On the ideal of Veronesean surfaces

We consider blowing up of P^2 at s sufficiently general distinct poits and its projective embedding by the linear system of the curves of a given degree through the points. We study the iodeal of the resulting (Veronesean) surface and find that it can be described by two matrices of linear forms, in the sense that it is generatedby the entries of the product matrix and the minors of complementary orders of the two matrices. By cuttingb twice with general hyperplanes, we also obtain information about the generation (or even the resolution) of certain classes of points in projective space.