Klyachko diagrams of monomial ideals

TL;DR

This paper introduces Klyachko diagrams for monomial ideals in Cox rings of toric varieties, providing computational tools and formulas for local cohomology, saturation, and Hilbert functions.

Contribution

It defines Klyachko diagrams for monomial ideals and develops algorithms to compute and use them for various algebraic invariants.

Findings

Computed the first local cohomology module using Klyachko diagrams

Derived a formula for the Hilbert function of saturated ideals

Characterized ideals with constant Hilbert polynomial via Klyachko diagrams

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 $I^{\mathrm{sat}}$ of $I$ from its Klyachko diagram. We use this description to compute the first local cohomology module $H^{1}_{B}(I)$. As an application, we find a formula for the Hilbert function of $I^{\mathrm{sat}}$, and a characterization of monomial ideals with constant Hilbert polynomial, in terms of their Klyachko diagram.