arXiv subjects

Ravi Vakil

Publications and source records attributed to Ravi Vakil.

47 records · Page 3Linked to original sources

arXiv2003

Schubert induction

We describe a Schubert induction theorem, a tool for analyzing intersections on a Grassmannian over an arbitrary base ring. The key ingredient in the proof is the Geometric Littlewood-Richardson rule, described in a companion paper. Schubert problems are among the most classical problems in enumerative geometry of continuing interest. As an application of Schubert induction, we address several long-standing natural questions related to Schubert problems, including: the "reality" of solutions; effective numerical methods; solutions over algebraically closed fields of positive characteristic; solutions over finite fields; a generic smoothness (Kleiman-Bertini) theorem; and monodromy groups of Schubert problems. These methods conjecturally extend to the flag variety.

math.AG↗

arXiv2003

A geometric Littlewood-Richardson rule

We describe an explicit geometric Littlewood-Richardson rule, interpreted as deforming the intersection of two Schubert varieties so that they break into Schubert varieties. There are no restrictions on the base field, and all multiplicities arising are 1; this is important for applications. This rule should be seen as a generalization of Pieri's rule to arbitrary Schubert classes, by way of explicit homotopies. It has a straightforward bijection to other Littlewood-Richardson rules, such as tableaux, and Knutson and Tao's puzzles. This gives the first geometric proof and interpretation of the Littlewood-Richardson rule. It has a host of geometric consequences, described in the companion paper "Schubert induction". The rule also has an interpretation in K-theory, suggested by Buch, which gives an extension of puzzles to K-theory. The rule suggests a natural approach to the open question of finding a Littlewood-Richardson rule for the flag variety, leading to a conjecture, shown to be true up to dimension 5. Finally, the rule suggests approaches to similar open problems, such as Littlewood-Richardson rules for the symplectic Grassmannian and two-flag varieties.

math.AG↗

arXiv2000

Hodge integrals and Hurwitz numbers via virtual localization

Ekedahl, Lando, Shapiro, and Vainshtein announced a remarkable formula expressing Hurwitz numbers (counting covers of the projective line with specified simple branch points, and specified branching over one other point) in terms of Hodge integrals. We give a proof of this formula using virtual localization on the moduli space of stable maps, and describe how the proof could be simplified by the proper algebro-geometric definition of a "relative space".

math.AG↗

arXiv1999

Twelve points on the projective line, branched covers, and rational elliptic fibrations

The following divisors in the space Sym^{12} P^1 of twelve points on P^1 are actually the same: (A) the possible locus of the twelve nodal fibers in a rational elliptic fibration (i.e. a pencil of plane cubic curves); (B) degree 12 binary forms that can be expressed as a cube plus a square; (C) the locus of the twelve tangents to a smooth plane quartic from a general point of the plane; (D) the branch locus of a degree 4 map from a hyperelliptic genus 3 curve to P^1; (E) the branch locus of a degree 3 map from a genus 4 curve to P^1 induced by a theta-characteristic; and several more. The corresponding moduli spaces are smooth, but they are not all isomorphic; some are finite etale covers of others. We describe the web of interconnections between these spaces, and give monodromy, rationality, and Prym-related consequences. Enumerative consequences include: (i) the degree of this locus is 3762 (e.g. there are 3762 rational elliptic fibrations with nodes above 11 given general points of the base); (ii) if C --> P^1 is a cover as in (D), then there are 135 different such covers branched at the same points; (iii) the general set of 12 tangent lines that arises in (C) turns up in 120 essentially different ways. Some parts of this story are well-known, and some other parts were known classically (to Zeuthen, Zariski, Coble, Mumford, and others). The unified picture is surprisingly intricate and connects many beautiful constructions, including Recillas' trigonal construction and Shioda's E8-Mordell-Weil lattice.

math.AG↗

arXiv1999

The Gromov-Witten potential of a point, Hurwitz numbers, and Hodge integrals

Hurwitz numbers, which count certain covers of the projective line (or, equivalently, factorizations of permuations into transpositions), have been extensively studied for over a century. The Gromov-Witten potential F of a point, the generating series for Hodge integrals on the moduli space of curves, has been a central object of study in Gromov-Witten theory. We define a slightly enriched Gromov-Witten potential G (including integrals involving one "lambda-class"), and show that after a non-trivial change of variables, G=H in positive genus, where H is a generating series for Hurwitz numbers. We prove a conjecture of Goulden and Jackson on higher genus Hurwitz numbers, which turns out to be an analogue of a genus expansion ansatz of Itzykson and Zuber. As consequences, we have new combinatorial constraints on F, and a much more direct proof of the ansatz of Itzykson and Zuber. We can produce recursions and explicit formulas for Hurwitz numbers; the algorithm presented should prove "all" such recursions. Furthermore, there are many more recursions than previously suspected from geometry (and indeed they should exist in all genera); as examples we present surprisingly simple new recursions in genus up to 3 that are geometrically mysterious. As we expect this paper also to be of interest to combinatorialists, we have tried to make it as self-contained as possible, including reviewing some results and definitions well known in algebraic and symplectic geometry, and mathematical physics.

math.AG↗

arXiv1998

Recursions, formulas, and graph-theoretic interpretations of ramified coverings of the sphere by surfaces of genus 0 and 1

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 and Jackson, and extend results of Hurwitz and many others.

math.CO↗

arXiv1998

The characteristic numbers of quartic plane curves

The characteristic numbers of smooth plane quartics are computed using intersection theory on a component of the moduli space of stable maps. This completes the verification of Zeuthen's prediction of characteristic numbers of smooth plane curves. A short sketch of a computation of the characteristic numbers of plane cubics is also given as an illustration.

math.AG↗

arXiv1998

Enumerative geometry of plane curves of low genus

We collect various known results (about plane curves and the moduli space of stable maps) to derive new recursive formulas enumerating low genus plane curves of any degree with various behaviors. Recursive formulas are given for the characteristic numbers of rational plane curves, elliptic plane curves, and elliptic plane curves with fixed complex structure. Recursions are also given for the number of elliptic (and rational) plane curves with various "codimension 1" behavior (cuspidal, tacnodal, triple pointed, etc., as well as the geometric and arithmetic sectional genus of the Severi variety). We compute the latter numbers for genus 2 and 3 plane curves as well. We rely on results of Caporaso, Diaz, Getzler, Harris, Ran, and especially Pandharipande.

math.AG↗

arXiv1997

The enumerative geometry of rational and elliptic curves in projective space

We study the geometry of varieties parametrizing degree d rational and elliptic curves in P^n intersecting fixed general linear spaces and tangent to a fixed hyperplane H with fixed multiplicities along fixed general linear subspaces of H. As an application, we derive recursive formulas for the number of such curves when the number is finite. These recursive formulas require as ``seed data'' only one input: there is one line in P^1 through two points. These numbers can be seen as top intersection products of various cycles on the Hilbert scheme of degree d rational or elliptic curves in P^n, or on certain components of $\mbar_0(P^n,d)$ or $\mbar_1(P^n,d)$, and as such give information about the Chow ring (and hence the topology) of these objects. The formula can also be interpreted as an equality in the Chow ring (not necessarily at the top level) of the appropriate Hilbert scheme or space of stable maps. In particular, this gives an algorithm for counting rational and elliptic curves in P^n intersecting various fixed general linear spaces. (The genus 0 numbers were found earlier by Kontsevich-Manin, and the genus 1 numbers were found for n=2 by Ran and Caporaso-Harris, and independently by Getzler for n=3.)

alg-geom↗

arXiv1997

Counting curves of any genus on rational ruled surfaces

In this paper we study the geometry of the Severi varieties parametrizing curves on the rational ruled surface $\fn$. We compute the number of such curves through the appropriate number of fixed general points on $\fn$, and the number of such curves which are irreducible. These numbers are known as Severi degrees; they are the degrees of unions of components of the Hilbert scheme. As (i) $\fn$ can be deformed to $\eff_{n+2}$, (ii) the Gromov-Witten invariants are deformation-invariant, and (iii) the Gromov-Witten invariants of $\eff_0$ and $\eff_1$ are enumerative, Theorem \ref{irecursion} computes the genus $g$ Gromov-Witten invariants of all $\fn$. (The genus 0 case is well-known.) The arguments are given in sufficient generality to also count plane curves in the style of L. Caporaso and J. Harris and to lay the groundwork for computing higher genus Gromov-Witten invariants of blow-ups of the plane at up to five points (in a future paper).

alg-geom↗

arXiv1997

Genus g Gromov-Witten invariants of Del Pezzo surfaces: Counting plane curves with fixed multiple points

As another application of the degeneration methods of [V3], we count the number of irreducible degree $d$ geometric genus $g$ plane curves, with fixed multiple points on a conic $E$, not containing $E$, through an appropriate number of general points in the plane. As a special case, we count the number of irreducible genus $g$ curves in any divisor class $D$ on the blow-up of the plane at up to five points (no three collinear). We then show that these numbers give the genus $g$ Gromov-Witten invariants of the surface. Finally, we suggest a direction from which the remaining del Pezzo surfaces can be approached, and give a conjectural algorithm to compute the genus g Gromov-Witten invariants of the cubic surface.

alg-geom↗