Abstract
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 ( M−42 ) + 1 and ( M−62 ) + 1 respectively.
Similar content being viewed by others
References
T. Eckl and A. Pukhlikov, “On the locus of nonrigid hypersurfaces,” in Springer Proc. Math. Statist., Vol. 79: Automorphisms in Birational and Affine Geometry (Springer, Cham, 2014), pp. 121–139.
A. V. Pukhlikov, “Birationally rigid double Fano hypersurfaces,” Mat. Sb. 191 (6), 101–126 (2000) [Sb. Math. 191 (5–6), 883–908 (2000)].
I. Cheltsov, “Double cubics and double quartics,” Math. Z. 253 (1), 75–86 (2006).
I. A. Chel’tsov, “Birationally rigid Fano varieties,” UspekhiMat. Nauk 60 (5 (365)), 71–160 (2005) [Russian Math. Surveys 60 (5), 875–965 (2005)].
A. V. Pukhlikov, “Birational geometry of algebraic varieties with a pencil of Fano cyclic covers,” Pure Appl. Math. Q. 5 (2), Special Issue, 641–700 (2009).
I. A. Chel’tsov, “Birationally super-rigid cyclic triple spaces,” Izv. Ross. Akad. Nauk Ser. Mat. 68 (6), 169–220 (2004) [Izv. Math. 68 (6), 1229–1275 (2004)].
R. Mullani, “Fano double spaces with a large singular locus,” Mat. Zametki 87 (3), 472–476 (2010) [Math. Notes 87 (3–4), 444–448 (2010)].
A. V. Pukhlikov, “Birationally rigid Fano fibrations. II,” Izv. Ross. Akad. Nauk Ser. Mat. 79 (4), 175–204 (2015) [Izv. Math. 79 (4), 809–937 (2015)].
F. Call and G. Lyubeznik, “A simple proof of Grothendieck’s theorem on the parafactoriality of local rings,” in Commutative Algebra: Syzygies, Multiplicities, and Birational Algebra (Amer. Math. Soc., Providence, RI, 1994), Vol. 159, pp. 15–18. Contemp. Math.
A. Pukhlikov, Birationally Rigid Varieties, in Math. Surveys Monogr. (Amer. Math. Soc., Providence, RI, 2013), Vol.190.
A. V. Pukhlikov, “Birational geometry of algebraic varieties fiberd into Fano double spaces,” Izv. Ross. Akad. Nauk Ser. Mat. 81 (3), 160–188 (2017) [Izv. Math. 81 (3), 618–644 (2017)].
A. V. Pukhlikov, “Birational automorphisms of a dual space and a dual quadric,” Izv. Akad. Nauk SSSR Ser. Mat. 52 (1), 229–239 (1988) [Math. USSR-Izv. 32 (1), 233–243 (1989)].
A. V. Pukhlikov, “On the self-intersection of a movable linear system,” Fundam. Prikl. Mat. 14 (6), 177–192 (2008) [J. Math. Sci. (New York) 164 (1), 119–130 (2010)].
Flips and Abundance for Algebraic Threefolds (Salt Lake City, UT, 1991), in Astérisque, Ed. by J. Kollár, (Soc. Math. France, Paris, 1992), Vol.211.
W. Fulton, Intersection Theory, in Ergeb. Math. Grenzgeb. (3) (Springer-Verlag, Berlin, 1998), Vol.2.
A. V. Pukhlikov, “Essentials of the method of maximal singularities,” in London Math. Soc. Lecture Note Ser., Vol. 281: Explicit Birational Geometry of 3-Folds (Cambridge Univ. Press, Cambridge, 2000), pp. 73–100.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © E. Johnstone, 2017, published in Matematicheskie Zametki, 2017, Vol. 102, No. 4, pp. 549–558.
Rights and permissions
About this article
Cite this article
Johnstone, E. Birationally rigid singular double quadrics and double cubics. Math Notes 102, 508–515 (2017). https://doi.org/10.1134/S000143461709022X
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S000143461709022X