65 research outputs found
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
Tropical types and associated cellular resolutions
An arrangement of finitely many tropical hyperplanes in the tropical torus
leads to a notion of `type' data for points, with the underlying unlabeled
arrangement giving rise to `coarse type'. It is shown that the decomposition of
the tropical torus induced by types gives rise to minimal cocellular
resolutions of certain associated monomial ideals. Via the Cayley trick from
geometric combinatorics this also yields cellular resolutions supported on
mixed subdivisions of dilated simplices, extending previously known
constructions. Moreover, the methods developed lead to an algebraic algorithm
for computing the facial structure of arbitrary tropical complexes from point
data.Comment: minor correction
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