Moduli of noncommutative Hirzebruch surfaces

TL;DR

This paper introduces three interconnected moduli stacks for noncommutative deformations of Hirzebruch surfaces, establishing their birational equivalence and providing classifications, decompositions, and a noncommutative McKay correspondence.

Contribution

It constructs and relates three moduli stacks of noncommutative Hirzebruch surfaces, offering new classifications, semiorthogonal decompositions, and a derived equivalence generalizing McKay correspondence.

Findings

The three moduli stacks are birationally equivalent.

Classification of locally free sheaf bimodules of rank 2 is provided.

A noncommutative McKay correspondence is established.

Abstract

We introduce three non-compact moduli stacks parametrizing noncommutative deformations of Hirzebruch surfaces; the first is the moduli stack of locally free sheaf bimodules of rank 2, which appears in the definition of noncommutative $\mathbb{P}^1$-bundle in the sense of Van den Bergh arXiv:math/0102005, the second is the moduli stack of relations of a quiver in the sense of arXiv:1411.7770, and the third is the moduli stack of quadruples consisting of an elliptic curve and three line bundles on it. The main result of this paper shows that they are naturally birational to each other. We also give an Orlov-type semiorthogonal decomposition for noncommutative $\mathbb{P}^1$-bundles, an explicit classification of locally free sheaf bimodules of rank 2, and a noncommutative generalization of the (special) McKay correspondence as a derived equivalence for the cyclic group $\left\langle \frac{1}{d}(1,1) \right\rangle$.