Browse > Article
http://dx.doi.org/10.4134/BKMS.b160682

A SIMPLE PROOF OF THE NON-RATIONALITY OF A GENERAL QUARTIC DOUBLE SOLID  

Prokhorov, Yuri (Steklov Mathematical Institute)
Publication Information
Bulletin of the Korean Mathematical Society / v.54, no.5, 2017 , pp. 1619-1625 More about this Journal
Abstract
The aim of this short note is to give a simple proof of the non-rationality of the double cover of the three-dimensional projective space branched over a sufficiently general quartic.
Keywords
double solid; intermediate Jacobian; Torelli theorem;
Citations & Related Records
연도 인용수 순위
  • Reference
1 M. Artin and D. Mumfordm Some elementary examples of unirational varieties which are not rational, Proc. London Math. Soc. (3) 25 (1972), 75-95.
2 A. Beauville, Varietes de Prym et jacobiennes intermediaires, Ann. Sci. Ecole Norm. Sup. (4) 10 (1977), no. 3, 309-391.   DOI
3 A. Beauville, Non-rationality of the symmetric sextic Fano threefold, In Geometry and arithmetic, EMS Ser. Congr. Rep., pages 57-60, Eur. Math. Soc., Zurich, 2012.
4 A. Beauville, Non-rationality of the S6-symmetric quartic threefolds, Rend. Semin. Mat. Univ. Politec. Torino 71 (2013), no. 3-4, 385-388.
5 I. Cheltsov, V. Przyjalkowski, and C. Shramov, Quartic double solids with icosahedral symmetry, Eur. J. Math. 2 (2016), no. 1, 96-119.   DOI
6 I. Cheltsov, V. Przyjalkowski, and C. Shramov, Which quartic double solids are rational?, arXiv preprint, 1508.07277(2015), to appear in J. Algebraic Geom.
7 C. H. Clemens, Double solids, Adv. in Math. 47 (1983), no. 2, 107-230.   DOI
8 C. H. Clemens, The quartic double solid revisited, In Complex geometry and Lie theory (Sundance, UT, 1989), volume 53 of Proc. Sympos. Pure Math., pages 89-101. Amer. Math. Soc., Providence, RI, 1991.
9 C. H. Clemens and P. A. Griths, The intermediate Jacobian of the cubic threefold, Ann. of Math. (2) 95 (1972), 281-356.   DOI
10 O. Debarre, Sur le theoreme de Torelli pour les solides doubles quartiques, Compos. Math. 73 (1990), no. 2, 161-187.
11 V. Przyjalkowski and C. Shramov, Double quadrics with large automorphism groups, Proc. Steklov Inst. Math. 294 (2016), 154-175.   DOI
12 A. S. Tikhomirov, The Abel-Jacobi mapping of sextics of genus three onto double $P^3 of index two, Soviet Math. Dokl. 33 (1986), no. 1, 204-206.
13 R. Varley, Weddle's surfaces, Humbert's curves, and a certain 4-dimensional abelian variety, Amer. J. Math. 108 (1986), no. 4, 931-952.   DOI
14 C. Voisin, Sur la jacobienne intermediaire du double solide d'indice deux, Duke Math. J. 57 (1988), no. 2, 629-646.   DOI
15 Y. Zarhin, Cubic surfaces and cubic threefolds, Jacobians and intermediate Jacobians, In Algebra, arithmetic, and geometry. In honor of Y. I. Manin on the occasion of his 70th birthday. Vol. II, pages 687-691. Boston, MA: Birkhauser, 2009.
16 C. Voisin, Unirational threefolds with no universal codimension 2 cycle, Invent. Math. 201 (2015), no. 1, 207-237.   DOI