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arXiv · 1705.06943

Construction of noncommutative surfaces with exceptional collections of length 4

Pieter Belmans·Dennis Presotto
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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 $m\geq 2$ 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.

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Pieter Belmans, Dennis Presotto. 2018-12-28. Construction of noncommutative surfaces with exceptional collections of length 4. https://doi.org/10.1112/jlms.12126

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