Geodesic
On $G$-Fano threefolds
Izvestiya. Mathematics , Tome 79 (2015) no. 4, pp. 795-808.

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

We study Fano threefolds with terminal Gorenstein singularities admitting a ‘minimal’ action of a finite group. We prove that under certain additional assumptions such a variety contains no planes. We also obtain upper bounds for the number of singular points of certain Fano threefolds with terminal factorial singularities.
Keywords: Fano variety, terminal singularity, divisor, linear system.
Mots-clés : birational map 
@article{IM2_2015_79_4_a5,
     author = {Yu. G. Prokhorov},
     title = {On $G${-Fano} threefolds},
     journal = {Izvestiya. Mathematics },
     pages = {795--808},
     publisher = {mathdoc},
     volume = {79},
     number = {4},
     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2015_79_4_a5/}
}
TY  - JOUR
AU  - Yu. G. Prokhorov
TI  - On $G$-Fano threefolds
JO  - Izvestiya. Mathematics 
PY  - 2015
SP  - 795
EP  - 808
VL  - 79
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2015_79_4_a5/
LA  - en
ID  - IM2_2015_79_4_a5
ER  - 
%0 Journal Article
%A Yu. G. Prokhorov
%T On $G$-Fano threefolds
%J Izvestiya. Mathematics 
%D 2015
%P 795-808
%V 79
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2015_79_4_a5/
%G en
%F IM2_2015_79_4_a5
Yu. G. Prokhorov. On $G$-Fano threefolds. Izvestiya. Mathematics , Tome 79 (2015) no. 4, pp. 795-808. http://geodesic.mathdoc.fr/item/IM2_2015_79_4_a5/

[1] Yu. Prokhorov, “Simple finite subgroups of the Cremona group of rank 3”, J. Algebraic Geom., 21:3 (2012), 563–600 | DOI | MR | Zbl

[2] Yu. Prokhorov, “$G$-Fano threefolds. I”, Adv. Geom., 13:3 (2013), 389–418 | MR | Zbl

[3] Yu. Prokhorov, “$p$-elementary subgroups of the Cremona group of rank 3”, Classification of algebraic varieties, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2011, 327–338 | MR | Zbl

[4] Yu. Prokhorov, “$G$-Fano threefolds. II”, Adv. Geom., 13:3 (2013), 419–434 | MR | Zbl

[5] Yu. Prokhorov, “On the degree of Fano threefolds with canonical Gorenstein singularities”, Sb. Math., 196:1 (2005), 77–114 | DOI | DOI | MR | Zbl

[6] A.-S. Kaloghiros, “The defect of {F}ano 3-folds”, J. Algebraic Geom., 20:1 (2011), 127–149 ; Errata: 21:2 (2012), 397–399 | DOI | MR | Zbl | DOI | Zbl

[7] H. F. Baker, A locus with linear self-transformations, Cambridge Tracts in Mathematics and Mathematical Physics, 39, University Press, Cambridge; The Macmillan Company, New York, 1946, xi+107 pp. | MR

[8] Y. Kawamata, “Crepant blowing-up of {$3$}-dimensional canonical singularities and its application to degenerations of surfaces”, Ann. of Math. (2), 127:1 (1988), 93–163 | DOI | MR | Zbl

[9] V. A. Iskovskikh, Yu. G. Prokhorov, “Fano varieties”, Algebraic geometry. V: Fano varieties, Encyclopaedia Math. Sci., 47, Springer, Berlin, 1999, 1–245 | MR

[10] Y. Namikawa, “Smoothing Fano 3-folds”, J. Algebraic Geom., 6:2 (1997), 307–324 | MR | Zbl

[11] P. Jahnke, I. Radloff, “Gorenstein Fano threefolds with base points in the anticanonical system”, Compos. Math., 142:2 (2006), 422–432 | DOI | MR | Zbl

[12] V. V. Przhyjalkowski, I. A. Cheltsov, K. A. Shramov, “Hyperelliptic and trigonal Fano threefolds”, Izv. Math., 69:2 (2005), 365–421 | DOI | DOI | MR | Zbl

[13] P. Jahnke, I. Radloff, “Terminal Fano threefolds and their smoothings”, Math. Z., 269:3–4 (2011), 1129–1136 | DOI | MR | Zbl

[14] R. Lazarsfeld, Positivity in algebraic geometry. II: Positivity for vector bundles, and multiplier ideals, Ergeb. Math. Grenzgeb., 49, Springer-Verlag, Berlin, 2004, xviii+385 pp. | MR | Zbl

[15] J. Kollár (Ed.), Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah (Salt Lake City, UT, 1991), Astérisque, 211, Soc. Math. France, Paris, 1992, 258 pp. | MR

[16] M. Kawakita, “Inversion of adjunction on log canonicity”, Invent. Math., 167 (2007), 129–133 | DOI | MR | Zbl

[17] V. A. Iskovskikh, “Double projection from a line on Fano threefolds of the first kind”, Math. USSR-Sb., 66:1 (1990), 265–284 | DOI | MR | Zbl

[18] V. V. Šokurov, “The existence of a straight line on Fano 3-folds”, Math. USSR-Izv., 15:1 (1980), 173–209 | DOI | MR | Zbl | Zbl

[19] J. Kollár, “Flops”, Nagoya Math. J., 113 (1989), 15–36 | MR | Zbl

[20] S. Cutkosky, “Elementary contractions of Gorenstein threefolds”, Math. Ann., 280:3 (1988), 521–525 | DOI | MR | Zbl

[21] J. Kollár, “Flips, flops, minimal models, etc.”, Surveys in differential geometry (Cambridge, MA, 1990), Lehigh Univ., Bethlehem, PA, 1991, 113–199 | MR | Zbl