Geodesic
Birational Rigidity and $\mathbb Q$-Factoriality of a Singular Double Cover of a Quadric Branched over a Divisor of Degree~4
Matematičeskie zametki, Tome 84 (2008) no. 2, pp. 300-311.

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We prove birational rigidity and calculate the group of birational automorphisms of a nodal $\mathbb Q$-factorial double cover $X$ of a smooth three-dimensional quadric branched over a quartic section. We also prove that $X$ is $\mathbb Q$-factorial provided that it has at most 11 singularities; moreover, we give an example of a non-$\mathbb Q$-factorial variety of this type with 12 simple double singularities.
Keywords: birational geometry, Mori fibration, birational rigidity, Fano variety, sextic, superrigidity.
Mots-clés : birational automorphism, quartic 
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K. A. Shramov. Birational Rigidity and $\mathbb Q$-Factoriality of a Singular Double Cover of a Quadric  Branched over a Divisor of Degree~4. Matematičeskie zametki, Tome 84 (2008) no. 2, pp. 300-311. http://geodesic.mathdoc.fr/item/MZM_2008_84_2_a10/

[1] K. Matsuki, Introduction to the Mori Program, Universitext, Springer-Verlag, New York, 2002 | MR | Zbl

[2] 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

[3] A. Pukhlikov, “Essentials of the method of maximal singularities”, Explicit birational geometry of $3$-folds, London Math. Soc. Lecture Note Ser., 281, Cambridge Univ. Press, Cambridge, 2000, 73–100 | MR | Zbl

[4] V. A. Iskovskikh, “Antikanonicheskie modeli trekhmernykh algebraicheskikh mnogoobrazii”, Itogi nauki i tekhniki. Sovrem. problemy matem., 12, VINITI, M., 1979, 59–157 | MR | Zbl

[5] V. A. Iskovskikh, Yu. I. Manin, “Trekhmernye kvartiki i kontrprimery k probleme Lyurota”, Matem. sb., 86:1 (1971), 140–166 | MR | Zbl

[6] V. A. Iskovskikh, “Biratsionalnye avtomorfizmy trekhmernykh algebraicheskikh mnogoobrazii”, Itogi nauki i tekhniki. Sovrem. problemy matem., 12, VINITI, M., 1979, 159–236 | MR | Zbl

[7] V. A. Iskovskikh, A. V. Pukhlikov, “Biratsionalnye avtomorfizmy mnogomernykh algebraicheskikh mnogoobrazii”, Itogi nauki i tekhniki. Sovrem. problemy matem., 19, VINITI, M., 2001, 5–139

[8] A. V. Pukhlikov, “Biratsionalnye avtomorfizmy trekhmernoi kvartiki s prosteishei osobennostyu”, Matem. sb., 135:4 (1988), 472–496 | MR | Zbl

[9] I. Cheltsov, J. Park, Sextic double solids, arXiv: math/0404452v2

[10] M. Mella, “Birational geometry of quartic 3-folds. II: The importance of being $\mathbb Q$-factorial”, Math. Ann., 330:1 (2004), 107–126 | DOI | MR | Zbl

[11] M. M. Grinenko, “Biratsionalnye avtomorfizmy trekhmernoi dvoinoi kvadriki s prosteishei osobennostyu”, Matem. sb., 189:1 (1998), 101–118 | MR | Zbl

[12] M. M. Grinenko,, “Biratsionalnye avtomorfizmy trekhmernogo dvoinogo konusa”, Matem. sb., 189:7 (1998), 37–52 | MR | Zbl

[13] I. Cheltsov, “Nonrational nodal quartic threefolds”, Pacific J. Math., 226:1 (2006), 65–81 | MR

[14] K. A. Shramov, “$\mathbb{Q}$-faktorialnye trekhmernye kvartiki”, Matem. sb., 198:8 (2007), 103–114 | MR

[15] I. A. Cheltsov, “Biratsionalno zhestkie mnogoobraziya Fano”, UMN, 60:5 (2005), 71–160 | MR

[16] I. Cheltsov, Points in projective spaces and applications, arXiv: math/0511578v8

[17] S. Cynk, “Defect of a nodal hypersurface”, Manuscripta Math., 104:3 (2001), 325–331 | DOI | MR | Zbl

[18] D. Eisenbud, J.-H. Koh, “Remarks on points in a projective space”, Commutative Algebra (Berkeley, CA, 1987), Math. Sci. Res. Inst. Publ., 15, Springer, New York, 1989, 157–172 | MR | Zbl