On the Twelve-Point Theorem for $\ell$-Reflexive Polygons

Dimitrios I Dais

doi:10.37236/8011

Abstract

It is known that, adding the number of lattice points lying on the boundary of a reflexive polygon and the number of lattice points lying on the boundary of its polar, always yields 12. Generalising appropriately the notion of reflexivity, one shows that this remains true for $\ell$-reflexive polygons. In particular, there exist (for this reason) infinitely many (lattice inequivalent) lattice polygons with the same property. The first proof of this fact is due to Kasprzyk and Nill. The present paper contains a second proof (which uses tools only from toric geometry) as well as the description of complementary properties of these polygons and of the invariants of the corresponding toric log del Pezzo surfaces.

Highlights

The purpose of this paper is to give a second proof of the so-called “Twelve-Point Theorem” for “ -reflexive polygons”, to explain where 12 comes from by taking a slightly different approach, and to provide some additional consequences of it from the point of view of toric geometry. Polygons
We recall the interrelation between lattice polygons and compact toric surfaces with a fixed ample divisor, and explain how one computes the area and the number of lattice points lying on the boundary of a lattice polygon via intersection numbers. (See Theorems 3.8 and 3.9.) In §4-§5 we indicate the manner in which we classify compact toric surfaces via the wve2c-graphs and, in particular, toric log del Pezzo surfaces via LDP-polygons. Giving priority to those log del Pezzo surfaces which are associated with -reflexive polygons we present in §6 the geometric meaning of the lattice change from [46, §2.2]: One may patch together canonical cyclic covers over the singularities in order to construct a finite holomorphic map of degree and to represent the surfaces under consideration as global quotients of Gorenstein del Pezzo surfaces by finite cyclic groups of order
No longer vanish, but are equal to the number of lattice points lying on the boundary of I(Q∗) and I(Q), i.e., of the polygons defined as convex hulls of the interior lattice points of Q∗ and Q, respectively

Summary

Introduction

The purpose of this paper is to give a second proof of the so-called “Twelve-Point Theorem” for “ -reflexive polygons” (see below Theorem 1.27), to explain where 12 comes from by taking a slightly different approach, and to provide some additional consequences of it from the point of view of toric geometry. Polygons. Giving priority to those log del Pezzo surfaces which are associated with -reflexive polygons we present in §6 the geometric meaning of the lattice change from [46, §2.2] (which, in a sense, seems to be the standard method of reducing -reflexivity to 1-reflexivity): One may patch together canonical cyclic covers over the singularities in order to construct a finite holomorphic map of degree and to represent the surfaces under consideration as global quotients of Gorenstein del Pezzo surfaces by finite cyclic groups of order Results of this geometric interpretation (e.g., Proposition 6.13, concerning the relation between the self-intersection numbers of the canonical divisors), combined with Noether’s formula and other information derived from the desingularization, give rise to a new proof of Theorem 1.27 in §7 and to various consequences of it (upper bound for (Vert(Q)), a proof of “oddness” of , a new approach of Suyama’s formula, number-theoretic identities involving types of singularities, combinatorial triples, Dedekind sums etc.). I], [18, §4], [28, §2.2-2.4, §7.1, and §15.2], and [53, §II(b)]), working over C and within the analytic category (with complex (analytic) spaces as objects, holomorphic maps as morphisms and biholomorphic maps as isomorphisms)

Two-dimensional lattice cones and toric surfaces

Compact toric surfaces and lattice polygons

WVE2C-graphs and classification up to isomorphism

Toric log del Pezzo surfaces

Families of combinatorial mirror pairs in the lowest dimension

Corti and