Klyachko Diagrams of Monomial Ideals

Abstract

In this paper, we introduce the notion of a Klyachko diagram for a monomial ideal I in a certain multi-graded polynomial ring, namely the Cox ring R of a smooth complete toric variety, with irrelevant maximal ideal B. We present procedures to compute the Klyachko diagram of I from its monomial generators, and to retrieve the B −saturation Isat of I from its Klyachko diagram. We use this description to compute the first local cohomology module ${H}^{1}_{B}(I)$ H B 1 ( I ) . As an application, we find a formula for the Hilbert function of Isat, and a characterization of monomial ideals with constant Hilbert polynomial, in terms of their Klyachko diagram.