Newton-Okounkov body, Rees algebra, and analytic spread of graded families of monomial ideals

TL;DR

This paper explores the use of Newton-Okounkov bodies to characterize the Noetherian property of Rees algebras and interpret the analytic spread of graded monomial ideal families, with applications to symbolic Rees algebras.

Contribution

It introduces a novel geometric approach using Newton-Okounkov bodies to analyze algebraic properties of monomial ideal families, including Noetherianity and analytic spread.

Findings

Characterizes Noetherian Rees algebras via Newton-Okounkov bodies.

Provides a combinatorial interpretation of analytic spread.

Investigates generation type and Veronese degree of symbolic Rees algebras.

Abstract

Let $\mathcal{I} = \{I_k\}_{k \in \mathbb{N}}$ be a graded family of monomial ideal. We use the Newton-Okounkov body of $\mathcal{I}$ to: (a) give a characterization for the Noetherian property of the Rees algebra of the family; and (b) present a combinatorial interpretation for the analytic spread of the family. We also apply these results to investigate the generation type and the Veronese degree of the symbolic Rees algebra of a monomial ideal.