634 research outputs found
On the intersection theory of the moduli space of rank two bundles
We give an algebro-geometric derivation of the known 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 from the moduli space to a Quot scheme, where
we obtain the intersections by equivariant localization with respect to a
natural torus action
Higher rank Segre integrals over the Hilbert scheme of points
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 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.Comment: The article has been greatly expanded to include the analysis of the
Segre integrals in the rank 2 case. A complete answer is found, giving a full
analogue of Lehn's line bundle formul
The combinatorics of Lehn's conjecture
Let S be a smooth 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 of this short note 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 smooth 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
- …