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

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 $H^0A$ and $H^0L\otimes A$. 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.

Author: Gentiana Danila

Source: https://archive.org/