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 $\le 8$ 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.

Cited by in corpus (48)

• Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves
• G-algebras, twistings, and equivalences of graded categories
• Naive Noncommutative Blowing Up
• Poisson Cohomology of Del Pezzo surfaces
• Noncommutative Blowups of Elliptic Algebras
• The Grothendieck Group of a Quantum Projective Space Bundle
• Hilbert squares: derived categories and deformations
• Admissible subcategories in derived categories of moduli of vector bundles on curves
• Weighted noncommutative regular projective curves
• An introduction to Noncommutative Projective Geometry
• Classification of categorical subspaces of locally noetherian schemes
• Serre duality for non-commutative P^1-bundles
• The image of Colmez's Montreal functor
• Calabi-Yau algebras and superpotentials
• Localisations and Completions of Skew Power Series Rings
• Non-commutative Geometry of Homogenized Quantum $\mathfrak{sl}(2,\mathbb{C})$
• The noncommutative schemes of generalized Weyl algebras
• Classifying Orders in the Sklyanin Algebra
• The Picard group of the graded module category of a generalized Weyl algebra
• Noncommutative Geometry in the Framework of Differential Graded Categories
• Noncommutative rigidity of the moduli stack of stable pointed curves
• Generalized double affine Hecke algebras of rank 1 and quantized Del Pezzo surfaces
• Well-closed subschemes of noncommutative schemes
• On some approaches towards non-commutative algebraic geometry
• On Continuous and Adjoint Morphisms Between Noncommutative Prime Spectra
• Moduli of noncommutative Hirzebruch surfaces
• The finite dual coalgebra as a quantization of the maximal spectrum
• Construction of noncommutative surfaces with exceptional collections of length 4
• Ring-theoretic blowing down: I
• "Classical" flag varieties for quantum groups: the standard quantum SL(n,C)
• ${\mathbb{Z}}^2$-algebras as noncommutative blow-ups
• Some Noncommutative Minimal Surfaces
• Degenerations of orbifold curves as noncommutative varieties
• Ring-theoretic blowing down II: Birational transformations
• Noncommutative Images of Commutative Spectra
• Blowup subalgebras of the Sklyanin algebra
• The Eilenberg-Watts theorem over schemes
• Noncommutative Tsen's theorem in dimension one
• Continuity is an adjoint functor
• Naive noncommutative blowups at zero-dimensional schemes
• Noetherian Skew Inverse Power Series Rings
• Serre finiteness and Serre vanishing for non-commutative P^1-bundles
• Ideal classes of three dimensional Artin-Schelter regular algebras
• Ideal classes of three dimensional Sklyanin algebras
• A Remark on Closed Noncommutative Subspaces
• Noetherianity of the Space of Irreducible Representations
• Residue Complexes over Noncommutative Rings
• Exotic Elliptic Algebras of dimension 4 (with an Appendix by Derek Tomlin)