A hyperplane restriction theorem and applications to reductions of ideals

TL;DR

This paper strengthens Green's hyperplane restriction theorem by characterizing the linear forms that do not satisfy the bound and applies this to extend the Eakin-Sathaye theorem on reductions of ideals.

Contribution

It improves Green's theorem by describing the linear forms that violate the bound and introduces a method to extend the Eakin-Sathaye theorem for reductions.

Findings

Linear forms not satisfying the bound lie in a finite union of proper linear spaces.

The method extends the Eakin-Sathaye theorem to complete and joint reductions.

The results recover and generalize previous theorems by O'Carroll.

Abstract

Green's general hyperplane restriction theorem gives a sharp upper bound for the Hilbert function of a standard graded algebra over and infinite field K modulo a general linear form. We strengthen Green's result by showing that the linear forms that do not satisfy such estimate belong to a finite union of proper linear spaces. As an application we give a method to derive variations of the Eakin-Sathaye theorem on reductions. In particular, we recover and extend results by O'Carroll on the Eakin-Sathaye theorem for complete and joint reductions.