Geodesic
On the twelve-point theorem for \(\ell\)-reflexive polygons
Dimitrios I. Dais1
1University of Crete, Department of Mathematics and Applied Mathematics, Division Algebra and Geometry, Voutes Campus, GR-70013, Heraklion, Crete, Greece
The electronic journal of combinatorics, Tome 26 (2019) no. 4
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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.
DOI : 10.37236/8011
Classification : 52B20, 14M25
Mots-clés : lattice polygons, toric varieties, twelve-point theorem, reflexive polygon, toric log del Pezzo surfaces

Dimitrios I. Dais  1

1 University of Crete, Department of Mathematics and Applied Mathematics, Division Algebra and Geometry, Voutes Campus, GR-70013, Heraklion, Crete, Greece 
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Dimitrios I. Dais. On the twelve-point theorem for \(\ell\)-reflexive polygons. The electronic journal of combinatorics, Tome 26 (2019) no. 4. doi: 10.37236/8011

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