Birationally Rigid Singular Double Quadrics and Double Cubics

E. Johnstone

Matematičeskie zametki, Tome 102 (2017) no. 4, pp. 549-558.

Résumé

In this paper it is shown that Fano double quadrics of index 1 and dimension at least 6 are birationally superrigid if the branch divisor has at most quadratic singularities of rank at least 6. Fano double cubics of index 1 and dimension at least 8 are birationally superrigid if the branch divisor has at most quadratic singularities of rank at least 8 and another minor condition of general position is satisfied. Hence, in the parameter spaces of these varieties the complement to the set of factorial and birationally superrigid varieties is of codimension at least $\binom{M-4}{2}+1$ and $\binom{M-6}{2}+1$ respectively.

Détail

Keywords: algebraic geometry, birational geometry, birational rigidity, Fano variety.

Bibliographie

[1] T. Eckl, A. Pukhlikov, “On the locus of nonrigid hypersurfaces”, Automorphisms in Birational and Affine Geometry, Springer Proc. Math. Stat., 79, Springer, Cham, 2014, 121–139 | MR | Zbl

[2] A. V. Pukhlikov, “Biratsionalno zhestkie dvoinye giperpoverkhnosti Fano”, Matem. sb., 191:6 (2000), 101–126 | DOI | MR | Zbl

[3] I. Cheltsov, “Double cubics and double quartics”, Math. Z., 253:1 (2006), 75–86 | DOI | MR | Zbl

[4] I. A. Cheltsov, “Biratsionalno zhestkie mnogoobraziya Fano”, UMN, 60:5 (365) (2005), 71–160 | DOI | MR | Zbl

[5] A. V. Pukhlikov, “Birational geometry of algebraic varieties with a pencil of Fano cyclic covers”, Pure Appl. Math. Q., 5:2, Special Issue (2009), 641–700 | DOI | MR | Zbl

[6] I. A. Cheltsov, “Biratsionalno sverkhzhestkie tsiklicheskie troinye prostranstva”, Izv. RAN. Ser. matem., 68:6 (2004), 169–220 | DOI | MR | Zbl

[7] R. Dzh. Mullani, “Dvoinye prostranstva Fano s bolshim singulyarnym lokusom”, Matem. zametki, 87:3 (2010), 472–476 | DOI | MR | Zbl

[8] A. V. Pukhlikov, “Biratsionalno zhestkie rassloeniya Fano. II”, Izv. RAN. Ser. matem., 79:4 (2015), 175–204 | DOI | MR | Zbl

[9] F. Call, G. Lyubeznik, “A simple proof of Grothendieck's theorem on the parafactoriality of local rings”, Commutative Algebra: Syzygies, Multiplicities, and Birational Algebra, Contemp. Math., 159, Amer. Math. Soc., Providence, RI, 1994, 15–18 | MR | Zbl

[10] A. Pukhlikov, Birationally Rigid Varieties, Math. Surveys Monogr., 190, Amer. Math. Soc., Providence, RI, 2013 | MR | Zbl

[11] A. V. Pukhlikov, “Biratsionalnaya geometriya algebraicheskikh mnogoobrazii, rassloennykh na dvoinye prostranstva Fano”, Izv. RAN. Ser. matem., 81:3 (2017), 160–188 | DOI | MR

[12] A. V. Pukhlikov, “Biratsionalnye avtomorfizmy dvoinogo prostranstva i dvoinoi kvadriki”, Izv. AN SSSR. Ser. matem., 52:1 (1988), 229–239 | MR | Zbl

[13] A. V. Pukhlikov, “O samoperesechenii podvizhnoi lineinoi sistemy”, Fundament. i prikl. matem., 14:6 (2008), 177–192 | MR

[14] Flips and Abundance for Algebraic Threefolds (Salt Lake City, UT, 1991), Astérisque, 211, ed. J. Kollár, Soc. Math. France, Paris, 1992 | Zbl

[15] W. Fulton, Intersection Theory, Ergeb. Math. Grenzgeb. (3), 2, Springer-Verlag, Berlin, 1998 | MR | Zbl

[16] A. V. 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