Fourier-Mukai Transform of Vector Bundles on Surfaces to Hilbert Scheme

Abstract

Let S be an irreducible smooth projective surface defined over an algebraically closed field k. For a positive integer d, let Hilb^d (S) be the Hilbert scheme parametrizing the zero-dimensional subschemes of S of length d. For a vector bundle E on S, let H(E) →Hilb^d (S) be its Fourier- Mukai transform constructed using the structure sheaf of the universal subscheme of S × Hilb^d (S) as the kernel. We prove that two vector bundles E and F on S are isomorphic if the vector bundles H(E) and H(F) are isomorphic.