Projectively unique polytopes and toric slack ideals

JOURNAL OF PURE AND APPLIED ALGEBRA

卷:224

Article

Gouveia, Joao¹  Macchia, Antonio²  Thomas, Rekha R.³  Wiebe, Amy³ 

[1] Univ Coimbra, Dept Math, CMUC, P-3001454 Coimbra, Portugal

[2] Free Univ Berlin, Discrete Geometry Grp, Arnimallee 2, D-14195 Berlin, Germany

[3] Univ Washington, Dept Math, Box 354350, Seattle, WA 98195 USA

关键词: Polytopes;    Slack matrix;    Slack ideal;    Realization spaces;    Toric ideal;    Projectively unique polytopes;   

DOI  :  10.1016/j.jpaa.2019.106229

来源: Elsevier

【 摘 要 】

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. (C) 2019 Elsevier B.V. All rights reserved.

【 授权许可】

Free