Two local inequalities
Izvestiya. Mathematics , Tome 78 (2014) no. 2, pp. 375-426.

Voir la notice de l'article provenant de la source Math-Net.Ru

We prove two new local inequalities for divisors on smooth surfaces and consider several applications of these inequalities.
Keywords: Cremona group.
Mots-clés : Tian's $\alpha$-invariant, del Pezzo surface
@article{IM2_2014_78_2_a5,
     author = {I. A. Cheltsov},
     title = {Two local inequalities},
     journal = {Izvestiya. Mathematics },
     pages = {375--426},
     publisher = {mathdoc},
     volume = {78},
     number = {2},
     year = {2014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2014_78_2_a5/}
}
TY  - JOUR
AU  - I. A. Cheltsov
TI  - Two local inequalities
JO  - Izvestiya. Mathematics 
PY  - 2014
SP  - 375
EP  - 426
VL  - 78
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2014_78_2_a5/
LA  - en
ID  - IM2_2014_78_2_a5
ER  - 
%0 Journal Article
%A I. A. Cheltsov
%T Two local inequalities
%J Izvestiya. Mathematics 
%D 2014
%P 375-426
%V 78
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2014_78_2_a5/
%G en
%F IM2_2014_78_2_a5
I. A. Cheltsov. Two local inequalities. Izvestiya. Mathematics , Tome 78 (2014) no. 2, pp. 375-426. http://geodesic.mathdoc.fr/item/IM2_2014_78_2_a5/

[1] J. Kollár, Sh. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Math., 134, Cambridge Univ. Press, Cambridge, 1998 | MR | Zbl

[2] D. Kosta, Del Pezzo surfaces with Du Val singularities, Ph. D. thesis, University of Edinburgh, 2009; arXiv: abs/0904.0943

[3] A. Corti, “Singularities of linear systems and 3-fold birational geometry”, Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser., 281, Cambridge Univ. Press, Cambridge, 2000, 259–312 | MR | Zbl

[4] V. A. Iskovskikh, “Birational rigidity of Fano hypersurfaces in the framework of Mori theory”, Russian Math. Surveys, 56:2 (2001), 207–291 | DOI | DOI | MR | Zbl

[5] J. Kollár, “Singularities of pairs”, Algebraic geometry (Santa Cruz, 1995), Proc. Sympos. Pure Math., 62, Amer. Math. Soc., Providence, RI, 1997, 221–287 | DOI | MR | Zbl

[6] V. V. Shokurov, “3-fold log flips”, Russian Acad. Sci. Izv. Math., 40:1 (1993), 95–202 | DOI | MR | Zbl

[7] R. Lazarsfeld, Positivity in algebraic geometry, v. II, Ergeb. Math. Grenzgeb. (3), 49, Springer-Verlag, Berlin, 2004 | MR | Zbl

[8] I. A. Cheltsov, “Birationally rigid Fano varieties”, Russian Math. Surveys, 60:5 (2005), 875–965 | DOI | DOI | MR | Zbl

[9] C. Birkar, P. Cascini, C. D. Hacon, J. McKernan, “Existence of minimal models for varieties of log general type”, J. Amer. Math. Soc., 23:2 (2010), 405–468 | DOI | MR | Zbl

[10] G. Tian, “On Kähler–Einstein metrics on certain Kahler manifolds with $C_1(M)>0$”, Invent. Math., 89:2 (1987), 225–246 | DOI | MR | Zbl

[11] J.-P. Demailly, J. Kollár, “Semi-continuity of complex singularity exponents and Kähler–Einstein metrics on Fano orbifolds”, Ann. Sci. École Norm. Sup. (4), 34:4 (2001), 525–556 | DOI | MR | Zbl

[12] I. A. Cheltsov, C. A. Shramov, “Log canonical thresholds of smooth Fano threefolds”, Russian Math. Surveys, 63:5 (2008), 859–958 | DOI | DOI | MR | Zbl

[13] I. Cheltsov, J. Park, C. Shramov, “Exceptional del Pezzo hypersurfaces”, J. Geom. Anal., 20:4 (2010), 787–816 | DOI | MR | Zbl

[14] I. Cheltsov, “On singular cubic surfaces”, Asian J. Math., 13:2 (2009), 191–214 | DOI | MR | Zbl

[15] I. Cheltsov, “Log canonical thresholds of del Pezzo surfaces”, Geom. Funct. Anal., 18:4 (2008), 1118–1144 | DOI | MR | Zbl

[16] T. A. Springer, Invariant theory, Lecture Notes in Math., 585, Springer-Verlag, Berlin–New York, 1977 | MR | Zbl

[17] A. V. Pukhlikov, “Birational geometry of Fano direct products”, Izv. Math., 69:6 (2005), 1225–1255 | DOI | DOI | MR | Zbl

[18] A. V. Pukhlikov, “Birational automorphisms of algebraic threefolds with a pencil of Del Pezzo surfaces”, Izv. Math., 62:1 (1998), 115–155 | DOI | DOI | MR | Zbl

[19] Q. Zhang, “Rational connectedness of log $\mathbb Q$-Fano varieties”, J. Reine Angew. Math., 590 (2006), 131–142 | DOI | MR | Zbl

[20] T. Graber, J. Harris, J. Starr, “Families of rationally connected varieties.”, J. Amer. Math. Soc., 16:1 (2003), 57–67 | DOI | MR | Zbl

[21] Yu. Kawamata, K. Matsuda, K. Matsuki, “Introduction to the minimal model problem”, Algebraic geometry (Sendai, 1985), Adv. Stud. Pure Math., 10, North-Holland, Amsterdam, 1987, 283–360 | MR | Zbl

[22] A. V. Pukhlikov, “Birational automorphisms of Fano hypersurfaces”, Invent. Math., 134 (1998), 401–426 | DOI | MR | Zbl

[23] A. Corti, M. Mella, “Birational geometry of terminal quartic 3-folds. I”, Amer. J. Math., 126:4 (2004), 739–761 | DOI | MR | Zbl

[24] J. Kollár, Flips and abundance for algebraic threefolds (Salt Lake City, Utah, 1991), Astérisque, 211, Soc. Math. de France, Paris, 1992 | MR | Zbl

[25] S. S.-T. Yau, Y. Yu, Gorenstein quotient singularities in dimension three, Mem. Amer. Math. Soc., 105, no. 505, Amer. Math. Soc., Providence, RI, 1993 | MR | Zbl

[26] J. Rauschning, P. Slodowy, “An aspect of icosahedral symmetry”, Canad. Math. Bull., 45:4 (2005), 686–696 | DOI | MR | Zbl

[27] I. V. Dolgachev, V. A. Iskovskikh, “Finite subgroups of the plane Cremona group”, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin, v. I, Progr. Math., 269, Birkhäuser, Boston, MA, 2009, 443–548 | MR | Zbl

[28] J. Blanc, “Sous-groupes algebriques du groupe de Cremona”, Transform. Group, 14:2 (2009), 249–285 | DOI | MR | Zbl

[29] S. Bannai, H. Tokunaga, “A note on embeddings of $\mathbb S_4$ and $\mathbb A_5$ into the two-dimensional Cremona group and versal Galois covers”, Publ. Res. Inst. Math. Sci., 43:4 (2007), 1111–1123 | DOI | MR | Zbl

[30] W. L. Edge, “A pencil of four-nodal plane sextics”, Math. Proc. Cambridge Philos. Soc., 89:3 (1981), 413–421 | DOI | MR | Zbl

[31] M. Rosenlicht, “Some basic theorems on algebraic groups”, Amer. J. Math., 78 (1956), 401–443 | DOI | MR | Zbl

[32] J. Kollár, “Birational rigidity of Fano varieties and field extensions”, Mnogomernaya algebraicheskaya geometriya, Tr. MIAN, 264, MAIK, M., 2009, 103–108 | MR

[33] I. Cheltsov, “Fano varieties with many selfmaps”, Adv. Math., 217:1 (2008), 97–124 | DOI | MR | Zbl

[34] N. Hitchin, “Spherical harmonics and the icosahedron”, Groups and symmetries, CRM Proc. Lecture Notes,, 47, Amer. Math. Soc., Providence, RI, 2009, 215–231 | MR | Zbl