Linear balls and the multiplicity conjecture

A linear ball is a simplicial complex whose geometric realization is homeomorphic to a ball and whose Stanley--Reisner ring has a linear resolution. It turns out that the Stanley--Reisner ring of the sphere which is the boundary complex of a linear ball satisfies the multiplicity conjecture. A class of shellable spheres arising naturally from commutative algebra whose Stanley--Reisner rings satisfy the multiplicity conjecture will be presented.Comment: 19 Page

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

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.Comment: Version 2, minor improvements, 20 pages. Package may be downloaded at http://www.math.uni-sb.de/ag/schreyer/jb/Macaulay2/CyclicPolytopeRes/html