Approximation at places of bad reduction for rationally connected varieties. (English) Zbl 1160.14040

In a previous paper, the authors proved that a rationally connected variety \(X\) defined over the function field of a projective complex curve \(B\) satisfies weak approximation at places of good reduction [Invent. Math. 163, No. 1, 171–190 (2006; Zbl 1095.14049)]. Weak approximation at places of good reduction can be formulated geometrically by letting \(\pi : {\mathcal X} \to B\) be a flat and proper model of \( X\), \(\{b_1,\ldots,b_m\}\subset B\) any finite set of points such that the corresponding fibers \(X_{b_i}=\pi^{-1}(b_i)\) are smooth, and for each \(b_i\) assign a jet \(j_i\) of a germ of a smooth curve transversal to the fiber \(X_{b_i}\) at \(x_i\) . The variety \(X\) is said to satisfy weak approximation at places of good reduction if there exists a section of \(\pi\) containing the prescribed jets. The existence of a section of \(\pi\) was proven by T. Graber, J. D. Harris and J. Starr [J. Am. Math. Soc. 16, No. 1, 57–67 (2003; Zbl 1092.14063)] and the existence of a section through prescribed points with smooth fibers was proven by J. Kollár, Y. Miyaoka and S. Mori [J. Algebr. Geom. 1, No. 3, 429–448 (1992; Zbl 0780.14026)].
In this paper the authors extend their previous result to include certain places of bad reduction. They no longer require the points \(\{b_1, \ldots, b_m\}\subset B\) to have smooth fibers \(X_{b_i}\). They prove that weak approximation holds as long as the smooth locus of the singular fibers of \(\pi\) are strongly rationally connected. The notion of strong rational connectivity is defined in this paper and various equivalent properties of strong rational connectivity are proven. One simple definition is that every point can be joined to a generic point by a rational curve. They conjecture that the smooth locus of a log del Pezzo surface is strongly rationally connected and prove the conjecture in the case of cubic surfaces with rational double points. As an application of their main theorem they prove weak approximation for cubic surfaces and Fano hypersurfaces of dimension at least three with square-free discriminant.


14M99 Special varieties
14G05 Rational points
14H05 Algebraic functions and function fields in algebraic geometry
11D25 Cubic and quartic Diophantine equations
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