Waldschmidt constant of monomial ideals and Simis ideals

TL;DR

This paper investigates the Waldschmidt constant of monomial ideals, confirming conjectures related to lower bounds and structural properties of Simis ideals, with results applicable to specific classes of these ideals.

Contribution

It verifies two conjectures about the Waldschmidt constant and the structure of Simis ideals for certain classes of monomial ideals.

Findings

Confirmed the 2017 conjecture for some monomial ideals.

Verified the structure conjecture for specific classes of Simis ideals.

Established connections between monomial ideals and weighted squarefree monomial ideals.

Abstract

In 2017, Cooper et al. proposed a conjecture providing a lower bound for the Waldschmidt constant of monomial ideals. We confirm this conjecture for some classes of monomial ideals. Recently, M\'endez, Pinto, and Villarreal formulated a conjecture stating that if $I$ is a monomial ideal without embedded associated primes, whose irreducible decomposition is minimal and which is a Simis ideal, then there exist a Simis squarefree monomial ideal $J$ and a standard linear weighting $w$ such that $I = J_{w}.$ In this work, we verify this conjecture for some classes of monomial ideals.