We study the Hilbert functions of fat points in P^1 x P^1. If Z subset of or equal to P^1 x P^1 is an arbitrary fat point scheme, 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 much greater than 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 point schemes using our description of the eventual behaviour. In fact, in the case that Z subset of or equal to P^1 x P^1 is ACM, then the entire Hilbert function and its minimal free resolution depend solely on knowing the eventual values of the Hilbert function.
Fat points in \mathbb{P}^1\times \mathbb{P}^1 and their Hilbert functions
GUARDO, ELENA MARIA;
2004-01-01
Abstract
We study the Hilbert functions of fat points in P^1 x P^1. If Z subset of or equal to P^1 x P^1 is an arbitrary fat point scheme, 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 much greater than 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 point schemes using our description of the eventual behaviour. In fact, in the case that Z subset of or equal to P^1 x P^1 is ACM, then the entire Hilbert function and its minimal free resolution depend solely on knowing the eventual values of the Hilbert function.File | Dimensione | Formato | |
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