Advertisement

Double quadrics with large automorphism groups

  • Victor V. Przyjalkowski
  • Constantin A. Shramov
Article

Abstract

We classify nodal Fano threefolds that are double covers of smooth quadrics branched over intersections with quartics and are acted on by finite simple non-abelian groups. We also study their rationality.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Artin and D. Mumford, “Some elementary examples of unirational varieties which are not rational, ” Proc. London Math. Soc., Ser. 3, 25, 75–95 (1972).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    P. S. Aspinwall and D. R. Morrison, “Stable singularities in string theory” (with an appendix by M. Gross), Commun.Math. Phys. 178 (1), 115–134 (1996).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    A. Beauville, “Variétés de Prym et jacobiennes intermediaires, ” Ann. Sci. Éc. Norm. Supér., Sér. 4, 10, 309–391 (1977).MathSciNetzbMATHGoogle Scholar
  4. 4.
    A. Beauville, “Non-rationality of the S6-symmetric quartic threefolds, ” Rend. Semin. Mat., Univ. Politec. Torino 71, 385–388 (2013).MathSciNetzbMATHGoogle Scholar
  5. 5.
    H. Burkhardt, “Untersuchungen aus dem Gebiete der hyperelliptischen Modulfunctionen. Zweiter Theil, ” Math. Ann. 38, 161–224 (1891).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    I. Cheltsov, V. Przyjalkowski, and C. Shramov, “Quartic double solids with icosahedral symmetry, ” Eur. J. Math. 2 (1), 96–119 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    I. Cheltsov and C. Shramov, “Two rational nodal quartic threefolds,” arXiv: 1511.07508 [math.AG].Google Scholar
  8. 8.
    I. Cheltsov and C. Shramov, CremonaGroups and the Icosahedron (CRC Press, Boca Raton, FL, 2016).zbMATHGoogle Scholar
  9. 9.
    J.-L. Colliot-Thélène and A. Pirutka, “Hypersurfaces quartiques de dimension 3: non-rationalité stable, ” Ann. Sci. Éc. Norm. Supér., Sér. 4, 49 (2), 371–397 (2016).zbMATHGoogle Scholar
  10. 10.
    J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlasof Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups (Clarendon Press, Oxford, 1985).zbMATHGoogle Scholar
  11. 11.
    S. Cynk, “Defect of a nodal hypersurface, ” Manuscr. Math. 104 (3), 325–331 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    W. Feit, “The current situation in the theory of finite simple groups, ” in Actes Congr. Int. Math., Nice, 1970 (Gauthier-Villars, Paris, 1971), Vol. 1, pp. 55–93.Google Scholar
  13. 13.
    S. O. Gorchinskiy and C. A. Shramov, “Unramified Brauer group and its applications,” arXiv: 1512.00874 [math.AG].Google Scholar
  14. 14.
    M. M. Grinenko, “Birational automorphisms of a three-dimensional double quadric with an elementary singularity, ” Mat. Sb. 189 (1), 101–118 (1998) [Sb. Math. 189, 97–114 (1998)].MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    M. M. Grinenko, “Birational automorphisms of a three-dimensional double cone, ” Mat. Sb. 189 (7), 37–52 (1998) [Sb. Math. 189, 991–1007 (1998)].MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    B. Hassett and Yu. Tschinkel, “On stable rationality of Fano threefolds and del Pezzo fibrations,” arXiv: 1601.07074 [math.AG].Google Scholar
  17. 17.
    A. Iliev, L. Katzakov, and V. Przyjalkowski, “Double solids, categories and non-rationality, ” Proc. Edinburgh Math. Soc., Ser. 2, 57 (1), 145–173 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    V. A. Iskovskikh and Yu. G. Prokhorov, FanoVarieties (Springer, Berlin, 1999), Encycl. Math. Sci. 47, Algebraic Geometry V.Google Scholar
  19. 19.
    V. A. Iskovskikh and A. V. Pukhlikov, “Birational automorphisms of multidimensional algebraic manifolds, ” in Algebraic Geometry–1 (VINITI, Moscow, 2001), Itogi Nauki Tekh., Ser.: Sovrem. Mat. Prilozh., Temat. Obzory 19, pp. 5–139 [J. Math. Sci. 82 (4), 3528–3613 (1996)].Google Scholar
  20. 20.
    F. Klein, “Ueber die Transformation siebenter Ordnung der elliptischen Functionen, ” Math. Ann. 14, 428–471 (1879).MathSciNetCrossRefGoogle Scholar
  21. 21.
    A. G. Kuznetsov, “On Küchle varieties with Picard number greater than 1, ” Izv. Ross. Akad. Nauk, Ser. Mat. 79 (4), 57–70 (2015) [Izv. Math. 79, 698–709 (2015)].MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    K. Pettersen, “On nodal determinantal quartic hypersurfaces in P4, ” PhD Thesis (Univ. Oslo, Oslo, 1998).Google Scholar
  23. 23.
    Yu. Prokhorov, “Simple finite subgroups of the Cremona group of rank 3, ” J. Algebr. Geom. 21 (3), 563–600 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Yu. G. Prokhorov, “On G-Fano threefolds, ” Izv. Ross. Akad. Nauk, Ser. Mat. 79 (4), 159–174 (2015) [Izv. Math. 79, 795–808 (2015)].MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Yu. G. Prokhorov, “Singular Fano threefolds of genus 12, ” Mat. Sb. 207 (7), 101–130 (2016) [Sb. Math. 207, 983–1009 (2016)].MathSciNetCrossRefGoogle Scholar
  26. 26.
    Yu. G. Prokhorov, “Q-Fano threefolds of index 7, ” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 294, 152–166 (2016) [Proc. Steklov Inst. Math. 294, 139–153 (2016)].Google Scholar
  27. 27.
    A. V. Pukhlikov, “Birational automorphisms of a double space and double quadric, ” Izv. Akad. Nauk SSSR, Ser. Mat. 52 (1), 229–239 (1988) [Math. USSR, Izv. 32 (1), 233–243 (1989)].Google Scholar
  28. 28.
    S. Schreieder and L. Tasin, “A very general quartic or quintic fivefold is not stably rational,” arXiv: 1510.02011v2 [math.AG].Google Scholar
  29. 29.
    C. A. Shramov, “Birational rigidity and Q-factoriality of a singular double cover of a quadric branched over a divisor of degree 4, ” Mat. Zametki 84 (2), 300–311 (2008) [Math. Notes 84, 280–289 (2008)].MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    J. A. Todd, “Configurations defined by six lines in space of three dimensions, ” Proc. Cambridge Philos. Soc. 29, 52–68 (1933).CrossRefzbMATHGoogle Scholar
  31. 31.
    J. A. Todd, “A note on two special primals in four dimensions, ” Q. J. Math. 6, 129–136 (1935).CrossRefzbMATHGoogle Scholar
  32. 32.
    J. A. Todd, “On a quartic primal with forty-five nodes, in space of four dimensions, ” Q. J. Math. 7, 168–174 (1936).CrossRefzbMATHGoogle Scholar
  33. 33.
    C. Voisin, HodgeTheory and Complex Algebraic Geometry. I (Cambridge Univ. Press, Cambridge, 2007), Cambridge Stud. Adv. Math. 76.Google Scholar
  34. 34.
    C. Voisin, “Unirational threefolds with no universal codimension 2 cycle, ” Invent. Math. 201 (1), 207–237 (2015).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  • Victor V. Przyjalkowski
    • 1
  • Constantin A. Shramov
    • 1
  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

Personalised recommendations