A generalization of a theorem of G. K. White

TL;DR

This paper generalizes White's theorem on empty lattice simplices from three dimensions to all odd dimensions, leading to a classification of certain Gorenstein cyclic quotient singularities.

Contribution

It extends White's theorem to odd dimensions, confirming a conjecture by Seb"H{o} and Borisov, and classifies specific Gorenstein cyclic quotient singularities.

Findings

Generalization of White's theorem to odd dimensions

Classification of 2d-dimensional Gorenstein cyclic quotient singularities

Implication for minimal log-discrepancy at least d

Abstract

An n-dimensional simplex \Delta in \R^n is called empty lattice simplex if \Delta \cap\Z^n is exactly the set of vertices of \Delta . A theorem of G. K. White shows that if n=3 then any empty lattice simplex \Delta \subset\R^3 is isomorphic up to an unimodular affine linear transformation to a lattice tetrahedron whose all vertices have third coordinate 0 or 1. In this paper we prove a generalization of this theorem for an arbitrary odd dimension n=2d-1 which in some form was conjectured by Seb\H{o} and Borisov. This result implies a classification of all 2d-dimensional isolated Gorenstein cyclic quotient singularities with minimal log-discrepancy at least d.