Support-2 monomial ideals that are Simis

TL;DR

The paper advances the classification of Simis ideals among monomial ideals by focusing on the broad class of support-$2$ ideals. It proves the Mendez-Pinto-Villarreal conjecture for this class, showing that a minimal irreducible decomposition together with absence of embedded primes yields $I$ is Simis precisely when $ ootedness(I)$ is Simis and $I$ carries standard linear weights. It then delivers cycle-based and whisker-graph results: even-length cycles yield Simis under the standard-weighting condition, while whiskered-graph radicals admit a complete Cohen–Macaulay characterization and admit criteria for when the second symbolic power coincides with the second ordinary power. The work also furnishes explicit obstructions to the Simis property via embedded primes and provides a suite of examples that illuminate the limits of the conjecture beyond the minimal-decomposition setting. Overall, it clarifies when symbolic and ordinary powers align for a rich family of monomial ideals and ties the Simis property to precise combinatorial and weighting data.

Abstract

A monomial ideal $I\subseteq \mathbb{K}[x_1,\ldots , x_n]$ is called a Simis ideal if $I^{(s)}=I^s$ for all $s\geq 1$, where $I^{(s)}$ denotes the $s$-th symbolic power of $I$. Let $I$ be a support-2 monomial ideal such that its irreducible primary decomposition is minimal. We prove that $I$ is a Simis ideal if and only if $\sqrt{I}$ is Simis and $I$ has standard linear weights. This result thereby proves a recent conjecture for the class of support-2 monomial ideals proposed by Mendez, Pinto, and Villarreal. Furthermore, we give a complete characterization of the Cohen-Macaulay property for support-2 monomial ideals whose radical is the edge ideal of a whiskered graph. Finally, we classify when these ideals are Simis in degree 2.

Theorems & Definitions (29)

• Conjecture 1.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Theorem 2.3
• proof
• Proposition 3.1
• proof
• Remark 3.2