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# Fat Points in $\mathbb{P}^1 \times \mathbb{P}^1$ and Their Hilbert Functions

Published:2004-08-01
Printed: Aug 2004
• Elena Guardo
• Adam Van Tuyl
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## Abstract

We study the Hilbert functions of fat points in $\popo$. If $Z \subseteq \popo$ is an arbitrary fat point scheme, then it can be shown that for every $i$ and $j$ the values of the Hilbert function $_{Z}(l,j)$ and $H_{Z}(i,l)$ eventually become constant for $l \gg 0$. We show how to determine these eventual values by using only the multiplicities of the points, and the relative positions of the points in $\popo$. 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 sing our description of the eventual behaviour. In fact, n the case that $Z \subseteq \popo$ is ACM, then the entire Hilbert function and its minimal free resolution depend solely on knowing the eventual values of the Hilbert function.
 Keywords: Hilbert function, points, fat points, Cohen-Macaulay, multi-projective space
 MSC Classifications: 13D40 - Hilbert-Samuel and Hilbert-Kunz functions; Poincare series 13D02 - Syzygies, resolutions, complexes 13H10 - Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05] 14A15 - Schemes and morphisms

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