Projectively unique polytopes and toric slack ideals

TL;DR

This paper explores the properties of slack ideals of polytopes, establishing connections to toric ideals and projective uniqueness, and provides new examples of slack ideals with unique algebraic features.

Contribution

It proves that projectively unique polytopes with toric slack ideals are related to bipartite graph toric ideals and introduces the first example of a reducible slack ideal.

Findings

Toric slack ideals correspond to bipartite graph toric ideals for projectively unique polytopes.

Polytopes without rational realizations cannot have toric slack ideals.

The slack ideal of the Perles polytope is reducible, not prime.

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.