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Some Families of Componentwise Linear Monomial Ideals

Published online by Cambridge University Press:  11 January 2016

Christopher A. Francisco  and
Adam Van Tuyl
Christopher A. Francisco
Affiliation:
Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK 74078chris@math.okstate.edu, http://www.math.okstate.edu/~chris
Adam Van Tuyl
Affiliation:
Department of Mathematical Sciences, Lakehead University, Thunder Bay, ON P7B 5E1, Canada, avantuyl@sleet.lakeheadu.ca, http://flash.lakeheadu.ca/~avantuyl
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Abstract

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Let R = k[x1,…,xn] be a polynomial ring over a field k. Let J = {j1,…,jt} be a subset of {1,…, n}, and let mJ ⊂ R denote the ideal (xj1,…,xjt). Given subsets J1,…,Js of {1,…, n} and positive integers a1,…,as, we study ideals of the form 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.

Keywords

13D4013D02
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 187 , September 2007 , pp. 115 - 156
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2007

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