Abstract

 
 
 We give a classification of toric log del Pezzo surfaces with two or three singular points. Our proofs are purely combinatorial, relying on the bijection between toric log del Pezzo surfaces and the so-called LDP-polygons introduced by Dais and Nill.
 
 
 
Highlights
A normal projective surface is called log del Pezzo surface if it has at worst log-terminal singularities and its anticanonical divisor is a Q-Cartier ample divisor
The bijection sends isomorphism classes of toric log del Pezzo surfaces to equivalence classes of LDP-polygons, where two LDP-polygons Q and Q are called equivalent if there is an automorphism of the lattice Z2 sending Q to Q
Among them: Picard number: The Picard number of XQ equals the number of edges of Q minus two
Summary
A normal projective surface is called log del Pezzo surface if it has at worst log-terminal singularities (that is, quotient singularities) and its anticanonical divisor is a Q-Cartier ample divisor. An edge of an LDP-polygon Q is nonsingular (that is, the corresponding torus-fixed point of the associated toric surface XQ is nonsingular) if and only if its two vertices form a basis of Z2. It is well known that there are exactly five nonsingular LDP-polygons (that is, five nonsingular toric del Pezzo surfaces) Theorems 2 and 5(2) imply that any LDP-polygon Q has at most four nonsingular edges except if Q is the polar of the unique smooth reflexive pentagon or hexagon. The structure of the paper is as follows: In Section 2, we recall the bijection between toric log del Pezzo surfaces and LDP-polygons, and we fix some notation. The three parts of Theorem 5 are proved as we go along
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