  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.
