In this paper we will think of certain abelian categories with favorable proper-
ties 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 at 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.
Received by the editor November 2, 1998, and in revised form July 10, 2000.
1991 Mathematics Subject Classification. Primary 16E40.
Key words and phrases. Blowing up, non-commutative geometry.
The author is a senior researcher at the FWO.
The author dedicates this monograph to Sarah and Bertold.
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