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

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.