Axial constants and sectional regularity of homogeneous ideals

TL;DR

This paper introduces a new measure called sectional regularity for homogeneous ideals, linking it to axial constants and showing their invariance and linear growth across powers of the ideal.

Contribution

It establishes the equivalence between sectional regularity, axial constants, and other invariants, and demonstrates their linear growth for powers of an ideal.

Findings

Sectional regularity is related to the exponents of monomials in generic initial ideals.

The invariants are shown to be equivalent and grow linearly with powers of the ideal.

The paper connects homological invariants with combinatorial properties of ideals.

Abstract

A notion of sectional regularity for a homogeneous ideal $I$, which measures the regularity of its generic sections with respect to linear spaces of various dimensions, is introduced. It is related to axial constants defined as the intercepts on the coordinate axes of the set of exponents of monomials in the reverse lexicographic generic initial ideal of $I$. The equivalence of these notions and several other homological and ideal-theoretic invariants is shown. It is also established that these equivalent invariants grow linearly for the family of powers of a given ideal.