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

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.

Reflexive Polytopes of Higher Index and the Number 12

We introduce reflexive polytopes of index $l$ as a natural generalisation of the notion of a reflexive polytope of index $1$. These $l$-reflexive polytopes also appear as dual pairs. In dimension two we show that they arise from reflexive polygons via a change of the underlying lattice. This allows us to efficiently classify all isomorphism classes of $l$-reflexive polygons up to index $200$. As another application, we show that any reflexive polygon of arbitrary index satisfies the famous "number $12$" property. This is a new, infinite class of lattice polygons possessing this property, and extends the previously known sixteen instances. The number $12$ property also holds more generally for $l$-reflexive non-convex or self-intersecting polygonal loops. We conclude by discussing higher-dimensional examples and open questions.