Volume and Lattice Points of Reflexive Simplices

Abstract

Using new number-theoretic bounds on the denominators of unit fractions summing up to one, we show that in any dimension d ≥ 4 there is only one d-dimensional reflexive simplex having maximal volume. Moreover, only these reflexive simplices can admit an edge that has the maximal number of lattice points possible for an edge of a reflexive simplex. In general, these simplices are also expected to contain the largest number of lattice points even among all lattice polytopes with only one interior lattice point. Translated in algebro-geometric language, our main theorem yields a sharp upper bound on the anticanonical degree of d-dimensional Q-factorial Gorenstein toric Fano varieties with Picard number one, e.g., of weighted projective spaces with Gorenstein singularities.

The Reflexive Dimension of a Lattice Polytope

“…There is precisely one reflexive three-polytope with an edge of length 12, and precisely one reflexive four-polytope with an edge of length 84 (cf. [12]). …”

Fano polytopes

“…This initiated an intense study of the geometric and combinatorial properties of reflexive polytopes [14,19,31,32,40,42]. Definition 2.1.…”

“…Much more is known about simplicial reflexive polytopes (corresponding to Q-factorial Gorenstein toric Fano varieties). Their combinatorics is quite restrictive: The vertex-edge-graph has diameter two, and given any vertex v 2 V(P ) there exist at most three other vertices of P not contained in a facet containing v [40]. Casagrande [14] showed that the maximal number of vertices is 3n if n is even, or 3n 1 if n is odd.…”

Ehrhart Series and Lattice Triangulations

“…6 Nill has constructed a sequence of reflexive simplices Q j such that dim Q j = j and the normalized volume vol Q j = h * 0 (Q j ) + · · · + h * j (Q j ) grows doubly exponentially with j [17]. Therefore, there exists a reflexive simplex Q such that h * i (Q) ≥ n for some i, and dim(Q) = O(log log n).…”