CHAPTER 1 Introduction Throughout this paper k will be an algebraically closed field. 1.1. Motivation Let X be a smooth connected surface over k and let q be a Poisson bracket on X. Since we are in the dimension two, q corresponds to a section of the anti-canonical bundle oo*x. Let p G X and let a : X X be the blowup of X at p. Prom the fact that X and X share the same function field it is easily seen that q extends to X if and only if q vanishes at p. Denote the extended Poisson bracket by q' and let Y resp. T be the zero divisors of q and qf. One verifies that as divisors : T = a~x(Y) L, where L = a~x(p) is the exceptional curve. In particular T contains the strict transform Y of Y, and if p G F is simple then actually T = Y. Our aim in this paper is to show that there exists a non-commutative version of this situation. That is we show that it is possible to view the blowup of a Poisson surface as the quasi-classical analogue of a blowup of a non-commutative surface. Our motivation for doing this is to provide a step in the ongoing project of classifying graded domains of low Gelfand-Kirillov dimension. Since the case of dimension two was completely solved in [5] the next interesting case will very likely be dimension three (leaving aside rings with fractional dimension which seem to be quite exotic). One may view three dimensional graded rings as homogeneous coordinate rings of non-commutative projective surfaces. Motivated by some heuristic evidence Mike Artin conjectures in [4] that, up to birational equivalence, there will be only a few classes, the largest one consisting of those algebras that are birational to a quantum P2 (see below). Once a birational classification exists, one might hope that there would be some version of Zariski's theorem saying that if two (non-commutative) surfaces are birationally equivalent then they are related through a sequence of blowing ups and downs. With the current level of understanding it seems rather unlikely that Artin's conjecture or a non-commutative version of Zariski's theorem will be proved soon, but this paper provides at least one piece of the puzzle. This being said, it is perhaps the right moment to point out that in this paper we won't really define the notion of a non-commutative surface. Instead we first introduce non-commutative schemes (or quasi-schemes, to follow the terminology of [28]). These will simply be abelian categories having sufficiently nice homological properties. t Then we will impose a few convenient additional hypotheses which would hold for a commutative smooth surface (see §5.1). To fix ideas we will first discuss two particular cases of quasi-schemes. If R is a ring then SpecR is the category of right i?-modules (the "affine case"). If A = AQ 0 Ai ® is a graded ring then Proj A is (roughly) the category of graded l
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