A generalization of a theorem of White

Abstract

An $m$-dimensional simplex $\mathrm{\Delta }$ in ${ℝ}^m$ is called empty lattice simplex if $\mathrm{\Delta }\cap {\mathbb{Z}}^m$ is exactly the set of vertices of $\mathrm{\Delta }$. A theorem of White states that if $m=3$ then, up to an affine unimodular transformation of the lattice ${\mathbb{Z}}^m$, any empty lattice simplex $\mathrm{\Delta }\subset {ℝ}^3$ is isomorphic to a tetrahedron whose vertices have third coordinate $0$ or $1$. We prove a generalization of this theorem for some special empty lattice simplices of arbitrary odd dimension $m=2d-1$ which was conjectured by Sebő and Borisov. Our result implies a classification of all $2d$-dimensional isolated Gorenstein cyclic quotient singularities with minimal $log$-discrepancy $\ge d$.