Dragos Oprea

The moduli space of stable quotients

A moduli space of stable quotients of the rank n trivial sheaf on stable curves is introduced. Over nonsingular curves, the moduli space is Grothendieck’s Quot scheme. Over nodal curves, a relative construction is made to keep the torsion of the quotient away from the singularities. New compactifications of classical spaces arise naturally: a nonsingular and irreducible compactification of the moduli of maps from genus 1 curves to projective space is obtained. Localization on the moduli of stable quotients leads to new relations in the tautological ring generalizing Brill‐Noether constructions. The moduli space of stable quotients is proven to carry a canonical 2‐term obstruction theory and thus a virtual class. The resulting system of descendent invariants is proven to equal the Gromov‐Witten theory of the Grassmannian in all genera. Stable quotients can also be used to study Calabi‐Yau geometries. The conifold is calculated to agree with stable maps. Several questions about the behavior of stable quotients for arbitrary targets are raised. 14N35; 14C17 Dedicated to William Fulton on the occasion of his 70th birthday

The level-rank duality for non-abelian theta functions

We prove that the strange duality conjecture of Beauville–Donagi–Tu holds for all curves. We establish first a more extended level-rank duality, interesting in its own right, from which the standard level-rank duality follows by restriction.

Generic strange duality for

Strange duality is shown to hold over generic K3 surfaces in a large number of cases. The isomorphism for elliptic K3 surfaces is established first via Fourier-Mukai techniques. Applications to Brill-Noether theory for sheaves on K3s are also obtained. The appendix written by Kota Yoshioka discusses the behavior of the moduli spaces under change of polarization, as needed in the argument.

K3

In the prequel to this paper, two versions of Le Potiers strange duality conjecture for sheaves over abelian surfaces were studied. A third version is considered here. In the current setup, the isomorphism involves moduli spaces of sheaves with fixed determinant and fixed determinant of the Fourier-Mukai transform on one side, and moduli spaces where both determinants vary, on the other side. We first establish the isomorphism in rank one using the representation theory of Heisenberg groups. For product abelian surfaces, the isomorphism is then shown to hold for sheaves with fiber degree 1 via Fourier-Mukai techniques. By degeneration to product geometries, the duality is obtained generically for a large number of numerical types. Finally, it is shown in great generality that the Verlinde sheaves encoding the variation of the spaces of theta functions are locally free over moduli.

surfaces

Abstract We determine generators for the codimension 1 Chow group of the moduli spaces of genus zero stable maps to flag varieties G | P. In the case of SL flags, we find all relations between our generators, showing that they essentially come from . In addition, we analyze the codimension 2 classes on the moduli spaces of stable maps to Grassmannians and prove a new codimension 2 relation. This will lead to a partial reconstruction theorem for the Grassmannian of 2 planes.

On the strange duality conjecture for abelian surfaces

Abstract We determine the splitting type of the Verlinde vector bundles in higher genus in terms of simple semihomogeneous factors. In agreement with strange duality, the simple factors are interchanged by the Fourier–Mukai transform, and their spaces of sections are naturally dual.

Divisors on the moduli spaces of stable maps to flag varieties and reconstruction

We show that the rational cohomology of the genus zero stable map spaces to SL flag varieties is entirely tautological.

The Verlinde bundles and the semihomogeneous Wirtinger duality

We study the splitting properties of the Verlinde bundles over elliptic curves. Our methods rely on the explicit description of the moduli space of semistable vector bundles on elliptic curves, and on the analysis of the symmetric powers of the Schrodinger representation of the Theta group.

The tautological rings of the moduli spaces of stable maps to flag varieties

We study the bundles of generalized theta functions constructed from moduli spaces of sheaves over abelian surfaces. The splitting type of these bundles is conjecturally expressed in terms of a new class of semihomogeneous bundles. The conjecture is confirmed in degree zero. Fourier-Mukai symmetries of the Verlinde bundles are found, consistently with strange duality. A version of level 1 strange duality is established.

A note on the Verlinde bundles on elliptic curves

We give a brief exposition of the 2d TQFT that captures the structure of the GL Verlinde numbers, following Witten.