Weak approximation for cubic hypersurfaces. (English) Zbl 1400.14133

Tian, Zhiyu

Duke Math. J. 164, No. 7, 1401-1435 (2015).

Let \(B\) be a smooth curve defined over an algebraically closed field \(k\) of characteristic zero, with function field \(F=k(B)\), and let \(X\) be a smooth projective variety defined over \(F\). A model of \(X/F\) is a flat surjective proper morphism \(\pi:\mathcal X\rightarrow B\) whose generic fiber is isomorphic to \(X\). It is of interest the study of the points of \(X\) rational over \(F\) (equivalently sections of \(\pi\)) when \(X\) is smooth proper rationally connected. In [T. Graber et al., J. Am. Math. Soc. 16, No. 1, 57–67 (2003; Zbl 1092.14063)] the authors proved that if \(X\) over \(F\) is rationally connected then \(X\) has a rational point. The result, in view of Theorem 2.13 in [J. Kollár et al., J. Algebraic Geom. 1, No. 3, 429–448 (1992; Zbl 0780.14026)], implies the stronger property that, for any model \(\pi: \mathcal X\rightarrow B\) of \(X/F\), given a finite set of places \(b_i \in B\), such that the fiber \(\mathcal X_{b_i}\) is smooth, and points \(x_i\in\mathcal X_{b_i}\) there exists a section \(s: B\rightarrow\mathcal X\) passing through the points \(x_i\). It is natural to investigate whether \(X\) satisfies weak aproximation. The geometric meaning of weak approximation is the existence of sections with prescribed jet data in a finite number of fibers. B. Hassett and Y. Tschinkel [Invent. Math. 163, No. 1, 171–190 (2006; Zbl 1095.14049)] proved that if \(X/F\) is smooth rationally connected the weak approximation holds at places of good reduction for \(X\) (equivalently smooth fibers in a good model of \(X\)). They also conjectured that any smooth rationally connected variety \(X\) over \(F\) satisfies weak approximation at all places. The conjecture is known in few spacial cases. In the paper under review, it is shown that if \(X/F\) is a smooth cubic hypersurface of dimension at least 2 then weak approximation holds at all places, confirming the conjecture for this well studied class of rationally connected varieties. The approach used by the author is to reduce first to the case of cubic surfaces, and then use composition law and deformation techniques.

Reviewer: Davide Fusi (Bluffton)

MSC:

14M22	Rationally connected varieties
14G27	Other nonalgebraically closed ground fields in algebraic geometry
14G05	Rational points

Keywords:

weak approximation; rationally connected varieties; cubic hypersurfaces

Citations:

Zbl 1092.14063; Zbl 0780.14026; Zbl 1095.14049

References:

[1]	J. W. Bruce and C. T. C. Wall, On the classification of cubic surfaces , J. Lond. Math. Soc. 19 (1979), 245-256. · Zbl 0393.14007 · doi:10.1112/jlms/s2-19.2.245
[2]	J. -L. Colliot-Thélène and P. Gille, “Remarques sur l’approximation faible sur un corps de fonctions d’une variable” in Arithmetic of Higher-Dimensional Algebraic Varieties (Palo Alto, Calif., 2002) , Progr. Math. 226 , Birkhäuser, Boston, 2004, 121-134.
[3]	A. Corti, Del Pezzo surfaces over Dedekind schemes , Ann. of Math. (2) 144 (1996), 641-683. · Zbl 0902.14026 · doi:10.2307/2118567
[4]	A. J. de Jong, X. He, and J. M. Starr, Families of rationally simply connected varieties over surfaces and torsors for semisimple groups , Publ. Math. Inst. Hautes Études Sci. 114 (2011), 1-85. · Zbl 1285.14053 · doi:10.1007/s10240-011-0035-1
[5]	T. Graber, J. Harris, and J. Starr, Families of rationally connected varieties , J. Amer. Math. Soc. 16 (2003), 57-67. · Zbl 1092.14063 · doi:10.1090/S0894-0347-02-00402-2
[6]	B. Hassett, “Weak approximation and rationally connected varieties over function fields of curves” in Variétés Rationnellement Connexes: Aspects Géométriques et Arithmétiques , Panor. Synthèses 31 , Soc. Math. France, Paris, 2010, 115-153.
[7]	B. Hassett and Y. Tschinkel, Weak approximation over function fields , Invent. Math. 163 (2006), 171-190. · Zbl 1095.14049 · doi:10.1007/s00222-005-0458-8
[8]	B. Hassett and Y. Tschinkel, Approximation at places of bad reduction for rationally connected varieties , Pure Appl. Math. Q. 4 (2008), 743-766. · Zbl 1160.14040 · doi:10.4310/PAMQ.2008.v4.n3.a6
[9]	B. Hassett and Y. Tschinkel, “Weak approximation for hypersurfaces of low degree” in Algebraic Geometry-Seattle, 2005: Part 2 , Proc. Sympos. Pure Math. 80 , Amer. Math. Soc., Providence, 2009, 937-955. · Zbl 1169.14306
[10]	S. Keel and J. McKernan, Rational curves on quasi-projective surfaces , Mem. Amer. Math. Soc. 140 , 1999. · Zbl 0955.14031 · doi:10.1090/memo/0669
[11]	A. Knecht, Weak approximation for general degree two del Pezzo surfaces , Proc. Amer. Math. Soc. 141 (2013), 801-811. · Zbl 1260.14062 · doi:10.1090/S0002-9939-2012-11376-0
[12]	J. Kollár, Rational Curves on Algebraic Varieties , Ergeb. Math. Grenzgeb. (3) 32 , Springer, Berlin, 1996.
[13]	J. Kollár, Y. Miyaoka, and S. Mori, Rationally connected varieties , J. Algebraic Geom. 1 (1992), 429-448. · Zbl 0780.14026
[14]	D. A. Madore, Approximation faible aux places de bonne réduction sur les surfaces cubiques sur les corps de fonctions , Bull. Soc. Math. France 134 (2006), 475-485.
[15]	P. Swinnerton-Dyer, “Weak approximation and \(R\)-equivalence on cubic surfaces”in Rational Points on Algebraic Varieties , Progr. Math. 199 , Birkhäuser, Basel, 2001, 357-404. · Zbl 1079.11035 · doi:10.1007/978-3-0348-8368-9_14
[16]	Z. Tian and R. Zong, Weak approximation for isotrivial families , preprint, [math.AG]. arXiv:1003.3502v4
[17]	C. A. Weibel, An Introduction to Homological Algebra , Cambridge Stud. Adv. Math. 38 , Cambridge Univ. Press, Cambridge, 1994.
[18]	C. Xu, Strong rational connectedness of surfaces , J. Reine Angew. Math. 665 (2012), 189-205. · Zbl 1246.14064 · doi:10.1515/CRELLE.2011.108
[19]	C. Xu, Weak approximation for low degree del Pezzo surfaces , J. Algebraic Geom. 21 (2012), 753-767. · Zbl 1314.14067 · doi:10.1090/S1056-3911-2012-00590-6