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**Every rationally connected variety over the function field of a curve has a rational point.**
*(English)*
Zbl 1063.14025

This is the main theorem of this paper: if \(f: P\to X\) is a flat morphism of algebraic varieties over an algebraically closed field \(k\) of arbitrary characteristic, with \(X\) a non-singular projective curve and the geometric generic fiber of \(f\) is a normal, separably rationally connected variety, then \(f\) has a section. Here, a normal variety is said to be separably rationally connected if there is a morphism \(C\to V\), with \(C\) a smooth rational curve, whose image does not contain any singular point of \(V\) and such that the pull-back via \(f\) of the tangent bundle of \(V\) is ample on \(C\). This result easily implies the statement of the title of the present article. The main theorem is closely related to a question posed by Kollar, Miyaoka and Mori, who asked whether the analagous statement, but using a different notion of rationally connected variety, is true. Both notions agree if \(V\) admits a resolution of singularities. When \(k\) is the field of complex numbers, the main theorem was proved by using a not purely algebraic argument in [T. Graber, J. Harris and J. Starr, J. Am. Math. Soc. 16, No. 1, 57–67 (2003; Zbl 1092.14063)].

The proof given in the present paper, partially based on that of the mentioned article, is purely algebraic and rather convoluted. By means of a normalized base change via a suitable finite, generally étale morphism \(Y\to X\) there is a reduction to the case where all the fibers of \(f: P\to X\) are reduced. In this situation, a key ingredient in the proof is a theorem saying that given a finite morphism \(Y\to X\) of irreducible smooth projective curves over \(k\) there is a family \(\{Y_t\}_{t\in X}\) of curves such that (among other properties) for suitable points \(0\), \(\infty\) in \(X\) we have that \(Y_0\) is nodal, with \(Y\) as normalization, and \(Y_\infty\) has an irreducible component which is naturally isomorphic to \(X\). Techniques from the theory of Hilbert schemes also play an important role in the proof. These and other auxiliary results and methods discussed in the paper may be of independent interest.

The proof given in the present paper, partially based on that of the mentioned article, is purely algebraic and rather convoluted. By means of a normalized base change via a suitable finite, generally étale morphism \(Y\to X\) there is a reduction to the case where all the fibers of \(f: P\to X\) are reduced. In this situation, a key ingredient in the proof is a theorem saying that given a finite morphism \(Y\to X\) of irreducible smooth projective curves over \(k\) there is a family \(\{Y_t\}_{t\in X}\) of curves such that (among other properties) for suitable points \(0\), \(\infty\) in \(X\) we have that \(Y_0\) is nodal, with \(Y\) as normalization, and \(Y_\infty\) has an irreducible component which is naturally isomorphic to \(X\). Techniques from the theory of Hilbert schemes also play an important role in the proof. These and other auxiliary results and methods discussed in the paper may be of independent interest.

Reviewer: Augusto Nobile (Baton Rouge)

### MSC:

14G05 | Rational points |

14A15 | Schemes and morphisms |

14H10 | Families, moduli of curves (algebraic) |

14D06 | Fibrations, degenerations in algebraic geometry |