A normal \((0,1)\)-polytope none of whose regular triangulations is unimodular
The authors study integral convex polytopes \({\mathcal P}\subset \mathbb{R}^N\). Such a polytope \({\mathcal P}\) is called normal if the subalgebra of \(K[t_1,t^{-1}_1,\dots, t_N, t^{-1}_N, s]\) which is generated by all monomials \(t^{\alpha_1}_1\cdots t^{\alpha_N}_N,s\) with \((\alpha_1,\dots,\alpha_N)\in {\mathcal P}\cap \mathbb{Z}^N\) is normal, i.e. integrally closed in its quotient field. The property that a polytope \({\mathcal P}\) possesses a unimodular regular covering is known to imply that \({\mathcal P}\) is normal. The authors show that the converse is false by constructing a normal \((0,1)\)-polytope none of whose regular triangulation is unimodular. The proof relies on earlier work by B. Sturmfels and the authors.
scientific article; zbMATH DE number 1271053