Blowing up non-commutative smooth surfaces

TL;DR

This paper develops a method to blow up points on non-commutative surfaces, creating new surfaces that are birational to the original, and explores their connections to non-commutative deformations of classical algebraic surfaces.

Contribution

It introduces a non-commutative blow-up procedure for abelian categories modeling surfaces, extending classical geometric concepts to the non-commutative setting.

Findings

Blowing up points yields new non-commutative surfaces birational to the original.

Constructs non-commutative deformations of Del-Pezzo and cubic surfaces.

Provides a formula for counting simple objects on these surfaces.

Abstract

In this paper we will think of certain abelian categories with favorable properties as non-commutative surfaces. We show that under certain conditions a point on a non-commutative surface can be blown up. This yields a new non-commutative surface which is in a certain sense birational to the original one. This construction is analogous to blowing up a Poisson surface in a point of the zero-divisor of the Poisson bracket. By blowing up $\le 8$ points in the elliptic quantum plane one obtains global non-commutative deformations of Del-Pezzo surfaces. For example blowing up six points yields a non-commutative cubic surface. Under a number of extra hypotheses we obtain a formula for the number of non-trivial simple objects on such non-commutative surfaces.