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2007 Some families of componentwise linear monomial ideals
Christopher A. Francisco, Adam Van Tuyl
Nagoya Math. J. 187: 115-156 (2007).

Abstract

Let $R = k[x_{1}, \dots, x_{n}]$ be a polynomial ring over a field $k$. Let $J = \{j_{1}, \dots, j_{t}\}$ be a subset of $\{1, \dots, n\}$, and let $\mathfrak{m}_{J} \subset R$ denote the ideal $(x_{j_{1}}, \dots, x_{j_{t}})$. Given subsets $J_{1}, \dots, J_{s}$ of $\{1, \dots, n\}$ and positive integers $a_{1}, \dots, a_{s}$, we study ideals of the form $I = \mathfrak{m}_{J_{1}}^{a_{1}} \cap \cdots \cap \mathfrak{m}_{J_{s}}^{a_{s}}$. 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 $s \le 3$ or when $J_{i} \cup J_{j} = [n]$ for all $i \neq j$. When $s \ge 4$, 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 $\operatorname{char}(k) = 0$, our work also yields new cases in which this conjecture holds.

Citation

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Christopher A. Francisco. Adam Van Tuyl. "Some families of componentwise linear monomial ideals." Nagoya Math. J. 187 115 - 156, 2007.

Information

Published: 2007
First available in Project Euclid: 4 September 2007

zbMATH: 1140.13012
MathSciNet: MR2354558

Subjects:
Primary: 13D02 , 13D40

Keywords: Betti numbers , componentwise linear , fat points , monomial ideals , multiprojective spaces , polymatroidal ideals , resolutions

Rights: Copyright © 2007 Editorial Board, Nagoya Mathematical Journal

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