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Tom Graber

Publications and source records attributed to Tom Graber.

14 recordsLinked to original sources
arXiv2026

Mirror Counts of Spectral Curves

Given a convex lattice polygon $\Delta\subset \mathbb R^2$, let $N_\Delta$ be the count of rational, nodal curves in the linear system of the ample line bundle $L_\Delta$ on the toric variety $\mathbb P_\Delta$ defined by $\Delta$, having fixed intersection with the toric boundary. We show that the relative Jacobian of the linear system defines an integrable system that has a mirror dual in the language of constructible sheaves on a two-torus microsupported on a Legendrian link. In this setting, we define the dual counting problem using an analogue of rulings of Legendrian links in three-space, then prove equivalence with $N_\Delta$. We perform calculations of $N_\Delta$ in several examples using constructible methods, tropical curve counting, and localization in logarithmic Gromov-Witten theory to demonstrate the equality of the different approaches. Moreover, the rulings give rise to a stratification of the moduli of constructible sheaves. We conjecture that this ruling decomposition recovers the refined tropical invariants of Block-G\"ottsche, and hence encodes higher-genus logarithmic Gromov--Witten invariants, by a theorem of Bousseau.

math.SG↗
arXiv2019

Local Gromov-Witten Invariants are Log Invariants

We prove a simple equivalence between the virtual count of rational curves in the total space of an anti-nef line bundle and the virtual count of rational curves maximally tangent to a smooth section of the dual line bundle. We conjecture a generalization to direct sums of line bundles.

math.AG↗
arXiv2007

The Crepant Resolution Conjecture

For orbifolds admitting a crepant resolution and satisfying a hard Lefschetz condition, we formulate a conjectural equivalence between the Gromov-Witten theories of the orbifold and the resolution. We prove the conjecture for the equivariant Gromov-Witten theories of the nth symmetric product of the complex plane and the Hilbert scheme of n points in the plane.

math.AG↗
arXiv2006

On The Global Quotient Structure of The Space of Twisted Stable Maps to a Quotient Stack

Let $\mathcal{X}$ be a tame proper Deligne-Mumford stack of the form $[M/G]$ where $M$ is a scheme and $G$ is an algebraic group. We prove that the stack $\mathcal{K}_{g,n}(\mathcal{X},d)$ of twisted stable maps is a quotient stack and can be embedded into a smooth Deligne-Mumford stack. When $G$ is finite, we give a more precise construction of $\mathcal{K}_{g,n}(\mathcal{X},d)$ using Hilbert schemes and admissible $G$-covers.

math.AG↗
arXiv2005

The orbifold quantum cohomology of C^2/Z_3 and Hurwitz-Hodge integrals

Let Z_3 act on C^2 by non-trivial opposite characters. Let X =[C^2/Z_3] be the orbifold quotient, and let Y be the unique crepant resolution. We show the equivariant genus 0 Gromov-Witten potentials of X and Y are equal after a change of variables -- verifying the Crepant Resolution Conjecture for the pair (X,Y). Our computations involve Hodge integrals on trigonal Hurwitz spaces which are of independent interest. In a self contained Appendix, we derive closed formulas for these Hurwitz-Hodge integrals.

math.AG↗
arXiv2005

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. (The proof of this result is quite short and naive.) 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.

math.AG↗
arXiv2002

A note on Hurwitz schemes of covers of a positive genus curve

We prove the irreducibility of the space parametrizing branched covers of a fixed Riemann surface $B$ of degree $d$, with at least 2d branch points, and with monodromy group equal to $S_d$. The result is classical for $g(B)=0$. The result is well-known for $g(B) > 0$, but we could find no reference.

math.AG↗
arXiv2001

Algebraic orbifold quantum products

The purpose of this note is to give an overview of our work on defining algebraic counterparts for W. Chen and Y. Ruan's Gromov-Witten Theory of orbifolds. This work will be described in detail in a subsequent paper. The presentation here is generally based on lectures given by two of us at the Orbifold Workshop in Madison, Wisconsin. Following the spirit of the workshop, we give a nontechnical introduction to stacks, a general discussion of twisted stable maps, and a run-through of the issues that arise when generalizing Gromov-Witten theory to Deligne-Mumford stacks. We make a special effort to make our constructions as canonical as we can, systematically using the language of algebraic stacks. Our efforts bear some concrete fruits; in particular, we are able to define the Chen-Ruan product in degree 0 (the so called stringy cohomology of the stack) with integer coefficients. Beware - this note may be vanishing in front of your eyes: first, a somewhat less technical version intended for the workshop proceedings will shortly (we hope) be submitted. Second, a much more detailed paper is in preparation, in which all the technicalities which are swept under the rug here are discussed, as well as some cool stuff about taking roots.

math.AG↗
arXiv2001

Open-String Gromov-Witten Invariants: Calculations and a Mirror "Theorem"

We propose localization techniques for computing Gromov-Witten invariants of maps from Riemann surfaces with boundaries into a Calabi-Yau, with the boundaries mapped to a Lagrangian submanifold. The computations can be expressed in terms of Gromov-Witten invariants of one-pointed maps. In genus zero, an equivariant version of the mirror theorem allows us to write down a hypergeometric series, which together with a mirror map allows one to compute the invariants to all orders, similar to the closed string model or the physics approach via mirror symmetry. In the noncompact example where the Calabi-Yau is $K_{\PP^2},$ our results agree with physics predictions at genus zero obtained using mirror symmetry for open strings. At higher genera, our results satisfy strong integrality checks conjectured from physics.

hep-th↗
arXiv2001

Descendant invariants and characteristic numbers

On a stack of stable maps, the psi classes are modified by subtracting certain boundary divisors. These modified psi classes are compatible with forgetful morphisms, and are well-suited to enumerative geometry: tangency conditions allow simple expressions in terms of modified psi classes. Topological recursion relations are established among their top products in genus zero, yielding effective recursions for characteristic numbers of rational curves in any projective homogeneous variety. In higher genus, the obtained numbers are only virtual, due to contributions from spurious components of the space of maps. For the projective plane, the necessary corrections are determined in genus 1 and 2 to give the characteristic numbers in these cases.

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↗
arXiv1998

Enumerative geometry of hyperelliptic plane curves

We recursively compute the Gromov-Witten invariants of the Hilbert scheme of two points in the plane. By studying the space of stable maps and computing virtual contributions, we use these invariants to enumerate hyperelliptic plane curves of degree d and genus g passing through 3d+1 general points.

math.AG↗