The Slack Realization Space of a Polytope
Connections for Women Workshop: Geometric and Topological Combinatorics August 31, 2017 - September 01, 2017
Location: SLMath: Eisenbud Auditorium
Polytopes
slack matrix
slack ideal
realization spaces
binomial ideal
toric ideal
projective uniqueness
1-Thomas
We introduce a new model of a realization space of a polytope that arises as the positive part of a real variety. The variety is determined by the slack ideal of the polytope, a saturated determinantal ideal of a sparse generic matrix that encodes the combinatorics of the polytope. The slack ideal offers a uniform computational framework for several classical questions about polytopes such as rational realizability, projectively uniqueness, non-prescribability of faces, and realizability of combinatorial polytopes. The simplest slack ideals are toric. We identify the toric ideals that arise from projectively unique polytopes. New and classical examples illuminate the relationships between projective uniqueness and toric slack ideals.
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