Fat Points in P^1 x P^1 and their Hilbert Functions

Details Fat Points in P^1 x P^1 and their Hilbert Functions Fat Points in P^1 x P^1 and their Hilbert Functions

2002-03-07

arxiv.org/abs/math/0203071v2

Mathematics

Commutative Algebra

23 Pages. Revised version to appear in Can. J. Math

Scientific paper

We study the Hilbert functions of fat points in P^1 x P^1. If Z is an arbitrary fat point subscheme of P^1 x P^1, then it can be shown that for every i and j the values of the Hilbert function H_Z(l,j) and H_Z(i,l) eventually become constant for l >> 0. We show how to determine these eventual values by using only the multiplicities of the points, and the relative positions of the points in P^1 x P^1. This enables us to compute all but a finite number values of H_Z without using the coordinates of points. We also characterize the ACM fat points schemes using our description of the eventual behaviour. In fact, in the case that Z is ACM, then the entire Hilbert function and its minimal free resolution depend solely on knowing the eventual values of the Hilbert function.

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