Del~Pezzo surfaces with log-terminal singularities.~III

V. V. Nikulin

Izvestiya. Mathematics , Tome 35 (1990) no. 3, pp. 657-675.

Résumé

Let $Z$ be a del Pezzo surface with log-terminal singularities and $\sigma\colon Y\to Z$ a minimal resolution of singularities. Then the Picard number of $Y$ satisfies $\rho(Y)$, where $e$ is the maximal multiplicity of the singularities of $Z$ and $N$ a certain function of $e$. Bibliography: 18 titles.

Bibliographie

[1] Alekseev V. A., “Drobnye indeksy log-poverkhnostei del Petstso”, Izv. AN SSSR. Ser. matem., 52:6 (1988), 1288–1304 | MR

[2] Alekseev V. A., Nikulin V. V., “Klassifikatsiya poverkhnostei del Petstso s log-terminalnymi osobennostyami indeksa $\leqslant2$, involyutsii na poverkhnostyakh K3 i gruppy otrazhenii v prostranstvakh Lobachevskogo”, Voprosy chistoi i prikladnoi matematiki (doklady po matematike i ee prilozheniyam), 2, no. 2, 1988 | MR

[3] Alekseev V. A., Nikulin V. V., “Klassifikatsiya poverkhnostei del Petstso s log-terminalnymi osobennostyami indeksa $\leqslant2$ i involyutsii na poverkhnostyakh K3”, Dokl. AN SSSR, 306:3 (1989), 525–528 | MR | Zbl

[4] Vinberg E. B., “Otsutstvie kristallograficheskikh grupp otrazhenii v prostranstvakh Lobachevskogo bolshoi razmernosti”, Tr. Mosk. matem. ob-va, 47, 1984, 68–102 | MR | Zbl

[5] Iliev A. I., “Log-terminalnye osobennosti algebraicheskikh poverkhnostei”, Vestn. Mosk. un-ta. Ser. 1. Matematika. Mekhanika, 1986, no. 3, 38–44 | MR

[6] Nikulin V. V., “O klassifikatsii arifmeticheskikh grupp, porozhdennykh otrazheniyami, v prostranstvakh Lobachevskogo”, Izv. AN SSSR. Ser. matem., 45:1 (1981), 113–142 | MR | Zbl

[7] Nikulin V. V., “Poverkhnosti del Petstso s log-terminalnymi osobennostyami”, Matem. sb., 180:2 (1989), 226–243 | MR | Zbl

[8] Nikulin V. V., “Poverkhnosti del Petstso s log-terminalnymi osobennostyami, II”, Izv. AN SSSR. Ser. matem., 52:5 (1988), 1032–1050 | MR

[9] Prokhorov M. N., “Otsutstvie kristallograficheskikh grupp otrazhenii s nekompaktnym fundamentalnym mnogogrannikom konechnogo ob'ema v prostranstve Lobachevskogo bolshoi razmernosti”, Izv. AN SSSR. Ser. matem., 50:2 (1986), 413–424 | MR | Zbl

[10] Khovanskii A. G., “Giperploskie secheniya mnogogrannikov, toricheskie mnogoobraziya i diskretnye gruppy v prostranstve Lobachevskogo”, Funkts. analiz i ego prilozh., 20 (1986), 50–61 | MR

[11] Shokurov V. V., “Teorema o neobraschenii v nul”, Izv. AN SSSR. Ser. matem., 49:3 (1985), 635–651 | MR

[12] Artin M., “On isolated rational singularities of surfaces”, Amer. J. Math., 88 (1966), 129–136 | DOI | MR | Zbl

[13] Brieskorn E., “Rationale Singularitaten komplexer Flähen”, Inv. Math., 4 (1967–1968), 336–358 | DOI | MR

[14] Kawamata Y., “The cone of the curves of algebraic varieties”, Ann. Math., 119:3 (1984), 603–633 | DOI | MR | Zbl

[15] Mori S., “Threefolds whose canonical bundles are not numerically effective”, Ann. Math., 116 (1982), 133–176 | DOI | MR | Zbl

[16] Nikulin V. V., Algebraic surfaces with log-terminal singularities and nef anticanonical class and reflection groups in Lobachevsky spaces, Preprint MPI-89-28, Max-Planck-Institut für Mathematik, Bonn, 1989

[17] Sakai F., “Classification of normal surfaces”, Proc. Sympos. Pure Mathem., 46 (1987), 451–465 | MR | Zbl

[18] Zarisky O., “The theorem of Riemann–Roch for high multiples of an effective divisor on an algebraic surface”, Ann. Math., 76:2 (1962), 560–615 | DOI | MR