A natural smooth compactification of the space of elliptic curves in

projective space via blowing up the space of stable maps Ravi Vakil and Aleksey Zinger

The moduli space of stable maps \overline{M}_{g,k}(X, β) to a complex projective manifold X (where g is the genus, k is the number of marked points, and β ∈ H_{2}(X, ℤ) is the image homology class) is the central tool and object of study in Gromov-Witten theory. The open subset corresponding to maps from smooth curves is denoted M_{g,k}(X, β).

The protean example is \overline{M}_{0,k}(ℙ^{n}, d). This space is wonderful in essentially all ways: it is irreducible, smooth, and contains M_{0,k}(ℙ^{n}, d) as a dense open subset. The boundary

Δ := \overline{M}_{0,k}(ℙ^{n}, d) \ M_{0,k}(ℙ^{n}, d)

is normal crossings. The divisor theory is fully understood, and combinatorially tractable [4]. In some sense, this should be seen as the natural generalization of the space of complete conics compactifying the space of smooth conics.

It is natural to wonder if such a beautiful structure exists in higher genus. In arbitrary genus, however, there is no reasonable hope: M_{g}(ℙ^{n}, d) is badly behaved. (We emphasize that even the interior of the moduli space of stable maps is badly-behaved.) More precisely, M_{g}(ℙ^{n}, d) (as g, n, and d vary) is arbitrarily singular in a well-defined sense — it can have essentially any singularity, and can have components of various dimension meeting in various ways with various nonreduced structures [6]. In short, there is no reasonable hope of describing a desingularization, as this would in essence involve describing a resolution of singularities.

In genus one, however, the situation remains remarkably beautiful. Although \overline{M}_{1,k}(ℙ^{n}, d) in general has many components, it is straightforward to show that M_{1,k}(ℙ^{n}, d) is irreducible and smooth. Let \overline{M}_{1,k}^{0}(ℙ^{n}, d) be the closure of this open subset (the “main component” of the moduli space).

We will describe a natural desingularization of this main component

M̃_{1,k}(ℙ^{n}, d) → \overline{M}_{1,k}^{0}(ℙ^{n}, d).

(Details appear in [7]. In particular, it is proved there that this construction actually gives a desingularization.) This desingularization has several desirable properties.

• It leaves the interior M_{1,k}(ℙ^{n}, d) unchanged.

• The boundary M̃_{1,k}(ℙ^{n}, d) \ M_{1,k}(ℙ^{n}, d) is simple normal crossings, with an explicitly described normal bundle.

• The points of the boundary have explicit geometric interpretations.

• The desingularization can be interpreted as blowing up “the most singular locus”, then “the next most singular locus”, and so on, but with an unusual twist.

• The divisor theory is explicitly describable, and the intersection theory is tractable. (For example, one can compute the top intersection of divisors using [9].)

• The compactification is natural in the following senses.

(i) The desingularization is equivariant: it behaves well with respect to the symmetries of ℙ^{n}. Hence we can apply Atiyah-Bott localization to this space — not just in theory, but in practice.

(ii) It behaves well with respect to the inclusion ℙ^{m} ↪ ℙ^{n}. (iii) It behaves well with respect to the marked points (forgetful maps,

ψ-classes, etc.). (iv) Consider the universal map π : C → ℙ^{n} over \overline{M}_{g,k}(ℙ^{n}, d), where

ρ : C → \overline{M}_{g,k}(ℙ^{n}, d) is the structure morphism. An important sheaf in Gromov-Witten theory is ρ_{∗}π^{∗}O_{ℙ^{n}}(a). When g > 0, this is not a vector bundle, which causes difficulty in theory and computation. However, in genus 1, “resolving \overline{M}_{1,k}^{0}(ℙ^{n}, d) also resolves this sheaf”: when the sheaf is pulled back to the desingularization, it “becomes” a vector bundle. More precisely, it contains a natural vector bundle, and is isomorphic to it on the interior. This vector bundle is explicitly describable.

We find it interesting that such a natural naive approach as we will describe actually works, and yields a desingularization with these nice properties. For example, if n > 2, this desingularization can be interpreted as a natural compactification of the Hilbert scheme of smooth degree d curves in projective space, and thus could be seen as the genus 1 version of the complete conics.

This construction also has a number of applications:

• enumerative geometry of genus 1 curves via localization.

• Gromov-Witten invariants in terms of enumerative invariants [8].

• the Lefschetz hyperplane property: effective computation of Gromov-Witten invariants of complete intersections [3] (see also [2] for the special case of the quintic threefold).

• algebraic version of “reduced” Gromov-Witten invariants in symplectic geometry [8].

• an approach to hopefully prove physicists’ predictions [1] about genus 1 Gromov-Witten invariants (work of Zinger, in progress).

We finally describe the construction explicitly. (In the lecture, the geography of \overline{M}_{1,k}(ℙ^{n}, d) was sketched as motivation.) It is straightforward to show that \overline{M}_{1,k}(ℙ^{n}, d) is nonsingular on the locus where there is no contracted genus 1 (possibly nodal) curve (for example, the proof of [5, Prop. 4.21] applies). We say a stable map is in the m-tail locus if there is an arithmetic genus 1 contracted curve, with precisely m points of the contracted curve that are either marked, or meet the rest of the curve. The algorithm is then as follows: blow up the one-tail component (which actually does nothing to \overline{M}_{1,k}^{0}(ℙ^{n}, d)), then the proper transform of all two-tail components, then the proper transform of all three-tail components, etc.

References

[1] M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa, Holomorphic anomalies in topological field theories, Nucl. Phys. B 405 (1993), 279–304.

[2] J. Li and A. Zinger, On the genus-one Gromov-Witten invariants of a quintic threefold, preprint 2004, math.AG/0406105.

[3] J. Li and A. Zinger, On the genus-one Gromov-Witten invariants of complete intersections, preprint 2005, math.AG/0507104.

[4] R. Pandharipande, Intersections of \overline{ℚ}-divisors on Kontsevich’s moduli space \overline{M}_{0,n}(ℙ^{r}, d) and enumerative geometry, Trans. Amer. Math. Soc. 351 (1999), no. 4, 1481–1505.

[5] R. Vakil, The enumerative geometry of rational and elliptic curves in projective space, J. Reine Angew. Math. (Crelle’s Journal) 529 (2000), 101–153.

[6] R. Vakil, Murphy’s Law in algebraic geometry: Badly-behaved moduli deformation spaces, Invent. Math., to appear (earlier version math.AG/0411469).

[7] R. Vakil and A. Zinger, A desingularization of the main component of the moduli space of genus-one stable maps to projective space, preprint 2006, math.AG/0603353.

[8] A. Zinger, Reduced genus-one Gromov-Witten invariants, preprint 2005, math.SG/0507103.

[9] A. Zinger, Intersections of tautological classes on blowups of moduli spaces of genus-one curves, preprint 2006, math.AG/0603357.