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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
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