Projectively unique polytopes and toric slack ideals
Abstract
The slack ideal of a polytope is a saturated determinantal ideal that gives rise to a new model for the realization space of the polytope. The simplest slack ideals are toric and have connections to projectively unique polytopes. We prove that if a projectively unique polytope has a toric slack ideal, then it is the toric ideal of the bipartite graph of vertex-facet non-incidences of the polytope. The slack ideal of a polytope is contained in this toric ideal if and only if the polytope is morally 2-level, a generalization of the 2-level property in polytopes. We show that polytopes that do not admit rational realizations cannot have toric slack ideals. A classical example of a projectively unique polytope with no rational realizations is due to Perles. We prove that the slack ideal of the Perles polytope is reducible, providing the first example of a slack ideal that is not prime.
Keywords
Cite
@article{arxiv.1808.01692,
title = {Projectively unique polytopes and toric slack ideals},
author = {João Gouveia and Antonio Macchia and Rekha R. Thomas and Amy Wiebe},
journal= {arXiv preprint arXiv:1808.01692},
year = {2019}
}
Comments
Parts of this paper originally appeared in the first version of arXiv:1708.04739 [math.CO]. That paper was subsequently split into two: arXiv:1708.04739[math.CO] describing the slack realization space of a polytope & applications of the slack ideal, and this paper focussing on toric slack ideals & projective uniqueness. In addition, this paper contains the first example of a reducible slack ideal