Hostname: page-component-7479d7b7d-rvbq7 Total loading time: 0 Render date: 2024-07-11T05:36:36.788Z Has data issue: false hasContentIssue false
  • English
  • Français

Del Pezzo Surfaces in Weighted Projective Spaces

Part of: Computational aspects in algebraic geometry Complex manifolds Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  04 April 2018

Erik Paemurru
Erik Paemurru*
Affiliation:
School of Mathematics, University of Edinburgh, Edinburgh EH9 3JZ, UK (E.Paemurru@lboro.ac.uk)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study singular del Pezzo surfaces that are quasi-smooth and well-formed weighted hypersurfaces. We give an algorithm to classify all of them.

Keywords

Fano varietydel Pezzo surfaceKähler–Einstein metric

MSC classification

Primary: 14J45: Fano varieties
Secondary: 32Q20: Kähler-Einstein manifolds 14J26: Rational and ruled surfaces 14Q10: Surfaces, hypersurfaces
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 2 , May 2018 , pp. 545 - 572
Copyright
Copyright © Edinburgh Mathematical Society 2018 

References

1.Alexeev, V. and Nikulin, V., Del Pezzo and K3 surfaces, MSJ Memoirs, Volume 15 (Mathematical Society Japan, 2006).CrossRefGoogle Scholar
2.Araujo, C., Kähler–Einstein metrics for some quasi-smooth log del Pezzo surfaces, Transactions of the American Mathematical Society, Volume 354, pp. 4303–3312 (American Mathematical Society, 2002).Google Scholar
3.Belousov, G., The maximal number of singular points on log del Pezzo surfaces, J. Math. Sci. Univ. Tokyo 16 (2009), 231–238.Google Scholar
4.Berman, R., K-polystability of ℚ-Fano varieties admitting Kähler–Einstein metrics, Invent. Math. to appear.Google Scholar
5.Boyer, C., Galicki, K. and Nakamaye, M., On the geometry of Sasakian–Einstein 5-manifolds, Math. Ann. 325 (2003), 485–524.CrossRefGoogle Scholar
6.Cheltsov, I., Log canonical thresholds of del Pezzo surfaces, Geom. Funct. Anal. 18 (2008), 1118–1144.CrossRefGoogle Scholar
7.Cheltsov, I., Park, J and Shramov, C., Exceptional del Pezzo hypersurfaces, J. Geom. Anal. 20 (2010), 787–816.CrossRefGoogle Scholar
8.Cheltsov, I. and Shramov, C., Log canonical thresholds of smooth Fano threefolds, Russian Math. Surveys 63 (2008), 859–958.CrossRefGoogle Scholar
9.Cheltsov, I. and Shramov, C., Del Pezzo zoo, Exp. Math. 22 (2013), 313–326.CrossRefGoogle Scholar
10.Chen, X.-X., Donaldson, S. and Sun, S., Kähler–Einstein metrics on Fano manifolds. I: approximation of metrics with cone singularities, J. Amer. Math. Soc. 28(1) (2015), 183–197.CrossRefGoogle Scholar
11.Chen, X.-X., Donaldson, S. and Sun, S., Kähler–Einstein metrics on Fano manifolds. II: limits with cone angle less than 2π, J. Amer. Math. Soc. 28(1) (2015), 199–234.CrossRefGoogle Scholar
12.Chen, X.-X., Donaldson, S. and Sun, S., Kähler–Einstein metrics on Fano manifolds. III: limits as cone angle approaches 2π and completion of the main proof, J. Amer. Math. Soc. 28(1) (2015), 199–234.CrossRefGoogle Scholar
13.Demailly, J.-P. and Kollár, J., Semi-continuity of complex singularity exponents and Kähler–Einstein metrics on Fano orbifolds, Ann. Sci. Éc. Norm. Supér. 34 (2001), 525–556.CrossRefGoogle Scholar
14.Donaldson, S., Lower bounds on the Calabi functional, J. Differential Geom. 70 (2005), 453–472.CrossRefGoogle Scholar
15.Elagin, A., Exceptional sets on del Pezzo surfaces with one log-terminal singularity, Math. Notes 82(1–2) (2007), 33–46.CrossRefGoogle Scholar
16.Gauntlett, J., Martelli, D., Sparks, J. and Yau, S.-T., Obstructions to the existence of Sasaki–Einstein metrics, Comm. Math. Phys. 273 (2007), 803–827.CrossRefGoogle Scholar
17.Hidaka, F. and Watanabe, K., Normal Gorenstein surfaces with ample anti-canonical divisor, Tokyo J. Math. 4 (1981), 319–330.CrossRefGoogle Scholar
18.Iano-Fletcher, A., Working with weighted complete intersections, London Mathematical Society Lecture Note Series, Volume 281, pp. 101–173 (Cambridge University Press, 2000).Google Scholar
19.Ishii, A. and Ueda, K., The special McKay correspondence and exceptional collection, preprint, 2011 (arXiv:1104.2381).Google Scholar
20.Johnson, J. and Kollár, J., Kähler–Einstein metrics on log del Pezzo surfaces in weighted projective 3-spaces, Ann. de l'Institut Fourier 51 (2001), 69–79.CrossRefGoogle Scholar
21.Kawamata, Y., Derived categories of toric varieties, Michigan Math. J. 54 (2006), 517–536.CrossRefGoogle Scholar
22.Keel, S. and McKernan, J., Rational curves on quasi-projective surfaces, Memoirs of the American Mathematical Society, Volume 669 (American Mathematical Society, 1999).Google Scholar
23.Kojima, H., Del Pezzo surfaces of rank one with unique singular points, Japan J. Math. 25 (1999), 343–374.CrossRefGoogle Scholar
24.Kollár, J., Singularities of pairs, Proc. Sympos. Pure Math. 62 (1997), 221–287.CrossRefGoogle Scholar
25.Miyanishi, M. and Zhang, D. Q., Gorenstein log del Pezzo surfaces of rank one, J. Algebra 118 (1988), 63–84.CrossRefGoogle Scholar
26.Shokurov, V., Complements on surfaces, J. Math. Sci. 102 (2000), 3876–3932.CrossRefGoogle Scholar
27.Spotti, C., Degenerations of Kähler–Einstein Fano manifolds, PhD thesis, Imperial College, London, UK (2012).Google Scholar
28.Stoppa, J., K-stability of constant scalar curvature Kähler manifolds, Adv. Math. 221 (2009), 1397–1408.CrossRefGoogle Scholar
29.Tian, G., On Kähler–Einstein metrics on certain Kähler manifolds with c 1(M) > 0, Inv. Math. 89 (1987), 225–246.CrossRefGoogle Scholar
30.Tian, G., On Calabi's conjecture for complex surfaces with positive first Chern class, Inv. Math. 101 (1990), 101–172.CrossRefGoogle Scholar
31.Tian, G., Kähler–Einstein metrics with positive scalar curvature, Inv. Math. 130 (1997), 1–37.CrossRefGoogle Scholar
32.Tian, G., K-stability and Kähler–Einstein metrics, Comm. Pure Appl. Math. 68(7) (2015), 1085–1156.CrossRefGoogle Scholar
33.Yau, S. and Yu, Y., Classification of 3-dimensional isolated rational hypersurface singularities with ℂ*-action, Rocky Mountain J. Math. 35 (2005), 1795–1802.CrossRefGoogle Scholar
34.Zhang, D.-Q., Logarithmic del Pezzo surfaces of rank one with contractible boundaries, Osaka J. Math. 25 (1988), 461–497.Google Scholar