} (= (Q^{β})^{2}) can be calculated as follows. We can calculate H · Q^{β}, so if we can evaluate (H − Q^{β}) · Q^{β} then we can find S^{2} = (Q^{β})^{2}. Fix a general hyperplane h containing Δ^{β}, and let (H − Q^{β}) be the multisection that is the pullback of h to F, minus the section Q^{β}. The cycle (H − Q^{β}) intersects Q^{β} if the curve is tangent to h along Δ^{β}, or if the curve breaks into two pieces, with Q^{β} on the rational piece. The first case happens

\frac{#W(d, Γ′, Δ \ \{Δ^{β}\})}{(d − 2)!} (12)

times, where Γ′_{1} contains d − 2 copies of h and Γ′_{2} contains one copy of Δ^{β} (which is contained in h). The second case happens

(13)  ∑_{\begin{matrix}d(0)+d(1)=d,d(0)≥2 \\ Δ(0) ∐ Δ(1)=Δ\Δ^{β} \end{matrix}}(d(0)d(1))^{δ_{n,2}} (\frac{#X (d(0), Γ(0), Δ(0))}{(d(0) − 2)!}) (\frac{#W(d(1),^{d(1)}Γ, Δ(1))}{d(1)!})

times, where Γ(0) consists of (d(0) − 1) copies of h and one copy of Δ^{β}. Hence (H − Q^{β}) · Q^{β} is the sum of (12) and (13). The denominator (d(0) − 2)! arises because we have a degree d(0) (rational) curve passing through a linear space Δ^{β} on h, and various incidence conditions Δ(0). The number of such curves with a choice of one of the other intersections of C(0) with h is (d(0) − 1)#X (d(0), Γ(0), Δ(0))/(d(0) − 1)!.

As an example, consider the elliptic quartics in ℙ^{2} passing through 11 fixed points, including q^{1}, q^{2}, q^{3}, q^{4}. How many such two-nodal quartics have O(1) = q^{1} + ⋯ + q^{4} in the Picard group of the normalization of the curve? We construct the fibration F over the (normalized) variety of two-nodal plane quartics through 11 fixed points. We have sections Q^{1}, …, Q^{11} and a multisection H. If β ≠ β′, Q^{β} · Q^{β′} = 3, H · Q^{β} = 30, H^{2} = 225 + 3\binom{11}{2} = 390, (H − Q^{1}) · Q^{1} = 185, so

S^{2} = H · Q^{1} − (H − Q^{1}) · Q^{1} = −155.

Let D = H − Q^{1} − Q^{2} − Q^{3} − Q^{4}. Then

D^{2} = H^{2} + 4S^{2} − 8H · Q^{1} + 12Q^{1} · Q^{2} = 390 + 4(−155) − 8(30) + 12(3) = −434

so the answer is S^{2} − D^{2}/2 = 62.

To determine the enumerative geometry of quartic elliptic space curves (see Section 8.3), various #Z_{2}(d, Δ)_{D} were needed with d = 3 and d = 4. When d = 3, Δ consists of 8 points (for dim Z to be 0) and some lines; the results are given in the Table 1. For convenience, we write p^{β} for the base points (in Δ) and l^{β} for the marked points on lines. These values were independently confirmed by M. Roth ([Ro]). When d = 4, Δ consists of 11 points and some lines; the results are given in the Table 2. For convenience again, we write p^{β} for points and l^{β} for lines.

\begin{matrix}#l^{β} & D & #Z_{2}(d, Δ)_{D} \\ 0 & p^{1} + p^{2} + p^{3} & 0 \\ 1 & p^{1} + p^{2} + l^{1} & 1 \\ 2 & p^{1} + l^{1} + l^{2} & 5 \\ 3 & l^{1} + l^{2} + l^{3} & 18 \\ 0 & p^{1} + 2p^{2} & 1 \\ 1 & 2p^{1} + l^{1} & 4 \\ 1 & p^{1} + 2l^{1} & 5 \\ 2 & l^{1} + 2l^{2} & 16 \\ 0 & 3p^{1} & 3 \\ 1 & 3l^{1} & 14 \\ 0 & p^{1} + p^{2} + p^{3} + p^{4} − p^{5} & 1 \\ 1 & p^{1} + p^{2} + p^{3} + p^{4} − l^{1} & 2 \\ 1 & p^{1} + p^{2} + p^{3} + l^{1} − p^{4} & 4 \\ 2 & p^{1} + p^{2} + p^{3} + l^{1} − l^{2} & 10 \\ 2 & p^{1} + p^{2} + l^{1} + l^{2} − p^{3} & 14 \\ 3 & p^{1} + p^{2} + l^{1} + l^{2} − l^{3} & 39 \\ 3 & p^{1} + l^{1} + l^{2} + l^{3} − p^{2} & 45 \\ 4 & p^{1} + l^{1} + l^{2} + l^{3} − l^{4} & 135 \\ 4 & l^{1} + l^{2} + l^{3} + l^{4} − p^{1} & 135 \\ 5 & l^{1} + l^{2} + l^{3} + l^{4} − l^{5} & 432 \end{matrix}

Table 1. Counting cubic elliptic plane curves with a divisorial condition

\begin{matrix}#l^{β} & D & #Z_{2}(d, Δ)_{D} \\ 0 & p^{1} + p^{2} + p^{3} + p^{4} & 62 \\ 1 & p^{1} + p^{2} + p^{3} + l^{1} & 464 \\ 2 & p^{1} + p^{2} + l^{1} + l^{2} & 2,522 \\ 3 & p^{1} + l^{1} + l^{2} + l^{3} & 11,960 \\ 4 & l^{1} + l^{2} + l^{3} + l^{4} & 52,160 \end{matrix}

Table 2. Counting quartic elliptic plane curves with a divisorial condition

8. Examples

8.1. Plane curves. Type IIc components in this case are never enumeratively relevant, as the elliptic curve C(0) must map to the line H with degree at least two. The recursive formulas we get are identical to the genus 1 recursive formulas of Caporaso and Harris in [CH].

8.2. Cubic elliptic space curves. The number of smooth cubic elliptic space curves through j general points and 12 − 2j general lines is 1500, 150, 14, and 1 for j = 0, 1, 2, and 3 respectively. (The number is 0 for j > 3 as cubic elliptic space curves must lie in a plane.) The degenerations involved in calculating the first case appeared in Section 2.4. As the Chow ring of the space of smooth elliptic cubics is not hard to calculate (see [H] p. 36), these results may be easily verified.

The number of cubic elliptics tangent to H, through j general points and 11 − 2j general lines is 4740, 498, 50, and 4 for j = 0, 1, 2, and 3 respectively. The number of cubic elliptics triply tangent to H through j general points and 10 − 2j general lines is 2790, 306, 33, and 3 for j = 0, 1, 2, and 3 respectively.

j # quartics
0 52,832,040
1 4,436,208
2  385,656
3  34,674
4  3,220
5  310
6  32
7  4
8  1

Table 3. Number of quartic elliptic space curves through j general points and 16 − 2j general lines

These numbers are needed for the next examples.

8.3. Quartic elliptic space curves. The number of smooth quartic elliptic space curves through j general points and 16 − 2j general lines is given in the Table 3. These numbers agree with those independently found by Getzler ([G], proof to appear in [GP]).

Other enumerative data can also be found en route. For example, Tables 4 and 5 give the number of smooth quartic elliptic space curves through i_{0} general points and i_{1} general lines, and h_{0} general points and h_{1} general lines in H, with

2i_{0} + i_{1} + 2h_{0} + h_{1} = 16.

At each stage, the number may be computed by degenerating a point or a line (assuming there is a point or line to degenerate). Each row is labelled, and the labels of the different degenerations that are also smooth quartics are given in each case, and a “+” is added if there are other degenerations (where the curve breaks into pieces). (This will help the reader to follow through the degenerations.) Keep in mind that these numbers are not quite what the algorithm of this article produces; in the algorithm, the intersections with H are labelled, so the number computed for (i_{0}, i_{1}, h_{0}, h_{1}) will be (4 − h_{0} − h_{1})! times the number in the table.

These computations are not as difficult as one might think (and were done by hand). For example, if i_{0} and i_{1} are both positive, it is possible to degenerate a point and then a line, or a line and then a point. Both methods must yield the same number, providing a means of double-checking.

As an example, some of the degenerations used to compute the 52,832,040 quartic space curves through 16 general lines are displayed in Figure 12, using the pictorial shorthand of Section 2. Degenerations involving irreducible quartic elliptic space curves are given (as well as a few more). The boldfaced numbers indicate the corresponding rows in Tables 4 and 5. The phrase “etc.” indicates that further degenerations appear. A full diagram of degenerations is available on request.

16 14 13 12 11 # general lines 10 915 87  ^{18}  ^{1}^{10} ^{40}^{4,436,268}^{26}^{4,436,268} ^{48}^{33}^{4,028,112} ^{55}  ^{68}^{75}^{2,849,436} ^{62}^{81}^{312,348}  ^{87}^{31,056}  ^{93} ^{150}  ^{756}  ^{225}^{2,519}  ^{98}^{11}  ^{62}  ^{756}  ^{231}^{52,832,010}^{52,832,010}^{48,395,772}^{39,347,736}^{23,962,326}  ^{385,656}  ^{22,500}^{384,156}  ^{1500}^{34,674}^{×2}^{etc.}  ^{×3}^{etc.}  ^{etc.}  ^{×2}^{etc.}  ^{×3}  ^{etc.}^{etc.} ^{etc.}  ^{×2} ^{×4}  ^{×2}  ^{×3}^{×6}  ^{×3}  ^{×3}  ^{×4}^{×4}^{175,500}

Figure 12. Counting quartic elliptic space curves through 16 general lines

(i_{0}, i_{1},  point  line  # curves
h_{0}, h_{1}) degen. degen.
1 (16,0,0,0)  10  52,832,040
2 (14,1,0,0)  40  11  4,436,268
3 (12,2,0,0)  41  12  385,656
4 (10,3,0,0)  42  13  34,674
5 (8,4,0,0)  43  14  3,220
6 (6,5,0,0)  44  15  310
7 (4,6,0,0)  45  16  32
8 (1,7,0,0)  46  17  4
9 (0,8,0,0)  47  1
10 (15,0,1,0)  18, 40 52,832,040
11 (13,1,1,0)  48  19, 41  4,436,268
12 (11,2,1,0)  49  20, 42  385,656
13 (9,3,1,0)  50  21, 43  34,674
14 (7,4,1,0)  51  22, 44  3,220
15 (5,5,1,0)  52  23, 45  310
16 (3,6,1,0)  53  24, 46  32
17 (1,7,1,0)  54  25, 47  4
18 (14,0,2,0)  26, 48+ 48,395,772
19 (12,1,2,0) 55+ 27, 49+ 4,050,612
20 (10,2,2,0) 56+ 28, 50+  350,982
21 (8,3,2,0)  57+ 29, 51+  31,454
22 (6,4,2,0)  58+  30, 52  2,910
23 (4,5,2,0)  59  31, 53  278
24 (2,6,2,0)  60  32, 54  28
25 (0,7,2,0)  61  3
26 (13,0,3,0)  33, 55+ 39,347,736
27 (11,1,3,0) 62+ 34, 56+ 3,266,100
28 (9,2,3,0)  63+ 35, 57+  280,752
29 (7,3,3,0)  64+ 36, 58+  24,972
30 (5,4,3,0)  65+ 37, 59+  2,290
31 (3,5,3,0)  66+ 38, 60+  214
32 (1,6,3,0)  67+ 39, 61+  20
33 (12,0,4,0)  62+  23,962,326
34 (10,1,4,0)  +  63+  1,939,857
35 (8,2,4,0)  +  64+  161,735
36 (6,3,4,0)  +  65+  13,908
37 (4,4,4,0)  +  66+  1,222
38 (2,5,4,0)  +  67+  104
39 (0,6,4,0)  +  8
40 (14,0,0,1)  48+  4,436,268
41 (12,1,0,1)  68  49  385,656
42 (10,2,0,1)  69  50  34,674
43 (8,3,0,1)  70  51  3,220
44 (6,4,0,1)  71  52  310
45 (3,5,0,1)  72  53  32
46 (2,6,0,1)  73  54  4
47 (0,7,0,1)  74  1
48 (13,0,1,1)  55, 68+ 4,436,268
49 (11,1,1,1) 75+ 56, 69+  385,656
50 (9,2,1,1)  76+ 57, 70+  34,674
51 (7,3,1,1)  77+ 58, 71+  3,220

Table 4. Quartic elliptic space curves with incidence conditions

(i_{0}, i_{1},  point  line  # curves
h_{0}, h_{1}) degen. degen.
52 (5,4,1,1)  78+  59, 72  310
53 (3,5,1,1)  79  60, 73  32
54 (1,6,1,1)  80  61, 74  4
55 (12,0,2,1)  62, 75 4,028,112
56 (10,1,2,1) 81+ 63, 76+  349,032
57 (8,2,2,1)  82+ 64, 77+  28,340
58 (6,3,2,1)  83+ 65, 78+  2,901
59 (4,4,2,1)  84+ 66, 79+  278
60 (2,5,2,1)  85+ 67, 80+  28
61 (0,6,2,1)  86+  3
62 (11,0,3,1)  81+  2,849,436
63 (9,1,3,1)  +  82+  243,507
64 (7,2,3,1)  +  83+  21,310
65 (5,3,3,1)  +  84+  1,909
66 (3,4,3,1)  +  85+  172
67 (1,5,3,1)  +  86+  14
68 (12,0,0,2)  75+  385,656
69 (10,1,0,2)  87  76+  34,674
70 (8,2,0,2)  88  77+  3,220
71 (6,3,0,2)  89  78+  310
72 (4,4,0,2)  90  79  32
73 (2,5,0,2)  91  80  4
74 (0,6,0,2)  92  1
75 (11,0,1,2)  81, 87+  384,156
76 (9,1,1,2)  93+ 82, 88+  34,524
77 (7,2,1,2)  94+ 83, 89+  3,206
78 (5,3,1,2)  95+ 84, 90+  309
79 (3,4,1,2)  96  85, 91+  32
80 (1,5,1,2)  97  86, 92+  4
81 (10,0,2,2)  93+  312,348
82 (8,1,2,2)  +  94+  28,340
83 (6,2,2,2)  +  95+  2,612
84 (4,3,2,2)  +  96+  246
85 (2,4,2,2)  +  97+  24
86 (0,5,2,2)  +  2
87 (10,0,0,3)  93+  34,674
88 (8,1,0,3)  98+  94+  3,220
89 (6,2,0,3)  99+  95+  310
90 (4,3,0,3) 100+  96  32
91 (2,4,0,3)  101  97  4
92 (0,5,0,3)  102  1
93 (9,0,1,3)  98+  31,056
94 (7,1,1,3)  +  99+  3,052
95 (5,2,1,3)  +  100+  304
96 (3,3,1,3)  +  101+  32
97 (1,4,1,3)  +  102+  4
98 (8,0,0,4)  +  2,519
99 (6,1,0,4)  +  +  277
100 (4,2,0,4)  +  +  31
101 (2,3,0,4)  +  +  4
102 (0,4,0,4)  +  1

Table 5. Quartic elliptic space curves with incidence conditions, cont’d

Appendix A. Background: The moduli space of stable maps

For the convenience of the reader we recall certain facts about moduli stacks of stable maps to smooth varieties, without proofs.

A family of m-pointed nodal curves over a base scheme S (or a nodal curve over S) is a proper flat morphism ρ : C → S whose geometric fibers are reduced and pure dimension 1, with at worst ordinary double points as singularities, with sections σ_{i} : S → C (1 ≤ i ≤ m) whose images are disjoint, and lie in the smooth locus of ρ. If X is a scheme, then a family of maps of nodal curves to X over S (or a map of a nodal curve to X over S) is a morphism π : C → X × S of schemes over S, where the induced morphism ρ : C → S is a family of nodal curves over S. A nodal curve (with no base scheme specified) is a nodal curve over Spec ℂ, and a map of a nodal curve to X is a map over Spec ℂ. Similar definitions hold for families of nodal curves (and maps) over Deligne-Mumford stacks (see [DMu] for definitions).

A stable map is a map π from a connected pointed nodal curve to a smooth variety X such that π has finite automorphism group. The arithmetic genus of a stable map is defined to be the arithmetic genus of the nodal curve C. If [C] ∈ H_{2}(C) is the fundamental class of C, and β = π_{∗}[C] ∈ H_{2}(X), then we say the image of C is in class β. If X is a projective space, and β is d times the class of a line, we say that d is the degree of the stable map. The requirement that the automorphism group be finite is equivalent to requiring each collapsed genus 0 component to have at least three special points and each contracted genus 1 component to have at least one special point, where a special point is defined to be either a marked point or a branch of a node.

A family of stable maps to X is a family of maps of connected pointed nodal curves to X whose fibers over closed points are stable maps. Let \overline{M}_{g,m}(X, β) be the stack whose category of sections of a scheme S is the category of families of stable maps to X over S of arithmetic genus g with m marked points, with image in class β. If X is a smooth projective variety, \overline{M}_{g,m}(X, β) is a fine moduli stack of Deligne-Mumford type. There is an open substack M_{g}(X, β) (possibly empty) that is a fine moduli stack of maps of smooth curves.

The stack \overline{M}_{g,m+1}(X, β) is the universal curve over \overline{M}_{g,m}(X, β). A morphism of smooth projective varieties f : X → Y induces a morphism of stacks \overline{M}_{g,m}(X, f^{∗}β) → \overline{M}_{g,m}(Y, β) ([BM], Remark after Theorem 3.14). There are natural evaluation maps ev_{i} : \overline{M}_{g,m}(X, β) → X (1 ≤ i ≤ m) that informally give “the image of the ith marked point”.

The versal deformation space to the map (C, \{p_{i}\}, π) in \overline{M}_{g,m}(X, β) is obtained from the complex

\underline{Ω}_{π} = (π^{∗}Ω_{X} → Ω_{C}(p_{1} + ⋯ + p_{m})) .

The vector space ℍom(\underline{Ω}_{π}, O_{C}) parametrizes infinitesimal automorphisms of the map (C, \{p_{i}\}, π) (denoted Aut(C, \{p_{i}\}, π)); if (C, \{p_{i}\}, π) is a stable map, ℍom(\underline{Ω}_{π}, O_{C}) = 0. The space of infinitesimal deformations to the map (C, \{p_{i}\}, π) (i.e. the Zariski tangent space to \overline{M}_{g,m}(X, β) at the point representing this stable map, denoted Def (C, \{p_{i}\}, π)) is given by 𝔼xt^{1}(\underline{Ω}_{π}, O_{C}) and the obstruction space (denoted Ob_{(C,\{p_{i}\},π)}) is given by 𝔼xt^{2}(\underline{Ω}_{π}, O_{C}).

By applying the functor Hom(·, O_{C}) to the exact sequence of complexes

0 → Ω_{C}(p_{1} + ⋯ + p_{m}) → \underline{Ω}_{π} → π^{∗}Ω_{X}[1] → 0

we obtain the long exact sequence

0 → ℍom(\underline{Ω}_{π}, O_{C}) → Hom(Ω_{C}(p_{1} + ⋯ + p_{m}), O_{C}) → H^{0}(C, π^{∗}T_{X})
→ 𝔼xt^{1}(\underline{Ω}_{π}, O_{C}) → Ext^{1}(Ω_{C}(p_{1} + ⋯ + p_{m}), O_{C}) → H^{1}(C, π^{∗}T_{X})
→ 𝔼xt^{2}(\underline{Ω}_{π}, O_{C}) → 0.

By the identifications given in the previous paragraph, and using Hom(Ω_{C}(p_{1}+⋯+p_{m}), O_{C}) = Aut(C, {p_{i}}) and Ext^{1}(Ω_{C}(p_{1} + ⋯ + p_{m}), O_{C}) = Def(C, {p_{i}}), this long exact sequence can be rewritten as

0 ⟶ Aut(C, {p_{i}}) ⟶ H^{0}(C, π^{∗}T_{X})
⟶ Def(C, {p_{i}}, π) ⟶ Def(C, {p_{i}}) ⟶ H^{1}(C, π^{∗}T_{X})
⟶ Ob(C, {p_{i}}, π) ⟶ 0.

The term H^{0}(C, π^{∗}T_{X}) can be interpreted as first order deformations of the map π, with the pointed curve (C, {p_{i}}) fixed.

If H^{1}(C, π^{∗}T_{X}) = 0 at a point (C, {p_{i}}, π), then the obstruction space vanishes, and the moduli stack is smooth. Thus \overline{M}_{0,m}(ℙ^{n}, d) is smooth of dimension (n + 1)(d + 1) + m − 4, and \overline{M}_{1,m}(ℙ^{n}, d) is smooth of dimension (n + 1)d + m − 1 at points (C, \{p_{i}\}, π) where d > 1 and C is smooth.

The construction of the versal deformation space from \underline{Ω}_{π} is discussed in [Ra], [Vi], and [LT]. All other facts described here appear in [BM] or in the comprehensive introduction [FP].

References

[A] D. Avritzer, Enumerative geometry of del Pezzo surfaces, ICTP preprint IC/93/53.

[AVa] D. Avritzer and I. Vainsencher, Compactifying the space of elliptic quartic curves, in Complex Projective Geometry, G. Ellingsrud et al. eds., Cambridge U.P.: Cambridge, 1992.

[BM] K. Behrend and Yu. Manin, Stacks of stable maps and Gromov-Witten invariants, Duke Math. J. 85 (1996), no. 1, 1–60.

[CH] L. Caporaso and J. Harris, Counting plane curves of any genus, Invent. Math. 131 (1998), no. 2, 345–392.

[DMu] P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–110.

[FP] W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, in Algebraic geometry Santa Cruz 1995, J. Kollár, R. Lazarsfeld, D. Morrison eds., AMS: Providence, 1997.

[Ga1] A. Gathmann, Gromov-Witten and degeneration invariants: Computation and enumerative significance, doctoral thesis, Hannover, 1998.

[Ga2] A. Gathmann, Rational curves on the quintic threefold, talk at Harvard algebraic geometry seminar; manuscript in preparation.

[G] E. Getzler, Intersection theory on \overline{M}_{1,4} and elliptic Gromov-Witten invariants, J. Amer. Math. Soc. 10 No. 4 (1997) 973–998.

[GP] E. Getzler and R. Pandharipande, The number of elliptic space curves, in preparation.

[H] J. Harris, Curves in Projective Space, Les Presses de l’Université de Montréal: Montréal, 1982.

[I] E.-N. Ionel, Genus one enumerative invariants in ℙ^{n} with fixed j-invariant, Duke Math. J. 94 (1998) no. 2, 279–324.

[IPa] E.-N. Ionel and T. Parker, Gromov-Witten invariants of symplectic sums, preprint 1998, math.SG/9806013.

[K1] S. Kleiman, On the transversality of a general translate, Compositio Math. 28 (1974), 287–297.

[K2] S. Kleiman, Problem 15: Rigorous foundation of Schubert’s enumerative calculus, in Mathematical developments arising from Hilbert problems, Proc. Sympos. Pure Math 28 (1976) 445–482.

[KSX] S. Kleiman, S.A. Strømme, and S. Xambó, Sketch of a verification of Schubert’s number 5819539783680 of twisted cubics, in Space Curves: Proceedings, Rocca di Papa 1985, F. Ghione, C. Peskine, E. Sernesi eds., LNM 1266, Springer-Verlag: New York, 1987.

[KoM] M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), 525–562.

[LT] J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998), no. 1, 119–174.

[LR] A.-M. Li and Y. Ruan, Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds I, preprint 1998, math.AG/9803036 v3.

[Ma] H. Matsumura, Commutative Ring Theory, M. Reid trans., Cambridge U.P.: Cambridge-New York, 1989.

[P] R. Pandharipande, Counting elliptic plane curves with fixed j-invariant, Proc. Amer. Math. Soc. 125 (1997), no. 12, 3471–3479.

[PiZ] M. Pieri and H.G. Zeuthen, Géométrie énumérative, in Encyclopédie des Sciences Mathématiques III, vol. 2, A. Schoenflies and A. Tresse eds., Teubner, 1911, pp 260–331.

[Ra] Z. Ran, Deformations of maps, in Algebraic Curves and Projective Geometry, E. Ballico and C. Ciliberto eds., Springer LNM 1389, 1989, pp 246–253.

[Ro] M. Roth, personal communication.

[RT] Y. Ruan and G. Tian, A mathematical theory of quantum cohomology, J.Diff. Geom. 42 (1995), no. 2, 259–367.

[R] Y. Ruan, Surgery, quantum cohomology and birational geometry, preprint 1998, math.AG/9810039, to appear in Berkeley-Stanford-Santa Cruz symplectic geometry conference proceedings.

[Th] M. Thaddeus, personal communication.

[V1] R. Vakil, Counting curves on rational surfaces, submitted for publication, available at http://wwwmath.mit.edu/˜ vakil/preprints.html. It contains the content of two preprints, Counting curves of any genus on rational ruled surfaces math.AG/9709003 and Genus g Gromov-Witten invariants of Del Pezzo surfaces: Counting plane curves with fixed multiple points math.AG/9709004.

[V2] R. Vakil, The characteristic numbers of plane quartic curves, preprint 1998, math.AG/9812018.

[V3] R. Vakil, Recursions, formulas, and graph-theoretic interpretations of ramified coverings of the sphere by surfaces of genus 0 and 1, preprint 1998, math.CO/9812105.

[Vi] A. Vistoli, The infinitesimal deformation theory of local complete intersections, preprint 1997, math.AG/9703008.

MIT Department of Mathematics 77 Massachusetts Ave. Rm. 2-248 Cambridge MA 02139 vakil@math.mit.edu