Fat Points in P1 × P1 and Their Hilbert Functions

Guardo, Elena ; Tuyl, Adam Van

Canadian journal of mathematics, Tome 56 (2004) no. 4, pp. 716-741

Résumé

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.

Détail

DOI : 10.4153/CJM-2004-033-0

Mots-clés : 13D40, 13D02, 13H10, 14A15, Hilbert function, points, fat points, Cohen-Macaulay multi-projective space

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