Systems with the integer rounding property in normal monomial subrings

TL;DR

This paper characterizes when certain normal monomial subrings have the integer rounding property, linking algebraic properties to combinatorial structures like graphs and Rees algebras, and identifies conditions for Gorenstein properties.

Contribution

It provides a description of the canonical module and a-invariant for normal subrings related to clutters and graphs, connecting algebraic and combinatorial properties.

Findings

Characterization of the integer rounding property for systems associated with clutters and graphs.

Conditions for the extended Rees algebra of bipartite graph edge ideals to be Gorenstein.

Description of algebraic invariants in terms of combinatorial properties.

Abstract

Let C be a clutter and let A be its incidence matrix. If the linear system x>=0;xA<=1 has the integer rounding property, we give a description of the canonical module and the a-invariant of certain normal subrings associated to C. If the clutter is a connected graph, we describe when the aforementioned linear system has the integer rounding property in combinatorial and algebraic terms using graph theory and the theory of Rees algebras. As a consequence we show that the extended Rees algebra of the edge ideal of a bipartite graph is Gorenstein if and only if the graph is unmixed.