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

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 " -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 [46]. 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. 1 2 denote the euclidean norm of any x ∈ R 2 . Proposition 1.1. For any nonempty subset N of R 2 the following conditions are equivalent:(i) N is a discrete subgroup of the additive group R 2 (i.e., n − n ∈ N for all n, n ∈ N, and for every n ∈ N there is an ε ∈ R >0 , s.t. B ε (n) ∩ N = {n}, where B ε (n) := x ∈ R 2 x − n ≤ ε ), and N spans the entire R 2 as R-vector space.(ii) There exists a set {b 1 , b 2 } of two R-linear independent vectors b 1 , b 2 ∈ R 2 such that N = {k 1 b 1 + k 2 b 2 | k 1 , k 2 ∈ Z} .