Alessandro Oneto: Hilbert functions of fat points in P1xP1

Abstract: We consider ideals of fat points (with same multiplicity and generic support) in the multi-projective space P1xP1 and we want to compute their bi-graded Hilbert function. In case of double points, this has been done by Catalisano-Geramita-Gimigliano by using a so-called multiprojective-affine-projective method to reduce the problem to the study of the dimension of particular linear systems of plane curves with multiple base points. I will describe the current status of the project. In particular, by following this method, we computed the Hilbert function in bi-degree (a,b), with a >= b, of ideals of s points of multiplicity m for low bidegrees (i.e., b <= m) and for large bidegrees (i.e., a >= m(k+1)-1, where k = [s/2]). 

This problem has been suggested by Ralf Fröberg and it is an on-going joint work with Maria Virginia Catalisano.