Graded Cohen–Macaulay Domains and Lattice Polytopes with Short h-Vector

Abstract

Let P be a lattice polytope with the \(h^{*}\)-vector \((1, h^*_1, \ldots , h^*_s)\). In this note we show that if \(h_s^* \le h_1^*\), then the Ehrhart ring \({\mathbb {k}}[P]\) is generated in degrees at most \(s-1\) as a \({\mathbb {k}}\)-algebra. In particular, if \(s=2\) and \(h_2^* \le h_1^*\), then P is IDP. To see this, we show the corresponding statement for semi-standard graded Cohen–Macaulay domains over algebraically closed fields.