Some Families of Componentwise Linear Monomial Ideals Academic Article uri icon

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abstract

  • AbstractLetR=k[x1,…,xn] be a polynomial ring over a fieldk. LetJ= {j1,…,jt} be a subset of {1,…,n}, and let mJRdenote the ideal (xj1,…,xjt). Given subsetsJ1,…,Jsof {1,…,n} and positive integersa1,…,as, we study ideals of the formThese 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 thatIis always componentwise linear whens≤ 3 or whenJiJj= [n] for allij. Whens≥ 4, we give examples to show thatImay 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 thes= 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.

publication date

  • September 2007