Non-commutative deformations of perverse coherent sheaves and rational curves

TL;DR

This paper develops tools for non-commutative deformation theory of sheaves on algebraic varieties, applying them to rational curves and confirming a conjecture in specific flop cases.

Contribution

It introduces methods to determine parameter algebras of versal non-commutative deformations for certain sheaves and applies these to universal flopping contractions, verifying a conjecture.

Findings

Confirmed Donovan-Wemyss conjecture for Laufer's flops

Developed tools for parameter algebras of non-commutative deformations

Applied methods to higher-length flopping contractions

Abstract

We consider non-commutative deformations of sheaves on algebraic varieties. We develop some tools to determine parameter algebras of versal non-commutative deformations for partial simple collections and the structure sheaves of smooth rational curves. We apply them to universal flopping contractions of length 2 and higher. We confirm Donovan-Wemyss conjecture in the case of deformations of Laufer's flops.