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.

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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

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