On the combinatorial classification of toric log del Pezzo surfaces

TL;DR

This paper classifies toric log del Pezzo surfaces by analyzing their associated convex lattice polygons, providing bounds on their volume and boundary points, and offering explicit classifications for certain indices.

Contribution

It introduces bounds on polygon volume and boundary points based on the index and provides explicit classifications for index two and all indices less than 17.

Findings

Bound on volume in terms of index l

Bound on boundary lattice points in terms of index l

Explicit classification for index two and for all l<17

Abstract

Toric log del Pezzo surfaces correspond to convex lattice polygons containing the origin in their interior and having only primitive vertices. An upper bound on the volume and on the number of boundary lattice points of these polygons is derived in terms of the index l. Techniques for classifying these polygons are also described: a direct classification for index two is given, and a classification for all l<17 is obtained.