A survey on fat points on a smooth quadric

Abstract. In this survey, which expands on a talk given at the XV ColoquiumLatinoamericano de Algebra, July 20–26, 2003, in Cocoyoc, Mexico, weillustrate 0-dimensional fat points schemes on a smooth quadric Q = P1 × P1.Given any fat point scheme Z, we construct two tuples \alpha_Z and \beta_Z and weshow how to compute all but a finite number of values of the Hilbert functionfrom \alpha_Z and \beta_Z. We show the characterization of those schemes which arearithmetically Cohen-Macaulay (ACM for short) as subschemes of Q givingtheir Hilbert matrix and bigraded Betti numbers.Moreover, we also find a minimal set of generators and the Hilbert functionfor those subschemes of double points whose support is ACM, whether ornot the subschemes are ACM.