On the defining ideal of a set of points in multi-projective space

2004/09/03 by Adam Van Tuyl, Van Tuyl, Adam
Mathematics · #13D02 #13D40 #14A05 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #math.AG #msc:13D02 #msc:13D40 #msc:14A05

paper · pdf · doi:10.48550/arxiv.math/0409056

14 pages

arxiv created 2004/09/03 · arxiv updated 2009/12/01

Abstract

We investigate the defining ideal of a set of points X in multi-projective space with a special emphasis on the case that X is in generic position, that is, X has the maximal Hilbert function. When X is in generic position, we determine the degrees of the generators of the associated ideal I_X. Letting ν(I_X) denote the minimal number of generators of I_X, we use this description of the degrees to construct a function v(s;n₁,...,n_k) with the property that ν(\Ix) >= v(s;n₁,...,n_k) always holds for s points in generic position in P^n₁ x ... x P^n_k. When k=1, v(s;n₁) equals the expected value for ν(I_X) as predicted by the Ideal Generation Conjecture. If k >= 2, we show that there are cases with ν(\Ix) > v(s;n₁,...,n_k). However, computational evidence suggests that in many cases ν(\Ix) = v(s;n₁,...,n_k).

Related

• Resolutions of Subsets of Finite Sets of Points in Projective Space
• The regularity of points in multi-projective spaces
• Hilbert series of generic ideals in products of projective spaces
• Gale Duality and Free Resolutions of Ideals of Points
• The (unexpected) importance of knowing α