Stable pairs on elliptic K3 surfaces

TL;DR

This paper explores the moduli spaces of semistable pairs on elliptic K3 surfaces, revealing their birational relationships and establishing isomorphisms with Hilbert schemes, thus advancing understanding of their geometric structures.

Contribution

It constructs a family of moduli spaces of pairs on elliptic K3 surfaces and analyzes wall crossing phenomena to connect different moduli spaces and Hilbert schemes.

Findings

Moduli spaces of pairs relate to sheaves and Hilbert schemes.

Wall crossing phenomena describe birational transformations.

In 4D, moduli space is isomorphic to the Hilbert scheme.

Abstract

We study semistable pairs on elliptic K3 surfaces with a section: we construct a family of moduli spaces of pairs, related by wall crossing phenomena, which can be studied to describe the birational correspondence between moduli spaces of sheaves of rank 2 and Hilbert schemes on the surface. In the 4-dimensional case, this can be used to get the isomorphism between the moduli space and the Hilbert scheme described by Friedman.