Symbolic powers of monomial ideals which are generically complete intersections

TL;DR

This paper classifies certain monomial ideals with specific properties related to their symbolic powers and provides bounds on the degrees of generators of associated algebras, advancing understanding of their algebraic structure.

Contribution

It offers a complete classification of unmixed codimension 2 monomial ideals that are generically complete intersections with standard graded symbolic power algebras, and establishes bounds on generator degrees.

Findings

Classification of all such monomial ideals with standard graded symbolic power algebra

Lower bound for the highest degree of generators in vertex cover ideal modifications

Upper bound for the highest degree of generators of the integral closure of the algebra

Abstract

We classify all unmixed monomial ideals I of codimension 2 which are generically a complete intersection and which have the property that the symbolic power algebra A(I) is standard graded. We give a lower bound for the highest degree of a generator of A(I) in the case that I is a modification of the vertex cover ideal of a bipartite graph, and show that this highest degree can be any given number. We furthermore give an upper bound for the highest degree of a generator of the integral closure of A(I) in the case that I is a monomial ideal which is generically a complete intersection.