The length classification of threefold flops via noncommutative algebras

Understand

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.