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Fat Points in ℙ1 × ℙ1 and Their Hilbert Functions

Published online by Cambridge University Press:  20 November 2018

Elena Guardo
Affiliation:
Dipartimento di Matematica e Informatica, Viale A. Doria 6 - 95100 – Catania, Italy e-mail: guardo@dmi.unict.it
Adam Van Tuyl
Affiliation:
Department of Mathematics, Lakehead University, Thunder Bay, ON, P7B 5E1 e-mail: avantuyl@sleet.lakeheadu.ca
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Abstract

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We study the Hilbert functions of fat points in ${{\mathbb{P}}^{1\,}}\times \,{{\mathbb{P}}^{1}}$ . If $Z\,\subseteq \,{{\mathbb{P}}^{1\,}}\times \,{{\mathbb{P}}^{1}}$ 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 $l\,\gg \,0$ . We show how to determine these eventual values by using only the multiplicities of the points, and the relative positions of the points in ${{\mathbb{P}}^{1\,}}\times \,{{\mathbb{P}}^{1}}$ . This enables us to compute all but a finite number values of ${{H}_{Z}}$ without using the coordinates of points. We also characterize the $\text{ACM}$ fat point schemes using our description of the eventual behaviour. In fact, in the case that $Z\,\subseteq \,{{\mathbb{P}}^{1\,}}\times \,{{\mathbb{P}}^{1}}$ is $\text{ACM}$ , then the entire Hilbert function and its minimal free resolution depend solely on knowing the eventual values of the Hilbert function.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2004

References

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