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

TL;DR

This paper extends the Twelve-Point Theorem to $$-reflexive polygons, providing a new proof using toric geometry and exploring related properties of associated toric surfaces.

Contribution

It offers a second proof of the theorem for $$-reflexive polygons using toric geometry and analyzes their properties and invariants.

Findings

The Twelve-Point Theorem holds for $$-reflexive polygons.

There are infinitely many lattice inequivalent $$-reflexive polygons.

Properties of associated toric log del Pezzo surfaces are characterized.

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.