Extremal behaviour in sectional matrices

TL;DR

This paper revisits the sectional matrix, a tool encoding Hilbert functions of hyperplane sections, providing new insights into their maximal growth and a generalization of Gotzmann's Persistence Theorem, with potential implications for algebraic geometry.

Contribution

It introduces new results on the maximal growth of sectional matrices and generalizes Gotzmann's Persistence Theorem, offering a fresh perspective on Hilbert function behavior.

Findings

New results on maximal growth of sectional matrices

A generalized Persistence Theorem extending Gotzmann's result

Reinterpretation of recent results using sectional matrices

Abstract

In this paper we want to revive the object sectional matrix which encodes the Hilbert functions of successive hyperplane sections of a homogeneous ideal. We translate and/or reprove recent results in this language. Moreover, some new results are shown about their maximal growth and Persistence Theorem, a gen- eralization of Gotzmann's persistence Theorem. This suggests that further investigation of this object might cast a new light in the study of geometric consequences of maximal growth of the Hilbert function.