Abstract
We prove that there are no del Pezzo surfaces with five log terminal singularities and the Picard number 1. In the course of the proof, we make use of fibrations with general fiber ℙ1.
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References
Y. Kawamata, K. Matsuda and J. Matsuki, “Introduction to the minimal model program,/rd in Algebraic Geometry, Adv. Stud. Pure Math. (North-Holland, Amsterdam, 1987), Vol. 10, pp. 283–360.
M. Furushima, “Singular del Pezzo surfaces and analytic compactifications of 3-dimensional complex affine space C 3,” Nagoya Math. J. 104, 1–28 (1986).
M. Miyanishi and D.-Q. Zhang, “Gorenstein log del Pezzo surfaces of rank one,” J. Algebra 118(1), 63–84 (1988).
H. Kojima, “Logarithmic del Pezzo surfaces of rang one with unique singular points,” Japan. J. Math. (N. S.) 25(2), 343–375 (1999).
V. A. Alekseev and V. V. Nikulin, “Classification of del Pezzo surfaces with log-terminal singularities of index ≤ 2, involutions on K3 surfaces and reflection groups in Lobachevskii spaces,” in Lectures in Mathematics and Its Applications (Ross. Akad. Nauk, Inst. Mat. im. Steklova, Moscow, 1988), Vol. 2, no. 2, pp. 51–150 [in Russian].
S. Keel and J. McKernan, Rational Curves on Quasi-projective Surfaces, in Mem. Amer. Math. Soc. (Amer. Math. Soc., Providence, RI, 1999), Vol. 140, 669.
J. Kollar, Is There a Topological Bogomolov-Miyaoka-Yau Inequality?, arXiv: math/0602562..
Y. Kawamata, “Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces,” Ann. of Math. (2) 127(1), 93–163 (1988).
D.-Q. Zhang, “Logarithmic del Pezzo surfaces one with rational double and triple singular points,” Tohoku Math. J. (2) 41(3), 399–452 (1989).
E. Brieskorn, “Rationale Singularitäten komplexer Flächen,” Invent. Math. 4(5), 336–358 (1968).
A. I. Iliev, “Log-terminal singularities of algebraic surfaces,” Vestnik Moskov. Univ. Ser. I Mat. Mekh., No. 3, 38–44 (1986) [in Russian].
D.-Q. Zhang, “Logarithmic del Pezzo surfaces of rang one with contractible boundaries,” Osaka J. Math. 25(2), 461–497 (1988).
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Original Russian Text © G. N. Belousov, 2008, published in Matematicheskie Zametki, 2008, Vol. 83, No. 2, pp. 170–180.
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Belousov, G.N. Del Pezzo surfaces with log terminal singularities. Math Notes 83, 152–161 (2008). https://doi.org/10.1134/S0001434608010185
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DOI: https://doi.org/10.1134/S0001434608010185