On the span of lattice points in a parallelepiped

Marcel Celaya

Published 2015-06-08Version 1

We find a good characterization for the following problem: Given an integral row vector $c=(c_{1}\ldots,c_{n})$ where each $c_{i}\in\{-1,0,1\}$ and a lattice $\Lambda\subset\mathbf{R}^{n}$ which contains the integer lattice $\mathbf{Z}^{n}$, do all lattice points of $\Lambda$ in the half-open unit cube $[0,1)^{n}$ lie on the hyperplane $\{x:cx=0\}$? The result is a direct generalization of the well-known Terminal Lemma of Reid, which in turn is based upon earlier work of Morrison and Stevens on the classification of terminal quotient singularities.