Ravi Vakil

Relative virtual localization and vanishing of tautological classes on moduli spaces of curves

We prove a localization formula for the moduli space of stable relative maps. As an application, we prove that all codimension i tautological classes on the moduli space of stable pointed curves vanish away from strata corresponding to curves with at least i − g + 1 genus 0 components. As consequences, we prove and generalize various conjectures and theorems about various moduli spaces of curves (due to Getzler, Ionel, Faber, Looijenga, Pandharipande, Diaz, and others). This theorem appears to be the geometric content behind these results; the rest is straightforward graph combinatorics. The theorem also suggests the importance of the stratification of the moduli space by number of rational components.

The equations for the moduli space of

A central question in invariant theory is that of determining the relations among invariants. Geometric invariant theory quotients come with a natural ample line bundle, and hence often a natural projective embedding. This question translates to determining the equations of the moduli space under this embedding. This article deals with one of the most classical quotients, the space of ordered points on the projective line. We show that under any weighting of the points, this quotient is cut out (scheme-theoretically) by a particularly simple set of quadric relations, with the single exception of the Segre cubic threefold (the space of six points with equal weight). We also show that the ideal of relations is generated in degree at most four, and give an explicit description of the generators. If all the weights are even (e.g. in the case of equal weight for odd n), we show that the ideal of relations is generated by quadrics. Unlike many facts in geometric invariant theory, these results are characteristic-independent, and indeed work over the integers.

n

We construct a natural smooth compactification of the space of smooth genus-one curves with k distinct points in a projective space. It can be viewed as an analogue of a well-known smooth compactification of the space of smooth genus-zero curves, that is, the space of stable genus-zero maps S M0;k.P n ;d/. In fact, our compactification is obtained from the singular space of stable genus-one maps S M1;k.P n ;d/ through a natural sequence of blowups along “bad” subvarieties. While this construction is simple to describe, it requires more work to show that the end result is a smooth space. As a bonus, we obtain desingularizations of certain natural sheaves over the “main” irreducible component S M 0;k .P n ;d/ of S M1;k.P n ;d/. A number of applications of these desingularizations in enumerative geometry and Gromov‐Witten theory are described in the introduction, including the second author’s proof of physicists’ predictions for genus-one Gromov‐Witten invariants of a quintic threefold. 14D20; 53D99

points on the line

The goal of this article is to motivate and describe how Gromov-Witten the- ory can and has provided tools to understand the moduli space of curves. For example, ideas and methods from Gromov-Witten theory have led to both conjectures and theorems showing that the tautological part of the cohomology ring has a remarkable and profound structure. As an illustration, we describe a new approach to Fabers intersection number conjecture via branched covers of the projective line (work with I.P. Goulden and D.M. Jackson, based on work with T. Graber). En route we review the work of a large number of mathematicians.

A desingularization of the main component of the moduli space of genus-one stable maps into ℙn

Abstract We give a short and direct proof of Getzler and Pandharipandes λg -conjecture. The approach is through the Ekedahl-Lando-Shapiro-Vainshtein theorem, which establishes the “polynomiality” of Hurwitz numbers, from which we pick off the lowest degree terms. The proof is independent of Gromov-Witten theory. We briefly describe the philosophy behind our general approach to intersection numbers and how it may be extended to other intersection number conjectures. Ideas from this paper feature in two independent recent enlightening proofs of Wittens conjecture by Kazarian [Adv. Math.] and Chen, Li, and Liu [Asian J. Math. 12: 511–518, 2009].

THE MODULI SPACE OF CURVES AND GROMOV-WITTEN THEORY

We derive a closed-form expression for all genus 1 Hurwitz numbers, and give a simple new graph-theoretic interpretation of Hurwitz numbers in genus 0 and 1. (Hurwitz numbers essentially count irreducible genus g covers of the sphere, with arbitrary specified branching over one point, simple branching over other specified points, and no other branching. The problem is equivalent to counting transitive factorisations of permutations into transpositions.) These results prove a conjecture of Goulden, Jackson and Vainshtein, and extend results of Hurwitz and many others.

A short proof of the λg -conjecture without Gromov-Witten theory: Hurwitz theory and the moduli of curves

Abstract The recurrence a ( n ) = a ( a ( n − 1)) + a ( n − a ( n − 1)), a (1) = a (2) = 1 defines an integer sequence, publicized by Conway and Mallows, with amazing combinatorial properties that cry out for explanation. We take a step towards unravelling this mystery by showing that a ( n ) can (and should) be viewed as a simple ‘compression’ operation on finite sets. This gives a combinatorial characterization of a ( n ) from which one can read off most of its properties. We prove a conjecture of Mallows on the asymptotic shape of a ( n ), and give an algorithm for solving Conways challenge problem about the epsilontics of ( a ( n )/ n −1/2). Along the way we encounter some beautiful constructions involving trees, recursively expanding finite strings, and refinements of Pascals triangle. Newmans generalizations of a ( n ) can be analyzed in the same way, and the results obtained point to possible relations with Conways theory of games.

Genus 0 and 1 Hurwitz numbers: Recursions, formulas, and graph-theoretic interpretations

Characteristic numbers of families of maps of nodal curves toP2 are defined as intersection of natural divisor classes. (This definition agrees with the usual definition for families of plane curves.) Simple recursions for characteristic numbers of genus one plane curves of all degrees are computed.

On Conway's recursive sequence

The study of the projective coordinate ring of the (geometric invariant theory) moduli space of n ordered points on P1 up to automorphisms began with Kempe in 1894, who proved that the ring is generated in degree one in the main (n even, unit weight) case. We describe the relations among the invariants for all possible weights. In the main case, we show that up to the Sn-symmetry, there is a single equation. For n 6 6, it is a simple quadratic binomial relation. (For n = 6, it is the classical Segre cubic relation.) For general weights, the ideal of relations is generated by quadratics inherited from the case of 8 points. This paper completes the program set out in (HMSV1).

Recursions for characteristic numbers of genus one plane curves

Using a blend of combinatorics and geometry, we give an algorithm for algebraically finding all flags in any zero-dimensional intersection of Schubert varieties with respect to three transverse flags, and more generally, any number of flags. The number of flags in a triple intersection is also a structure constant for the cohomology ring of the flag manifold. Our algorithm is based on solving a limited number of determinantal equations for each intersection (far fewer than the naive approach in the case of triple intersections). These equations may be used to compute Galois and monodromy groups of intersections of Schubert varieties. We are able to limit the number of equations by using the permutation arrays of Eriksson and Linusson, and their permutation array varieties, introduced as generalizations of Schubert varieties. We show that there exists a unique permutation array corresponding to each realizable Schubert problem and give a simple recurrence to compute the corresponding rank table, giving in particular a simple criterion for a Littlewood-Richardson coefficient to be 0. We describe pathologies of Eriksson and Linusson’s permutation array varieties (failure of existence, irreducibility, equidimensionality, and reducedness of equations), and define the more natural permutation array schemes. In particular, we give several counterexamples to the Readability Conjecture based on classical projective geometry. Finally, we give examples where Galois/monodromy groups experimentally appear to be smaller than expected.