Blowing up non-commutative smooth 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 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.
Cite
@article{arxiv.math/9809116,
title = {Blowing up non-commutative smooth surfaces},
author = {Michel Van den Bergh},
journal= {arXiv preprint arXiv:math/9809116},
year = {2007}
}