Tom Graber

Families of rationally connected varieties

Recall that a proper variety X is said to be rationally connected if two general points p, q ∈ X are contained in the image of a map g : P → X. This is clearly a birationally invariant property. When X is smooth, this turns out to be equivalent to the a priori weaker condition that two general points can be joined by a chain of rational curves and also to the a priori stronger condition that for any finite subset Γ ⊂ X, there is a map g : P → X whose image contains Γ and such that gTX is an ample bundle.

Gromov-Witten theory of Deligne-Mumford stacks

Given a smooth complex Deligne-Mumford stack

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

{\cal X}

The orbifold quantum cohomology of ℂ²/ℤ₃ and Hurwitz-Hodge integrals

with a projective coarse moduli space, we introduce Gromov-Witten invariants of

On the global quotient structure of the space of twisted stable maps to a quotient stack

{\cal X}

Descendant invariants and characteristic numbers

and prove some of their basic properties, including the WDVV equation.

Arithmetic Questions Related to Rationally Connected Varieties

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.

Jumps in Mordell-Weil Rank and Arithmetic Surjectivity

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.

LOCALIZATION OF VIRTUAL CLASSES

Let 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 Kg,n(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 Kg,n(X , d) using Hilbert schemes and admissible G-covers.

Algebraic orbifold quantum products

On a stack of stable maps, the cotangent line classes are modified by subtracting certain boundary divisors. These modified cotangent line classes are compatible with forgetful morphisms, and are well-suited to enumerative geometry: tangency conditions allow simple expressions in terms of modified cotangent line classes. Topological recursion relations are established among their top products in genus 0, 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.