Some families of componentwise linear monomial ideals

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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 [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.