Construction of non-commutative surfaces with exceptional collections of length 4

Abstract

Recently de Thanhoffer de Volcsey 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 P1xP1 (and non-commutative quadrics), and an infinite family indexed by the natural numbers. For m=0,1 there are commutative and non-commutative surfaces having this Euler form, whilst for m2 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.