Reduced arithmetically Gorenstein schemes and simplicial polytopes with maximal Betti numbers

TL;DR

This paper constructs specific arithmetically Gorenstein schemes with prescribed Hilbert functions and maximal Betti numbers, and applies these results to simplicial polytopes with fixed h-vectors.

Contribution

It introduces a method to build arithmetically Gorenstein schemes with maximal Betti numbers for given SI-sequences and applies this to simplicial polytopes.

Findings

Constructed Gorenstein schemes with maximal Betti numbers

Established existence of polytopes with maximal Betti numbers for fixed h-vectors

Demonstrated the Weak Lefschetz Property in constructed schemes

Abstract

An SI-sequence is a finite sequence of positive integers which is symmetric, unimodal and satisfies a certain growth condition. These are known to correspond precisely to the possible Hilbert functions of Artinian Gorenstein algebras with the Weak Lefschetz Property, a property shared by most Artinian Gorenstein algebras. Starting with an arbitrary SI-sequence, we construct a reduced, arithmetically Gorenstein configuration $G$ of linear varieties of arbitrary dimension whose Artinian reduction has the given SI-sequence as Hilbert function and has the Weak Lefschetz Property. Furthermore, we show that $G$ has maximal graded Betti numbers among all arithmetically Gorenstein subschemes of projective space whose Artinian reduction has the Weak Lefschetz Property and the given Hilbert function. As an application we show that over a field of characteristic zero every set of simplicial polytopes with fixed $h$-vector contains a polytope with maximal graded Betti numbers.