Moduli of non-pure sheaves and contractions of Hilbert schemes

TL;DR

This paper develops moduli spaces for rank one sheaves on smooth projective varieties, applying them to Hilbert scheme contractions, and demonstrates a non-GIT wall-crossing approach to prove cases of Kawamata's DK-hypothesis.

Contribution

It introduces new stacks of sheaves with proper moduli spaces and connects wall-crossing phenomena to Kawamata's conjecture in the context of Hilbert schemes.

Findings

Proper moduli spaces for certain sheaves are constructed.

A surgery diagram via wall-crossing is developed.

Instances of Kawamata's DK-hypothesis are proved.

Abstract

We define certain stacks of rank one sheaves on a smooth projective variety, and show that they admit proper good moduli spaces. We offer several applications to contractions of subschemes inside Hilbert schemes of points. We construct a surgery diagram via non-GIT wall-crossing, and use the interpretation of the surgery as a fine moduli of sheaves to prove instances Kawamata's DK-hypothesis in this setting.