A categorical flop in dimension one

TL;DR

This paper explores the categorical structure of flops in one dimension, linking non-commutative resolutions of nodal curves with Landau-Ginzburg models and introducing an intermediate crepancy condition.

Contribution

It identifies the categorical flop structure in non-commutative resolutions of nodal curves and provides a geometric interpretation via Landau-Ginzburg models, along with a new crepancy condition.

Findings

Categorical flop structure observed in non-commutative resolutions.

Description of flop-flop spherical twists.

Introduction of an intermediate crepancy condition.

Abstract

In this note we observe that the categorical structure of a flop occurs for some well-known non-commutative resolutions of a nodal curve. We describe the flop-flop spherical twists, and give a geometric interpretation in terms of Landau--Ginzburg models. The resolutions are all weakly crepant but not strongly crepant, and we formulate an intermediate condition that distinguishes the smaller ones.