  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.
