1998/09/21 by Michel Van den Bergh, Bergh, Michel Van den · 1 citation
Mathematics · #16E40 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #math.QA #msc:16E40
paper · pdf · doi:10.48550/arxiv.math/9809116
arxiv created 1998/09/21 · openalex publication_date 1998/09/21 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ≤ 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.