The Euclid-Fourier-Mukai algorithm for elliptic surfaces

TL;DR

This paper explores the birational relationships between moduli spaces of semistable sheaves on elliptic surfaces using Fourier-Mukai transforms, identifying conditions for isomorphisms and describing cases involving Mukai flops.

Contribution

It introduces explicit criteria for when Fourier-Mukai induced correspondences are isomorphisms and constructs new compactifications of moduli spaces.

Findings

Identifies conditions for isomorphisms between moduli spaces

Describes cases where birational maps are Mukai flops

Provides examples of new compactifications of moduli spaces

Abstract

We describe the birational correspondences, induced by the Fourier-Mukai functor, between moduli spaces of semistable sheaves on elliptic surfaces with sections, using the notion of $P$-stability in the derived category. We give explicit conditions to determine whether these correspondences are isomorphisms. This is indeed not true in general and we describe the cases where the birational maps are Mukai flops. Moreover, this construction provides examples of new compactifications of the moduli spaces of vector bundles via sheaves with torsion and via complexes. We finally get for any fixed dimension an isomorphism between the Picard groups of the moduli spaces.