Fat Points in P¹ x P¹ and their Hilbert Functions

2002/03/07 by Elena Guardo, Guardo, Elena, Adam Van Tuyl +1 · 1 citation
Computer Science · Mathematics · #13D40 #14A15 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #math.AC #math.AG #msc:13D40 #msc:14A15

paper · pdf · doi:10.48550/arxiv.math/0203071

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

openalex publication_date 2002/03/07 · arxiv created 2002/11/04 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the Hilbert functions of fat points in P¹ x P¹. If Z is an arbitrary fat point subscheme of P¹ x P¹, 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 P¹ x P¹. 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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• Splittings of Ideals of Points in ℙ¹×ℙ¹

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