Construction of noncommutative surfaces with exceptional collections of length 4

Abstract

Recently de Thanhoffer de Völcsey and Van den Bergh classified the Euler forms on a free abelian group of rank 4 having the properties of the Euler form of a smooth projective surface. There are two types of solutions: one corresponding to ${\mathbb{P}}^1\times {\mathbb{P}}^1$ (and noncommutative quadrics), and an infinite family indexed by the natural numbers. For $m=0,1$ there are commutative and noncommutative surfaces having this Euler form, whilst for $m\ge 2$ there are no commutative surfaces. In this paper we construct sheaves of maximal orders on surfaces having these Euler forms, giving a geometric construction for their numerical blowups.

24 pages, see also companion paper arXiv:1811.08810

References in corpus (1)

• Derived categories of curves as components of Fano manifolds