# Construction of noncommutative surfaces with exceptional collections of length 4

## TL;DR

This paper constructs sheaves of maximal orders on surfaces with specific Euler forms, providing a geometric approach to their numerical blowups, expanding the understanding of noncommutative surfaces with exceptional collections.

## Contribution

It introduces a geometric construction of noncommutative surfaces with Euler forms classified by de Thanhoffer de V"olcsey and Van den Bergh, including their numerical blowups.

## Key findings

- Constructed sheaves of maximal orders on surfaces with given Euler forms.
- Provided geometric constructions for noncommutative quadrics and related surfaces.
- Extended the classification to include surfaces with no commutative counterparts.

## Abstract

Recently de Thanhoffer de V\"olcsey 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.