# K-stability of smooth del Pezzo surfaces

Article

First Online:

Received:

Revised:

- 53 Downloads

## Abstract

In a new algebro-geometric way we completely determine whether smooth del Pezzo surfaces are K-(semi)stable or not.

In the present article, all varieties are defined over an algebraically closed field

kof characteristic 0.

### References

- 1.Blum, H., Jonsson, M.: Thresholds, valuations, and K-stability. arXiv:1706.04548
- 2.Cheltsov, I.: Log canonical thresholds of del Pezzo surfaces. Geom. Funct. Anal.
**11**, 1118–1144 (2008)MathSciNetCrossRefMATHGoogle Scholar - 3.Cheltsov, I., Park, J.: Global log-canonical thresholds and generalized Eckardt points. Sb. Math.
**193**(5–6), 779–789 (2002)MathSciNetCrossRefMATHGoogle Scholar - 4.Cheltsov, I., Park, J.: Sextic double solids. In: Bogomolov, F., Tschinkel, Yu. (eds.) Cohomological and geometric approaches to rationality problems, Progr. Math., vol. 282, pp. 75–132. Birkhäuser, Boston (2010)Google Scholar
- 5.Cheltsov, I., Park, J., Won, J.: Log canonical thresholds of certain Fano hypersurfaces. Math. Z.
**276**(1–2), 51–79 (2014)MathSciNetCrossRefMATHGoogle Scholar - 6.Cheltsov, I., Shramov, K.: Log-canonical thresholds for nonsingular Fano threefolds. With an appendix by J.-P. Demailly. Russ. Math. Surv.
**63**(5), 859–958 (2008)CrossRefGoogle Scholar - 7.Chen, X., Donaldson, S., Sun, S.: Kähler–Einstein metrics on Fano manifolds I: approximation of metrics with cone singularities. J. Am. Math. Soc.
**28**(1), 183–197 (2015)CrossRefMATHGoogle Scholar - 8.Chen, X., Donaldson, S., Sun, S.: Kähler–Einstein metrics on Fano manifolds, II: limits with cone angle less than \(2\pi \). J. Am. Math. Soc.
**28**(1), 199–234 (2015)CrossRefMATHGoogle Scholar - 9.Chen, X., Donaldson, S., Sun, S.: Kähler–Einstein metrics on Fano manifolds III: limits as cone angle approaches \(2\pi \) and completion of the main proof. J. Am. Math. Soc.
**28**(1), 235–278 (2015)CrossRefMATHGoogle Scholar - 10.Fujita, K.: On K-stability and the volume functions of \(\mathbb{Q}\)-Fano varieties. Proc. Lond. Math. Soc.
**113**(5), 541–582 (2016)MathSciNetCrossRefMATHGoogle Scholar - 11.Fujita, K.: K-stability of Fano manifolds with not small alpha invariants. To appear in J. Inst. Math. Jussieu. doi: 10.1017/S1474748017000111
- 12.Fujita, K.: On Berman–Gibbs stability and K-stability of \(\mathbb{Q}\)-Fano varieties. Compositio Math.
**152**, 288–298 (2016)MathSciNetCrossRefMATHGoogle Scholar - 13.Fujita, K., Odaka, Yu.: On the K-stability of Fano varieties and anticanonical divisors. To appear in Tohoku Math. J. arXiv:1602.01305
- 14.Hacking, P., Prokhorov, Yu.: Smoothable del Pezzo surfaces with quotient singularities. Compositio Math.
**146**(1), 169–192 (2010)MathSciNetCrossRefMATHGoogle Scholar - 15.Hwang, J.-M., Kim, H., Lee, Y., Park, J.: Slopes of smooth curves on Fano manifolds. Bull. Lond. Math. Soc.
**43**(5), 827–839 (2011)MathSciNetCrossRefMATHGoogle Scholar - 16.Kempf, G.: Instability in invariant theory. Ann. Math. (2)
**108**(2), 299–316 (1978)MathSciNetCrossRefMATHGoogle Scholar - 17.Kollár, J.: Singularities of pairs. In: Kollr, J., Lazarsfeld, R., Morrison, D. (eds.) Algebraic geometry (Santa Cruz, 1995) Part 1, Proc. Sympos. Pure Math., vol. 62, pp. 221–287. Amer. Math. Soc. (1997)Google Scholar
- 18.Manetti, M.: Normal degenerations of the complex projective plane. J. Reine Angew. Math.
**419**, 89–118 (1991)MathSciNetMATHGoogle Scholar - 19.Manetti, M.: Normal projective surfaces with \(\rho =1\), \(P_{-1}\ge 5\). Rend. Sem. Mat. Univ. Padova
**89**, 195–205 (1993)MathSciNetMATHGoogle Scholar - 20.Odaka, Yu., Sano, Yu.: Alpha invariant and K-stability of \(\mathbb{Q}\)-Fano varieties. Adv. Math.
**229**(5), 2818–2834 (2012)MathSciNetCrossRefMATHGoogle Scholar - 21.Odaka, Yu., Spotti, C., Sun, S.: Compact moduli spaces of del Pezzo surfaces and Kähler–Einstein metrics. J. Differ. Geom.
**102**(1), 127–172 (2016)CrossRefMATHGoogle Scholar - 22.Panov, D., Ross, J.: Slope stability and exceptional divisors of high genus. Math. Ann.
**343**(1), 79–101 (2009)MathSciNetCrossRefMATHGoogle Scholar - 23.Prokhorov, Yu.: A note on degenerations of del Pezzo surfaces. Ann. Inst. Fourier (Grenoble)
**65**(1), 369–388 (2015)MathSciNetCrossRefMATHGoogle Scholar - 24.Ross, J., Thomas, R.: A study of the Hilbert–Mumford criterion for the stability of projective varieties. J. Algebraic Geom.
**16**(2), 201–255 (2007)MathSciNetCrossRefMATHGoogle Scholar - 25.Tian, G.: On Kähler–Einstein metrics on certain Kähler manifolds with \(c_1(M)>0\). Invent. Math.
**89**, 225–246 (1987)MathSciNetCrossRefMATHGoogle Scholar - 26.Tian, G.: K-stability and Kähler–Einstein metrics. Commun. Pure Appl. Math.
**68**(7), 1085–1156 (2015). [Corrigendum: K-Stability and Kähler–Einstein Metrics.**68**(11), 2082–2083 (2015)]Google Scholar

## Copyright information

© Springer-Verlag GmbH Deutschland 2017