The border of the Hilbert function of a set of points in P^n₁ x ... x P^n_k

2001/12/18 by Adam Van Tuyl, Van Tuyl, Adam
Mathematics · #13D40 (Primary) 05A17 14M05 (Secondary) #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #math.AG #math.CO #msc:05A17 #msc:13D40 #msc:14M05

paper · pdf · doi:10.48550/arxiv.math/0112185

22 pages

arxiv created 2001/12/18 · arxiv updated 2009/11/30

Abstract

We describe the eventual behaviour of the Hilbert function of a set of distinct points in P^n₁ x ... x P^n_k. As a consequence of this result, we show that the Hilbert function of a set of points in P^n₁ x ... x P^n_k can be determined by computing the Hilbert function at only a finite number of values. Our result extends the result that the Hilbert function of a set of points in Pⁿ stabilizes at the cardinality of the set of points. Motivated by our result, we introduce the notion of the_border_ of the Hilbert function of a set of points. By using the Gale-Ryser Theorem, a classical result about (0,1)-matrices, we characterize all the possible borders for the Hilbert function of a set of distinct points in P¹ x P¹.

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