Fourier-Mukai transform of vector bundles on surfaces to Hilbert scheme

TL;DR

This paper demonstrates that the Fourier-Mukai transform on a surface induces an isomorphism of vector bundles on the surface to an isomorphism of their transforms on the Hilbert scheme, establishing a strong correspondence.

Contribution

It proves that the Fourier-Mukai transform on a surface preserves the isomorphism class of vector bundles, linking surface bundles to their Hilbert scheme transforms.

Findings

Isomorphic vector bundles on the surface have isomorphic transforms on the Hilbert scheme.

The Fourier-Mukai transform is faithful on the category of vector bundles.

This establishes a criterion for bundle isomorphism via their transforms.

Abstract

Let $S$ be an irreducible smooth projective surface defined over an algebraically closed field $k$. For a positive integer $d$, let ${\rm Hilb}^d(S)$ be the Hilbert scheme parametrizing the zero-dimensional subschemes of $S$ of length $d$. For a vector bundle $E$ on $S$, let ${\mathcal H}(E)\, \longrightarrow\, {\rm Hilb}^d(S)$ be its Fourier--Mukai transform constructed using the structure sheaf of the universal subscheme of $S\times {\rm Hilb}^d(S)$ 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.