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Del Pezzo surfaces with log-terminal singularities. (Russian) Zbl 0674.14024
In the two-dimensional complex case a log-terminal singularity is a singularity of the form $${\mathbb{C}}^ 2/G$$, where G is a finite subgroup of GL(2,$${\mathbb{C}})$$. A normal projective surface Z over $${\mathbb{C}}$$ is called del Pezzo surface with log-terminal singularities, or for short log-del Pezzo surface, if all singularities of Z are log-terminal and for some positive integer k the multiple $$-kK_ Z$$ of the anticanonical Weil divisor $$-K_ Z$$ of Z is an ample Cartier divisor. The smallest such number k is called the index of the log-del Pezzo surface Z. For a minimal desingularization $$\sigma:\quad Y\to Z$$ of Z we have $$K_ Y=\sigma^*K_ Z+\sum_{i}\alpha_ iF_ i$$ where $$F_ i$$ are irreducible components of the exceptional divisor of $$\sigma$$ (all of them are non-singular and rational) and $$\alpha_ i\in {\mathbb{Q}}$$, $$- 1<\alpha_ i\leq 0$$. Then the smallest common denominator of the numbers $$\alpha_ i$$ is the index k. One shows that Y is a rational surface. A complete classification of log-del Pezzo surfaces, which lies in giving the list of all possible intersection diagrams of the curves $$F_ i$$, is known in the log-singular case and in case of indices $$k=1\quad and\quad 2$$ only.
The author proves some facts about various invariants of log-del Pezzo surfaces and their minimal desingularizations such as the intersection diagrams, the rank $$\rho$$ (Y) of the Picard lattice of Y, the multiplicities of the singular points of Z, the numbers $$\alpha_ i$$ and the $$index\quad k.$$ Here is his main result: Let for every singular point of Z of an index $$>1$$ the corresponding $$\alpha_ i$$ be smaller than - 1/2. Denote by e the maximal multiplicity of all singular points of Z. Then $$\rho (Y)\leq 352e+1284\leq 704k+1284.$$
The author also gives a conjecture which, as he writes in a note added in proof, has been proved in another paper: For a fixed k, $$\rho$$ (Y) is bounded from above by a constant depending on k only, in particular the number of the intersection diagrams is then finite. Apart from this the author gives other conjectures and many results which are too technical, however, to be described in a review.
Reviewer: K.Dabrowski

##### MSC:
 14J17 Singularities of surfaces or higher-dimensional varieties 14J28 $$K3$$ surfaces and Enriques surfaces 32J25 Transcendental methods of algebraic geometry (complex-analytic aspects) 14C22 Picard groups 14J15 Moduli, classification: analytic theory; relations with modular forms 51F15 Reflection groups, reflection geometries 14E15 Global theory and resolution of singularities (algebro-geometric aspects)
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