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Abstract

Recently de Thanhoffer de Völcsey and Van den Bergh classified the Euler forms on a freeabelian 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 $\mathbb{P}^1 × \mathbb{P}^1$ (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 form $m \geq 2$ there are nocommutative surfaces. In this paper, we construct sheaves of maximal orders on surfaces having these Eulerforms, giving a geometric construction for their numerical blowups.