Del Pezzo surfaces with log-terminal singularities.

*(Russian)*Zbl 0674.14024In 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.

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) |