Klyachko Diagrams of Monomial Ideals

Abstract

AbstractIn this paper, we introduce the notion of aKlyachko diagramfor a monomial idealIin a certain multi-graded polynomial ring, namely the Cox ringRof a smooth complete toric variety, with irrelevant maximal idealB. We present procedures to compute the Klyachko diagram ofIfrom its monomial generators, and to retrieve theB−saturationIsatofIfrom its Klyachko diagram. We use this description to compute the first local cohomology module${H}^{1}_{B}(I)$HB1(I). As an application, we find a formula for the Hilbert function ofIsat, and a characterization of monomial ideals with constant Hilbert polynomial, in terms of their Klyachko diagram.