Local Cohomology, Arrangement of Subspaces and Monomial Ideals

Abstract

Let $k$ be a field of characteristic zero and I an ideal defining an arrangement of linear subspaces in the affine space $A^n_k$. We compute the D-module theoretic characteristic cycle of the local cohomology modules $H^r_I(k[x_1,...,x_n])$ in terms of the poset defined by the arrangement. In case I is a monomial ideal, we relate the multiplicities of the characteristic cycle with the Betti numbers of the Alexander dual ideal of I, we also study some extension problems attached to the modules $H^r_I(k[x_1,...,x_n])$. If $k$ is the field of complex numbers we study the ubiquity of these local cohomology modules in the category of perverse sheaves in $A^n_k$ with respect to the stratification given by the coordinate hyperplanes.