Moduli of noncommutative Hirzebruch surfaces

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 $⟨\frac{1}{d}(1,1)⟩$.

arXiv comment34 pages

References in corpus (6)

• Blowing up non-commutative smooth surfaces
• Smooth and proper noncommutative schemes and gluing of DG categories
• Hochschild homology and semiorthogonal decompositions
• Compact moduli of noncommutative projective planes
• Noncommutative quadric surfaces and noncommutative conifolds
• Quantum entanglement, Calabi-Yau manifolds, and noncommutative algebraic geometry