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Log canonical thresholds on Gorenstein canonical del Pezzo surfaces

Part of: Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  28 October 2010

Jihun Park  and
Joonyeong Won
Jihun Park
Affiliation:
Department of Mathematics, Pohang University of Science Technology (POSTECH), Hyoja-Dong San 31, Nam-Gu, Pohang, 790–784, Gyungbuk, Republic of Korea (wlog@postech.ac.kr; leonwon@postech.ac.kr)
Joonyeong Won
Affiliation:
Department of Mathematics, Pohang University of Science Technology (POSTECH), Hyoja-Dong San 31, Nam-Gu, Pohang, 790–784, Gyungbuk, Republic of Korea (wlog@postech.ac.kr; leonwon@postech.ac.kr)
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Abstract

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We classify all the effective anticanonical divisors on weak del Pezzo surfaces. Through this classification we obtain the smallest number among the log canonical thresholds of effective anticanonical divisors on a given Gorenstein canonical del Pezzo surface.

Keywords

del Pezzo surfaceeffective anticanonical divisorlog canonical threshold

MSC classification

Secondary: 14J26: Rational and ruled surfaces 14J17: Singularities
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 54 , Issue 1 , February 2011 , pp. 187 - 219
Copyright
Copyright © Edinburgh Mathematical Society 2010

References

1.Cheltsov, I., Log canonical thresholds of del Pezzo surfaces, Geom. Funct. Analysis 11 (2008), 1118–1144.CrossRefGoogle Scholar
2.Cheltsov, I., On singular cubic surfaces, Asian J. Math. 13 (2009), 191–214.CrossRefGoogle Scholar
3.Cheltsov, I. and Shramov, K., Log canonical thresholds of smooth Fano threefolds, Russ. Math. Surv. 63 (2008), 859–958.CrossRefGoogle Scholar
4.Demailly, J.-P. and Kollár, J., Semi-continuity of complex singularity exponents and Kähler–Einstein metrics on Fano orbifolds, Annales Scient. Éc. Norm. Sup. 34 (2001), 525–556.CrossRefGoogle Scholar
5.Demazure, M., Surfaces de del Pezzo: séminaire sur les singularités des surfaces, Lecture Notes in Mathematics, Volume 777, pp. 23–69 (Springer, 1980).Google Scholar
6.Furushima, M., Singular del Pezzo surfaces and analytic compactifications of 3-dimensional complex affine space ℂ3, Nagoya Math. J. 104 (1986), 1–28.CrossRefGoogle Scholar
7.Hidaka, F. and Watanabe, K., Normal Gorenstein surfaces with ample anti-canonical divisor, Tokyo J. Math. 4(2) (1981), 319–330.CrossRefGoogle Scholar
8.Miyanishi, M. and Zhang, D.-Q., Gorenstein log del Pezzo surfaces of rank one, J. Alg. 118(1) (1988), 63–84.CrossRefGoogle Scholar
9.Miyanishi, M. and Zhang, D.-Q., Gorenstein log del Pezzo surfaces, II, J. Alg. 156(1) (1993), 183–193.CrossRefGoogle Scholar
10.Park, J., Birational maps of del Pezzo fibrations, J. Reine Angew. Math. 538 (2001), 213–221.Google Scholar
11.Park, J., A note on del Pezzo fibrations of degree 1, Commun. Alg. 31(12) (2003), 5755–5768.CrossRefGoogle Scholar
12.Pinkham, H. C., Simple elliptic singularities, del Pezzo surfaces and Cremona transformations, Proceedings of Symposia in Pure Mathematics, Volume 30 (American Mathematical Society, Providence, RI, 1977), pp.69–70.Google Scholar
13.Pukhlikov, A. V., Birational geometry of Fano direct products, Izv. Math. 69 (2005), 1225–1255.CrossRefGoogle Scholar
14.Urabe, T., On singularities on degenerate del Pezzo surfaces of degree 1, 2, in Singularities, Part 2, pp. 587–591 (American Mathematical Society, Providence, RI, 1983).CrossRefGoogle Scholar
15.Zhang, D.-Q., Logarithmic del Pezzo surfaces of rank one with contractible boundaries, Osaka J. Math. 25(2) (1988), 461–497.Google Scholar