  Let R=k[x_1,...,x_n] be a polynomial ring over a field k. Let J={j_1,...,j_t}
be a subset of [n]={1,...,n}, and let m_J denote the ideal
(x_{j_1},...,x_{j_t}) of R. Given subsets J_1,...,J_s of [n] and positive
integers a_1,...,a_s, we study ideals of the form I=m_{J_1}^{a_1} \cap ... \cap
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 <= 3 or when J_i \cup J_j = [n] for all i \neq 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.
