On factoriality of nodal threefolds
Author:
Ivan Cheltsov
Journal:
J. Algebraic Geom. 14 (2005), 663-690
DOI:
https://doi.org/10.1090/S1056-3911-05-00405-4
Published electronically:
May 11, 2005
MathSciNet review:
2147353
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Abstract: We prove the $\mathbb {Q}$-factoriality of a nodal hypersurface in $\mathbb {P}^{4}$ of degree $n$ with at most ${\frac {(n-1)^{2}}{4}}$ nodes and the $\mathbb {Q}$-factoriality of a double cover of $\mathbb {P}^{3}$ branched over a nodal surface of degree $2r$ with at most ${\frac {(2r-1)r}{3}}$ nodes.
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Am99 F. Ambro, Ladders on Fano varieties, J. Math. Sci. (New York) 94 (1999), 1126–1135.
Ba96 W. Barth, Two projective surfaces with many nodes, admitting the symmetries of the icosahedron, J. Algebraic Geometry 5 (1996), 173–186.
Bes83 E. Bese, On the spannedness and very ampleness of certain line bundles on the blow-ups of $\mathbb {P}_{{\mathbb C}}^2$ and $\mathbb {F}_{r}$, Math. Ann. 262 (1983), 225–238.
Ch03b I. Cheltsov, Non-rationality of a four-dimensional smooth complete intersection of a quadric and a quartic not containing a plane, Mat. Sbornik 194 (2003), 95–116.
Ch04a I. Cheltsov, Non-rational nodal quartic threefolds, arXiv:math.AG/0405150 (2004).
Ch04b I. Cheltsov, Double spaces with isolated singularities, arXiv:math.AG/0405194 (2004).
ChPa04 I. Cheltsov, J. Park, Sextic double solids, arXiv:math.AG/0404452 (2004).
CiGe03 C. Ciliberto, V. di Gennaro, Factoriality of certain hypersurfaces of ${\mathbb P}^4$ with ordinary double points, Encyclopaedia of Mathematical Sciences 132 Springer-Verlag, Berlin, (2004), 1–7.
Cl83 H. Clemens, Double solids, Adv. in Math. 47 (1983), 107–230.
Col79 A. Collino, A cheap proof of the irrationality of most cubic threefolds, Boll. Un. Mat. Ital. 16 (1979), 451–465.
Co95 A. Corti, Factorizing birational maps of threefolds after Sarkisov, J. Alg. Geom. 4 (1995), 223–254.
Co00 A. Corti, Singularities of linear systems and 3-fold birational geometry, L.M.S. Lecture Note Series 281 (2000), 259–312.
Cy01 S. Cynk, Defect of a nodal hypersurface, Manuscripta Math. 104 (2001), 325–331.
De80 M. Demazure, Surfaces de del Pezzo, Lecture Notes in Math. 777 Springer, Berlin–Heidelberg–New York (1989), 21–69.
Di90 A. Dimca, Betti numbers of hypersufaces and defects of linear systems, Duke Math. Jour. 60 (1990), 285–298.
FiWe89 H. Finkelnberg, J. Werner, Small resolutions of nodal cubic threefolds, Nederl. Akad. Wetensch. Indag. Math. 51 (1989), 185–198.
GeMa84 A. V. Geramita, P. Maroscia, The ideal of forms vanishing at a finite set of points in $\mathbb {P}^{n}$, J. Algebra 90 (1984), 528–555.
Har80 R. Hartshorne, Stable reflexive sheaves, Math. Ann. 254 (1980), 121–176.
JaRu97 D. Jaffe, D. Ruberman, A sextic surface cannot have $66$ nodes, J. Alg. Geometry 6 (1997), 151–168.
IsMa71 V. Iskovskikh, Yu. Manin, Three-dimensional quartics and counterexamples to the Lüroth problem, Mat. Sbornik 86 (1971), 140–166.
JSV90 A. J. de Jong, N. Shepherd-Barron, A. V. de Ven, On the Burkhardt quartic, Math. Ann. 286 (1990), 309–328.
KMM Y. Kawamata, K. Matsuda, K. Matsuki, Adv. Stud. Pure Math. 10 (1987), 283–360.
Ko91 J. Kollár et al., Flips and abundance for algebraic threefolds Astérisque 211 (1992).
Me03 M. Mella, Birational geometry of quartic 3-folds II: the importance of being ${\mathbb Q}$-factorial, Math. Ann. 330 (2004), 107–126.
Mi68 J. Milnor, Singular points of complex hypersurfaces, (Princeton Univ. Press, New Jersey, 1968).
Pu00 A. Pukhlikov, Essentials of the method of maximal singularities, L.M.S. Lecture Note Series 281 (2000), 73–100.
Sh92 V. V. Shokurov, Three-dimensional log perestroikas, Izv. Ross. Akad. Nauk 56 (1992), 105–203.
vSt93 D. van Straten, A quintic hypersurface in $\mathbb {P}^{4}$ with $130$ nodes, Topology 32 (1993), 857–864.
Va83 A. Varchenko, On semicontinuity of the spectrum and an upper bound for the number of singular points of projective hypersurfaces, Dokl. Aka. Nauk USSR 270 (1983), 1294–1297.
Wa98 J. Wahl, Nodes on sextic hypersurfaces in $\mathbb {P}^3$, J. Diff. Geom. 48 (1998), 439–444.
We87 J. Werner, Kleine Auflösungen spezieller dreidimensionaler Varietäten, Bonner Mathematische Schriften 186 (1987) Universität Bonn, Mathematisches Institut, Bonn.
Za52 O. Zariski, Complete linear systems on normal varieties and a generalization of a lemma of Enriques–Severi, Ann. of Math. 55 (1952), 552–552.
Additional Information
Ivan Cheltsov
Affiliation:
Steklov Institute of Mathematics, 8 Gubkin street, Moscow 117966, Russia
Address at time of publication:
School of Mathematics, The University of Edinburgh, Kings Buildings, Mayfield Road, Edinburgh EH9 3JZ, UK
MR Author ID:
607648
Email:
cheltsov@yahoo.com, I.Cheltsov@ed.ac.uk
Received by editor(s):
June 17, 2004
Received by editor(s) in revised form:
October 6, 2004, October 18, 2004, and November 21, 2004
Published electronically:
May 11, 2005
Additional Notes:
The author is very grateful to A. Corti, M. Grinenko, V. Iskovskikh, S. Kudryavtsev, V. Kulikov, M. Mella, J. Park, Yu. Prokhorov, A. Pukhlikov, V. Shokurov, L. Wotzlaw for fruitful conversations. Special thanks goes to V. Kulikov for Lemma 38. The author would also like to thank the referee for useful comments.
