Vol. 10, No. 4, 2021

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A generalization of a theorem of White

Victor Batyrev and Johannes Hofscheier

Vol. 10 (2021), No. 4, 281–296
DOI: 10.2140/moscow.2021.10.281
Abstract

An m-dimensional simplex Δ in m is called empty lattice simplex if Δ m is exactly the set of vertices of Δ. A theorem of White states that if m = 3 then, up to an affine unimodular transformation of the lattice m , any empty lattice simplex Δ 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 d.

Keywords
empty lattice simplices, Ehrhart theory, $h^*$-polynomial, Bernoulli functions, quotient singularity
Mathematical Subject Classification
Primary: 52B20
Secondary: 14B05, 11B68
Milestones
Received: 24 February 2021
Revised: 29 May 2021
Accepted: 12 June 2021
Published: 17 January 2022
Authors
Victor Batyrev
Mathematisches Institut
Universität Tübingen
Tübingen
Germany
Johannes Hofscheier
School of Mathematical Sciences
University of Nottingham
Nottingham
United Kingdom