Abstract
We prove that in a family of projective threefolds defined over an algebraically closed field, the locus of rational fibers is a countable union of closed subsets of the locus of separably rationally connected fibers. When the ground field has characteristic zero, this implies that the locus of rational fibers in a smooth family of projective threefolds is the union of at most countably many closed subfamilies.
Similar content being viewed by others
References
Abramovich, D., Karu, K., Matsuki, K., Włodarczyk, J.: Torification and factorization of birational maps. J. Am. Math. Soc. 15(3), 531–572 (2002)
Beauville, A., Donagi, R.: La variété des droites d’une hypersurface cubique de dimension 4. C. R. Acad. Sci. Paris Sér. I Math. 301(14), 703–706 (1985) (French, with English summary)
de Fernex, T.: Birationally rigid hypersurfaces. Invent. Math. (2012). doi:10.1007/s00222-012-0417-0
de Jong, A.J., Starr, J.: Every rationally connected variety over the function field of a curve has a rational point. Am. J. Math. 125(3), 567–580 (2003)
Fano, G.: Alcune questioni sulla forma cubica dello spazio a cinque dimensioni. Comment. Math. Helv. 16, 274–283 (1944) (Italian)
Graber, T.: Rational Curves and Rational Points. International Congress of Mathematicians, vol. II, pp. 603–611. Eur. Math. Soc., Zürich (2006)
Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, No. 52. Springer, New York (1977)
Hassett, B.: Some rational cubic fourfolds. J. Algebr. Geom. 8(1), 103–114 (1999)
Hassett, B.: Special cubic fourfolds. Compos. Math. 120(1), 1–23 (2000)
Kollár, J.: Rational Curves on Algebraic Varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics (Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics), vol. 32. Springer, Berlin (1996)
Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties. Cambridge Tracts in Mathematics, vol. 134. Cambridge University Press, Cambridge (1998). With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original
Kuznetsov, A.: Derived Categories of Cubic Fourfolds. Cohomological and Geometric Approaches to Rationality Problems. Progress in Mathematics, vol. 282, pp. 219–243. Birkhauser Boston Inc., Boston (2010)
Lipman, J.: Desingularization of two-dimensional schemes. Ann. Math. (2) 107(1), 151–207 (1978)
Morin, U.: Sulla razionalità dell’ipersuperficie cubica generale dello spazio lineare \(S_{5}\). Rend. Sem. Mat. Univ. Padova 11, 108–112 (1940) (Italian)
Timmerscheidt, K.: On deformations of three-dimensional rational manifolds. Math. Ann. 258(3), 267–275 (1981/1982)
Tregub, S.L.: Three constructions of rationality of a cubic fourfold. Vestnik Moskov. Univ. Ser. I Mat. Mekh. 3, 8–14 (1984) (Russian)
Voisin, C.: Théorème de Torelli pour les cubiques de \({\bf P}^{5}\). Invent. Math. 86(3), 577–601 (1986) (French)
Włodarczyk, J.: Toroidal varieties and the weak factorization theorem. Invent. Math. 154(2), 223–331 (2003)
Zarhin, Y.G.: Algebraic cycles over cubic fourfolds. Boll. Un. Mat. Ital. B (7) 4(4), 833–847 (1990) (English, with Italian summary)
Zhu, Y.: Fano hypersurfaces in positive characteristic. Preprint 2011. arXiv: 1111.2964
Acknowledgments
T. de Fernex would like to express his gratitude to Paolo Francia who first got him interested in Question 1.1; we dedicate this paper to his memory. We would like to thank Emanuele Macrì, Massimiliano Mella, and Alessandro Verra for valuable comments, and Claire Voisin for explaining to us the argument of the proof of Proposition 2.3 given below. We thank János Kollár from bringing the paper [15] to our attention, and the referees for useful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
T. de Fernex is partially supported by NSF CAREER Grant DMS-0847059.
Rights and permissions
About this article
Cite this article
de Fernex, T., Fusi, D. Rationality in families of threefolds. Rend. Circ. Mat. Palermo 62, 127–135 (2013). https://doi.org/10.1007/s12215-013-0110-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12215-013-0110-1