Splittings and symbolic powers of square-free monomial Ideals

TL;DR

This paper investigates the properties of symbolic powers of square-free monomial ideals, demonstrating their algebraic structures, convergence behaviors, and conditions for equality with ordinary powers, with implications for optimization.

Contribution

It introduces a Frobenius-like splitting technique for symbolic Rees algebras in all characteristics and links algebraic conditions to linear optimization problems.

Findings

Symbolic Rees algebra and associated graded algebra are split via a Frobenius-like morphism.

Normalized $a$-invariants and Castelnuovo-Mumford regularity of symbolic powers converge.

Provides a sufficient condition for equality of symbolic and ordinary powers, relating to a conjecture and optimization.

Abstract

We study the symbolic powers of square-free monomial ideals via symbolic Rees algebras and methods in prime characteristic. In particular, we prove that the symbolic Rees algebra and the symbolic associated graded algebra are split with respect to a morphism which resembles the Frobenius map and that exists in all characteristics. Using these methods, we recover a result by Hoa and Trung which states that the normalized $a$-invariants and the Castelnuovo-Mumford regularity of the symbolic powers converge. In addition, we give a sufficient condition for the equality of the ordinary and symbolic powers of this family of ideals, and relate it to Conforti-Cornu\'ejols conjecture. Finally, we interpret this condition in the context of linear optimization.