Dragos Oprea

Divisors on the moduli spaces of stable maps to flag varieties and reconstruction PDF Crelle's Journal, 586 (2005), 169-205.

The tautological rings of the moduli spaces of stable maps PDF Journal of Algebraic Geometry, 15 (2006), 623-655.

We define and study the tautological systems of the moduli spaces of stable maps. We show that the rational cohomology of the moduli space of genus 0 stable maps to SL-flag varieties is generated by tautological classes. This result generalizes a well-known theorem of Keel, valid when the target is a point. As a corollary, we conclude that the Gromov-Witten invariants capture the entire intersection theory of the moduli space of stable maps to flag varieties, in genus 0.

Tautological classes on the moduli spaces of stable maps to projective spaces via torus actions PDF Advances in Mathematics, 207 (2006), 661-690.

We study the Bialynicki-Birula stratification of the moduli space of genus 0 stable maps to projective spaces for certain non-generic torus actions. The Bialynicki-Birula closed cells are described in terms of the Li-Gathmann spaces of relative stable morphisms. We show that the decomposition is filterable by explicitly describing an ordering of the cells. As a corollary, we prove that the rational cohomology and the rational Chow ring of the moduli spaces of stable maps are isomorphic, and in fact generated by tautological classes.

On the intersection theory on the moduli space of rank two bundles (with Alina Marian) PDF Topology, 45 (2006), 531-541.

We give an algebro-geometric derivation of the intersection theory on the moduli space of stable rank 2 bundles of odd degree over a smooth curve of genus g. We lift the computation computation from the moduli space to a Quot scheme, where the answer is evaluated via equivariant localization.

Virtual intersections on the Quot scheme and Vafa-Intriligator formulas (with Alina Marian) PDF Duke Math Journal, 136 (2007), 81-113.

We define a virtual fundamental class on the Quot scheme of arbitrary rank quotients of a fixed bundle over a smooth projective curve. We use the virtual localization formula to calculate virtual intersection numbers on Quot. We obtain closed formulas for some intersections of tautological classes, generalizing the ones written down by Vafa and Intriligator. Finally, we present an application of our method to the study of the Pontrjagin ring of the moduli space of stable bundles over a curve.

Counts of maps to Grassmannians and intersections on the moduli space of bundles (with Alina Marian) PDF Journal of Differential Geometry, 76 (2007), 155 - 175.

We show that intersection numbers on the moduli space of stable bundles of coprime rank and degree over a smooth complex curve can be recovered as highest degree asymptotics in formulas of Vafa-Intriligator type, i.e. formulas which capture some of the enumerative geometry of the space of maps to the Grassmannian. In particular, we evaluate algebraically all intersection numbers appearing in the Verlinde formula. Our results are in agreement with previous computations of Witten, Jeffrey-Kirwan and Liu. Moreover, we prove the vanishing of certain intersections on the Quot scheme, which can be interpreted as giving equations between counts of maps to the Grassmannian.

The level-rank duality for nonabelian theta functions (with Alina Marian) PDF Inventiones Mathematicae, 168 (2007), 225-247.

We prove that the strange duality conjecture of Beauville-Donagi-Tu holds for all curves, all ranks and degrees. This conjecture relates non-abelian theta functions, that is sections of the determinant bundle, on two different moduli spaces; the rank and the level are exchanged in the process. The conjecture follows from a more general level-rank duality which is the nonabelian generalization of the classical Wirtinger duality on the space of 2-Theta functions.

A tour of theta dualities on moduli spaces of sheaves (with Alina Marian), Curves and abelian varieties, 175-202, Contemporary Mathematics, 465, American Mathematical Society, Providence, Rhode Island (2008) - PDF.

The purpose of this paper is twofold. First, we survey known results about theta dualities on the moduli space of bundles on curves and surfaces. Secondly, we establish new dualities on the moduli spaces of sheaves over surfaces. Among others, the case of elliptic K3 surfaces is studied in detail; we propose an explicit approach/reduction of the strange duality conjecture.

Sheaves on abelian surfaces and Strange Duality (with Alina Marian), Math Annalen, 343 (2009), 1-33 - PDF.

We formulate three versions of a strange duality conjecture for sections of the theta bundles on different moduli spaces of sheaves on abelian surfaces. As supporting evidence, we check the equality of dimensions on dual moduli spaces. This sets the stage for strange duality for abelian surfaces, and also answers a question raised by Gottsche-Nakajima-Yoshioka.

A note on the Verlinde bundles on elliptic curves, accepted to Trans. Amer. Math. Soc. - PDF.

The non-abelian theta functions can be organized into Verlinde vector bundles over the Jacobian. We determine the splitting type of the Verlinde bundles on elliptic curves in terms of the indecomposable Atiyah bundles. The argument relies on the analysis of the Heisenberg action on the symmetric powers of the Schrodinger representation.

The Verlinde bundles and the semihomogeneous Wirtinger duality, submitted - PDF.

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 Fourier-Mukai, and their spaces of sections are naturally dual.

On the strange duality conjecture for elliptic K3 surfaces (with Alina Marian), submitted - PDF.

We consider moduli spaces of semistable sheaves on an elliptically fibered K3 surface, so that the first Chern class of the sheaves is a numerical section. For pairs of complementary such moduli spaces, we establish the strange duality isomorphism on sections of theta line bundles.

The moduli space of stable quotients (with Alina Marian and Rahul Pandharipande), preprint - PDF.

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.

Tautological relations on Kontsevich spaces, draft - This will most likely be rewritten to include some new results and streamline the exposition.

The cohomology of the Kontsevich spaces of genus $0$ stable maps to flag varieties is generated by tautological classes. We study relations between the tautological generators. We conjecture that all relations between these generators are tautological, i.e. they are essentially obtained from the pullbacks of Keel's relations on the moduli space of curves, with the aid of the pushforwards by the natural morphisms. We offer supporting evidence for this conjecture in low codimensions.