On projective toric varieties whose defining ideals have minimal generators of the highest degree.
Given an \(n\)-dimensional, projectively normal, toric variety \(X\), then it is well-known by Sturmfels that its (binomial) ideal is generated in degree \(\leq(n+1)\). Generalizing \textit{R.J. Koelman}'s papers [Beitr. Algebra Geom. 34, No. 1, 57--62 (1993; Zbl 0781.14025); Tohoku Math. J., II. Ser. 45, No. 3, 385--392 (1993; Zbl 0809.14042)] (dealing with the case \(n=2\)), the author classifies those cases, i.e. lattice polytopes, that require generators of degree \(n+1\). The result is that a lattice polytope enforces generators of degree \(n+1\) if and only if it is a simplex with ``smooth'' facets but containing interior lattice points. However, reading p.~2245 of the paper, I got the impression that the author has used the fact that so-called empty simplices (the vertices are the only lattice points) are smooth (isomorphic to the standard simplex), which is not true.
scientific article; zbMATH DE number 2093471
scientific article