Modular Serre Correspondence via stable pairs

TL;DR

This paper explores the structure of moduli spaces of stable pairs on smooth projective threefolds, revealing two stability chambers and their relation via wall-crossings, generalizing Serre correspondence.

Contribution

It introduces a new modular framework for stable pairs, connecting different moduli spaces through explicit wall-crossings and extending Serre correspondence to higher dimensions.

Findings

Identified two stability chambers with morphisms to known moduli spaces.

Described explicit wall-crossings and their effects on moduli space connectedness.

Generalized Serre correspondence for semistable pairs on threefolds.

Abstract

A stable pair on a projective variety consists of a sheaf and a global section subject to stability conditions parameterized by rational polynomials. We will show that for a smooth projective threefold and a class of a rank 2 sheaf, there are two stability chambers (in the space of rational polynomials under the lexicographic order) for which the moduli spaces of semistable pairs admit morphisms to a Gieseker moduli space of rank 2 semistable sheaves and a Hilbert scheme, respectively. In the latter moduli space, every semistable pair corresponds to a closed sub-scheme of codimension 2 with an extension class, providing a generalization of the Serre correspondence. These two moduli spaces are related by finitely many wall-crossings. We provide explicit descriptions of those wall-crossings for certain fixed numerical classes. In particular, these wall-crossings preserve the connectedness of the moduli space of semistable pairs.