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 ${\mathrm{Hilb}}^d(S)$ be the Hilbert scheme parametrizing the zero-dimensional subschemes of $S$ of length $d$. For a vector bundle $E$ on $S$, let $\mathcal{H}(E)⟶{\mathrm{Hilb}}^d(S)$ be its Fourier--Mukai transform constructed using the structure sheaf of the universal subscheme of $S\times {\mathrm{Hilb}}^d(S)$ as the kernel. We prove that two vector bundles $E$ and $F$ on $S$ are isomorphic if the vector bundles $\mathcal{H}(E)$ and $\mathcal{H}(F)$ are isomorphic.

To appear in JRMS