The length classification of threefold flops via noncommutative algebras

TL;DR

This paper translates the classification of threefold flops by length into noncommutative algebra, introducing universal flopping algebras as quivers with relations to construct and analyze examples of threefold flops.

Contribution

It introduces universal flopping algebras of each length as quivers with relations, enabling explicit constructions and analysis of threefold flops via noncommutative algebra.

Findings

Universal flopping algebras can be realized as quivers with relations.

Construction of NCCRs associated to threefold flops of any length.

Recovery of matrix factorization descriptions of universal flops.

Abstract

Smooth threefold flops with irreducible centres are classified by the length invariant, which takes values 1, 2, 3, 4, 5 or 6. This classification by Katz and Morrison identifies 6 possible partial resolutions of Kleinian singularities that can occur as generic hyperplane sections, and the simultaneous resolutions associated to such a partial resolution produce the universal flop of length $l$. In this paper we translate these ideas into noncommutative algebra. We introduce the universal flopping algebra of length $l$ from which the universal flop of length $l$ can be recovered by a moduli construction, and we present each of these algebras as the path algebra of a quiver with relations. This explicit realisation can then be used to construct examples of NCCRs associated threefold flops of any length as quiver with relations defined by superpotentials, to recover the matrix factorisation description of the universal flop conjectured by Curto and Morrison, and to realise examples of contraction algebras.