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.

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

Let S be a nonsingular projective surface equipped with a line bundle H. Lehn's conjecture is a formula for the top Segre class of the tautological bundle associated to H on the Hilbert scheme of points of S. Voisin has recently reduced Lehn's conjecture to the vanishing of certain coefficients of special power series. The first result here is a proof of the vanishings required by Voisin by residue calculations (A. Szenes and M. Vergne have independently found the same proof). Our second result is an elementary solution of the parallel question for the top Segre class on the symmetric power of a nonsingular projective curve C associated to a higher rank vector bundle V on C. Finally, we propose a complete conjecture for the top Segre class on the Hilbert scheme of points of S associated to a higher rank vector bundle on S in the K-trivial case.

We construct a virtual fundamental class on the Quot scheme parametrizing quotients of a trivial bundle on a smooth projective curve. We use the virtual localization formula to calculate virtual intersection numbers on Quot. As a consequence, we reprove the Vafa-Intriligator formula; our answer is valid even when the Quot scheme is badly behaved. More intersections of Vafa-Intriligator type are computed by the same method. Finally, we present an application to the non-vanishing of the Pontrjagin ring of the moduli space of bundles.

Let S be a nonsingular projective surface. Each vector bundle V on S of rank s induces a tautological vector bundle over the Hilbert scheme of n points of S . When s=1 , the top Segre classes of the tautological bundles are given by a recently proven formula conjectured in 1999 by M. Lehn. We calculate here the Segre classes of the tautological bundles for all ranks s over all K -trivial surfaces. Furthermore, in rank s=2 , the Segre integrals are determined for all surfaces, thus establishing a full analogue of Lehn's formula. We also give conjectural formulas for certain series of Verlinde Euler characteristics over the Hilbert schemes of points.