Journal of the Mathematical Society of Japan

Classification of log del Pezzo surfaces of index three

Abstract

A normal projective non-Gorenstein log-terminal surface $S$ is called a log del Pezzo surface of index three if the three-times of the anti-canonical divisor $-3K_S$ is an ample Cartier divisor. We classify all log del Pezzo surfaces of index three. The technique for the classification is based on the argument of Nakayama.

Article information

Source
J. Math. Soc. Japan, Volume 69, Number 1 (2017), 163-225.

Dates
First available in Project Euclid: 18 January 2017

https://projecteuclid.org/euclid.jmsj/1484730023

Digital Object Identifier
doi:10.2969/jmsj/06910163

Mathematical Reviews number (MathSciNet)
MR3597552

Zentralblatt MATH identifier
1375.14121

Subjects
Primary: 14J26: Rational and ruled surfaces
Secondary: 14E30: Minimal model program (Mori theory, extremal rays)

Citation

FUJITA, Kento; YASUTAKE, Kazunori. Classification of log del Pezzo surfaces of index three. J. Math. Soc. Japan 69 (2017), no. 1, 163--225. doi:10.2969/jmsj/06910163. https://projecteuclid.org/euclid.jmsj/1484730023

References

• V. A. Alexeev and V. V. Nikulin, Classification of del Pezzo surfaces with log-terminal singularities of index $\leq 2$, involutions on $K3$ surfaces, and reflection groups in Lobachevskiĭ spaces (Russian), Lectures in mathematics and its applications, 2, (Russian), 51–150, Ross. Akad. Nauk, Inst. Mat. im. Steklova, Moscow, 1988.
• V. A. Alexeev and V. V. Nikulin, Classification of del Pezzo surfaces with log-terminal singularities of index $\leq 2$ and involutions on $K3$ surfaces (Russian), Dokl. Akad. Nauk. SSSR, 306 (1989), 525–528; translation in Soviet Math. Dokl., 39 (1989), 507–511.
• V. A. Alexeev and V. V. Nikulin, Del Pezzo and $K3$ surfaces, MSJ Memoirs, 15, Math. Soc. of Japan, Tokyo, 2006.
• L. Brenton, On singular complex surfaces with negative canonical bundle, with applications to singular compactifications of $\C^2$ and to $3$-dimensional rational singularities, Math. Ann., 248 (1980), 117–124.
• M. Demazure, Surfaces de del Pezzo II–V, in Séminaire sur les Singularités des Surfaces (eds. M. Demazure, H. Pinkham and B. Teissier), Lecture Notes in Math., 777 (1980), Springer, Berlin, 35–68.
• K. Fujita, Log del Pezzo surfaces with not small fractional indices, Math. Nachr., 289 (2016), 34–59.
• K. Fujita, Log del Pezzo surfaces with large volumes, Kyushu J. Math., 70 (2016), 131–147.
• F. Hidaka and K-i. Watanabe, Normal Gorenstein surfaces with ample anti-canonical divisor, Tokyo J. Math., 4 (1981), 319–330.
• J. Kollár and S. Mori, Birational geometry of algebraic varieties, With the collaboration of C. H. Clemens and A. Corti. Cambridge Tracts in Math., 134, Cambridge University Press, Cambridge, 1998.
• S. Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. (2), 116 (1982), 133–176.
• N. Nakayama, Classification of log del Pezzo surfaces of index two, J. Math. Sci. Univ. Tokyo, 14 (2007), 293–498.
• H. Ohashi and S. Taki, $K3$ surfaces and log del Pezzo surfaces of index three, Manuscripta Math., 139 (2012), 443–471.
• D. Testa, A. Várilly-Alvarado and M. Velasco, Big rational surfaces, Math. Ann., 351 (2011), 95–107.