Powers of complete intersections: graded Betti numbers and applications

2004/09/06 by Elena Guardo, Guardo, Elena, Adam Van Tuyl +1 · 3 citations
Computer Science · Mathematics · #13D02 #13D40 #13H10 #14A15 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #math.AC #math.AG #msc:13D02 #msc:13D40 #msc:13H10 #msc:14A15

paper · pdf · doi:10.48550/arxiv.math/0409090

15 pages, minor corrections, to appear in Ill. J. Math

openalex publication_date 2004/09/06 · arxiv created 2005/03/21 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let I = (F₁,...,F_r) be a homogeneous ideal of R = k[x₀,...,x_n] generated by a regular sequence of type (d₁,...,d_r). We give an elementary proof for an explicit description of the graded Betti numbers of I^s for any s ≥ 1. These numbers depend only upon the type and s. We then use this description to: (1) write H_R/I^s, the Hilbert function of R/I^s, in terms of H_R/I; (2) verify that the k-algebra R/I^s satisfies a conjecture of Herzog-Huneke-Srinivasan; and (3) obtain information about the numerical invariants associated to sets of fat points in Pⁿ whose support is a complete intersection or a complete intersection minus a point.

Cited by

• On lower bounds for the F-pure threshold of equigenerated ideals
• Betti numbers and almost complete intersection monomial ideals
• Ordinary and symbolic powers of matroids via polarization

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