$\mathbb Q$-factorial quartic threefolds
Sbornik. Mathematics, Tome 198 (2007) no. 8, pp. 1165-1174 Cet article a éte moissonné depuis la source Math-Net.Ru

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It is proved that a nodal quartic threefold $X$ containing no planes is $\mathbb Q$-factorial if it has at most 12 singular points. An exception here is a quartic with precisely 12 singularities containing a quadric surface. Some geometric constructions relating to such a quartic are presented. Bibliography: 14 titles.
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K. A. Shramov. $\mathbb Q$-factorial quartic threefolds. Sbornik. Mathematics, Tome 198 (2007) no. 8, pp. 1165-1174. http://geodesic.mathdoc.fr/item/SM_2007_198_8_a5/

[1] V. A. Iskovskikh, Yu. I. Manin, “Trekhmernye kvartiki i kontrprimery k probleme Lyurota”, Matem. sb., 86(128):1(9) (1971), 140–166 | MR | Zbl

[2] A. V. Pukhlikov, “Biratsionalnye avtomorfizmy trekhmernoi kvartiki s prosteishei osobennostyu”, Matem. sb., 135(177):4 (1988), 472–496 ; A. V. Pukhlikov, “Birational automorphisms of a three-dimensional quartic with a quadratic singularity”, Math. USSR-Sb., 63:2 (1989), 457–482 | MR | Zbl | DOI | Zbl

[3] 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

[4] I. Cheltsov, “Nonrational nodal quartic threefolds”, Pacific J. Math., 226:1 (2006), 65–82 | MR | Zbl

[5] C. Ciliberto, V. Di Gennaro, “Factoriality of certain hypersurfaces of $\mathbb P^4$ with ordinary double points”, Algebraic transformation groups and algebraic varieties, Proceedings of the conference on interesting algebraic varieties arising in algebraic transformation group theory (Vienna, 2001), Encyclopaedia Math. Sci., 132, Springer-Verlag, Berlin, 2004, 1–7 | MR | Zbl

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

[7] E. M. Bese, “On the spannedness and very ampleness of certain line bundles on the blow-ups of $\mathbb P^2_{\mathbb C}$ and $F_r$”, Math. Ann., 262:2 (1983), 225–238 | DOI | MR | Zbl

[8] S. Cynk, “Defect of a nodal hypersurface”, Manuscripta Math., 104, no. 3, 2001, 325–331 | DOI | MR | Zbl

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

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

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

[12] G. Brown, A. Corti, F. Zucconi, “Birational geometry of 3-fold Mori fibre spaces”, Proceedings of the Fano Conference (Univ. Torino, Turin, 2002), Univ. Torino, Turin, 2004, 235–275 ; arXiv: math.AG/0307301 | MR | Zbl

[13] V. A. Iskovskikh, “Biratsionalnye avtomorfizmy trekhmernykh algebraicheskikh mnogoobrazii”, Itogi nauki i tekhniki. Sovrem. problemy matem., 12, VINITI, M., 1979, 159–236 ; V. A. Iskovskikh, “Birational automorphisms of three-dimensional algebraic varieties”, J. Soviet Math., 13 (1980), 815–868 | MR | Zbl | DOI | Zbl

[14] V. A. Iskovskikh, A. V. Pukhlikov, “Biratsionalnye avtomorfizmy mnogomernykh algebraicheskikh mnogoobrazii”, Itogi nauki i tekhniki. Sovrem. problemy matem. Algebraicheskaya geometriya\.–\.1, 19, VINITI, M., 2001, 5–139