Del Pezzo surfaces with log-terminal singularities. III.

*(English. Russian original)*Zbl 0711.14018
Math. USSR, Izv. 35, No. 3, 657-675 (1990); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 6, 1316-1334 (1989).

[For part I of this paper see Math. USSR, Sb. 66, No.1, 231-248 (1990); translation from Mat. Sb. 180, No.2, 226-243 (1989; Zbl 0674.14024); for part II see Math. USSR, Izv. 33, No.2, 335-372 (1989); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 52, No.5, 1032-1050 (1988; Zbl 0677.14008).]

For the definitions and notation we refer to our review of part I. In part II the author proved that for a minimal desingularisation \(\sigma: Y\to Z\) of a complex log-del Pezzo surface Z with a fixed index \(k,\) the Picard number \(\rho\) (Y) of Y could be estimated from above by a constant depending on k only. In particular the number of the intersection diagrams of exceptional curves on Y is finite.

In this part III of the paper the author shows that the theorem formulated above is true if we replace k with the multiplicity e of Z. This result is stronger than the previous one and the author considers e more geometrical than k. It is also shown that all the results of the previous parts are true over any algebraically closed field. Another theorem says that a similar estimation of \(\rho\) (Y) takes place for a wider class of surfaces which contains for example non-minimal K3 surfaces.

The methods used are as in the previous parts, they come from the theory of reflection group in Lobachevsky spaces. The author formulates a conjecture, that these methods can be generalised for higher dimensions and that similar results can be obtained for Fano manifolds of dimension higher than 2 and with log-terminal singularities. Finally he mentions his preprint in which he generalises the present results to projective surfaces with log-terminal singularities and with a nef anticanonical class.

For the definitions and notation we refer to our review of part I. In part II the author proved that for a minimal desingularisation \(\sigma: Y\to Z\) of a complex log-del Pezzo surface Z with a fixed index \(k,\) the Picard number \(\rho\) (Y) of Y could be estimated from above by a constant depending on k only. In particular the number of the intersection diagrams of exceptional curves on Y is finite.

In this part III of the paper the author shows that the theorem formulated above is true if we replace k with the multiplicity e of Z. This result is stronger than the previous one and the author considers e more geometrical than k. It is also shown that all the results of the previous parts are true over any algebraically closed field. Another theorem says that a similar estimation of \(\rho\) (Y) takes place for a wider class of surfaces which contains for example non-minimal K3 surfaces.

The methods used are as in the previous parts, they come from the theory of reflection group in Lobachevsky spaces. The author formulates a conjecture, that these methods can be generalised for higher dimensions and that similar results can be obtained for Fano manifolds of dimension higher than 2 and with log-terminal singularities. Finally he mentions his preprint in which he generalises the present results to projective surfaces with log-terminal singularities and with a nef anticanonical class.

Reviewer: K.Dabrowski

##### MSC:

14J17 | Singularities of surfaces or higher-dimensional varieties |

14C20 | Divisors, linear systems, invertible sheaves |

14C22 | Picard groups |

14J25 | Special surfaces |

51F15 | Reflection groups, reflection geometries |

14E15 | Global theory and resolution of singularities (algebro-geometric aspects) |