On the structure of Stanley-Reisner rings associated to cyclic polytopes

TL;DR

This paper explores the algebraic structure of Stanley-Reisner rings linked to cyclic polytopes, employing unprojection theory and the Kustin-Miller complex to derive minimal free resolutions and Betti numbers.

Contribution

It introduces a recursive method to express Stanley-Reisner rings of cyclic polytopes in terms of smaller cases and applies complex constructions to determine their resolutions.

Findings

Derived recursive relations for Stanley-Reisner rings of cyclic polytopes.

Identified minimal graded free resolutions using Kustin-Miller complexes.

Reproduced known Betti number results for these rings.

Abstract

We study the structure of Stanley-Reisner rings associated to cyclic polytopes, using ideas from unprojection theory. Consider the boundary simplicial complex Delta(d,m) of the d-dimensional cyclic polytope with m vertices. We show how to express the Stanley-Reisner ring of Delta(d,m+1) in terms of the Stanley-Reisner rings of Delta(d,m) and Delta(d-2,m-1). As an application, we use the Kustin-Miller complex construction to identify the minimal graded free resolutions of these rings. In particular, we recover results of Schenzel, Terai and Hibi about their graded Betti numbers.