On the combinatorial classification of toric log del Pezzo surfaces
  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. 
Alexander M. Kasprzyk, Maximilian Kreuzer, and Benjamin Nill