2005/08/30 by Christopher A. Francisco, Francisco, Christopher A., Adam Van Tuyl +1 · 1 citation
Computer Science · Mathematics · #13D02 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.math/0508589
openalex publication_date 2005/08/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let R=k[x1,...,xn] be a polynomial ring over a field k. Let J=j1,...,jt be a subset of [n]=1,...,n, and let mJ denote the ideal (xj1,...,xjt) of R. Given subsets J1,...,Js of [n] and positive integers a1,...,as, we study ideals of the form I=mJ1a1 ∩ ... ∩ mJsas. These ideals arise naturally, for example, in the study of fat points, tetrahedral curves, and Alexander duality of squarefree monomial ideals. Our main focus is determining when ideals of this form are componentwise linear. Using polymatroidality, we prove that I is always componentwise linear when s <= 3 or when Ji ∪ Jj = [n] for all i ≠ j. When s >= 4, we give examples to show that I may or may not be componentwise linear. We apply these results to ideals of small sets of general fat points in multiprojective space, and we extend work of Fatabbi, Lorenzini, Valla, and the first author by computing the graded Betti numbers in the s=2 case. Since componentwise linear ideals satisfy the Multiplicity Conjecture of Herzog, Huneke, and Srinivasan when char(k)=0, our work also yields new cases in which this conjecture holds.