Jets and principal components of monomial ideals, and very well-covered graphs

TL;DR

This paper explores the structure of jets in monomial ideals and graphs, establishing new connections between combinatorial properties and algebraic invariants, with applications to very well-covered graphs and principal jets.

Contribution

It extends the concept of jets to clutters, links jets of graphs to symbolic powers, and provides formulas for algebraic invariants of principal jets of monomial ideals.

Findings

Jets of very well-covered graphs are also very well-covered.

Established a connection between cover ideals of jets and symbolic powers.

Derived formulas for Hilbert series, Betti numbers, and other invariants.

Abstract

Motivated by using combinatorics to study jets of monomial ideals, we extend a definition of jets from graphs to clutters. We offer some structural results on their vertex covers, and show an interesting connection between the cover ideal of the jets of a clutter and the symbolic powers of the cover ideal of the original clutter. We use this connection to prove that jets of very well-covered graphs are very well-covered. Next, we turn our attention to principal jets of monomial ideals, describing their primary decomposition and minimal generating sets. Finally, we give formulas to compute various algebraic invariants of principal jets of monomial ideals, including their Hilbert series, Betti numbers, multiplicity and regularity.