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Classification of del Pezzo surfaces with log-terminal singularities of index $$\leq 2$$, and involutions on K3 surfaces. (English. Russian original) Zbl 0705.14038
Sov. Math., Dokl. 39, No. 3, 507-511 (1989); translation from Dokl. Akad. Nauk SSSR 306, No. 3, 525-528 (1989).
Let Z be a normal algebraic surface over $${\mathbb{C}}$$, and let $$K_ Z$$ be the canonical class of Z (defined as Weil divisor). Z is called a Del Pezzo surface if some multiple $$-NK_ Z$$ is an ample Cartier divisor. A singular point of Z is called log-terminal if it is locally (analytically) isomorphic to $${\mathbb{C}}^ 2/G$$, where $$G\subset GL(2,{\mathbb{C}})$$ is a finite subgroup. A log-terminal singularity $$z\in Z$$ has $$index\quad k$$ if k is the least number for which $$kK_ Z$$ is a Cartier divisor in a neighborhood of z (it is known that $$k=[G:G\cap SL(2,{\mathbb{C}})])$$, and a surface Z with log-terminal singularities has $$index\quad k$$ if k is the least common multiple of the singular points of Z. In the paper under review the authors give a classification of Del Pezzo surfaces with log-terminal singularities of index $$\leq 2$$ (classification of non-singular Del Pezzo surfaces is classical, and classification of Del Pezzo surfaces with log-terminal singularities of $$index\quad 1$$ was known previously). In particular, they show that up to isomorphism there exist exactly $$18\quad Del Pezzo$$ surfaces Z with log- terminal singularities of $$index\quad 2$$ such that $$Pic(Z)={\mathbb{Z}}$$.
Reviewer: F.L.Zak

##### MSC:
 14J26 Rational and ruled surfaces 14J17 Singularities of surfaces or higher-dimensional varieties 14J10 Families, moduli, classification: algebraic theory 14B05 Singularities in algebraic geometry
##### Keywords:
Del Pezzo surface; log-terminal singularity