Sur la cohomologie d'un fibre tautologique sur le schema de Hilbert d'une surface

Abstract

We compute the cohomology spaces for the tautological bundle tensor the determinant bundle on the punctual Hilbert scheme H of subschemes of length n of a smooth projective surface X. We show that for L and A invertible vector bundles on X, and w the canonical bundle of X, if $w^{-1}\otimes L$, $w^{-1}\otimes A$ and A are ample vector bundles, then the higher cohomology spaces on H of the tautological bundle associated to L tensor the determinant bundle associated to A vanish, and the space of global sections is computed in terms of $H^0(A)$ and $H^0(L\otimes A)$. This result is motivated by the computation of the space of global sections of the determinant bundle on the moduli space of rank 2 semi-stable sheaves on the projective plane, supporting Le Potier's Strange duality conjecture on the projective plane.

27 pages, part of PhD thesis

Cited by in corpus (5)

• Vanishing theorems and character formulas for the Hilbert scheme of points in the plane
• On the Cobordism Class of the Hilbert Scheme of a Surface
• Higher symmetric powers of tautological bundles on Hilbert schemes of points on a surface
• Rank-one sheaves and stable pairs on surfaces
• Topological properties of punctual Hilbert schemes of almost-complex fourfolds (I)