Weak approximation for isotrivial families (Q2001380)

Let \(B\) be a smooth, irreducible algebraic curve over the field of complex numbers. Let \(\pi: X \to B\) be a family of projective varieties. We say \(\pi\) satisfies weak approximation at points (or places) \(b_1,\ldots,b_m\) of \(B\) if for any positive integers \(N\), and any collection of sections \(\widehat{s}_i\) of \[ \pi|_{\mathrm{Spec}(\widehat{\mathcal{O}}_{B,b_i})} : X \times_B \mathrm{Spec}(\widehat{\mathcal{O}}_{B,b_i}) \to \mathrm{Spec}(\widehat{\mathcal{O}}_{B,b_i}), \] there is a section \(s\) of \(\pi\), such that \(s\) agrees with \(\widehat{s}_i\) after restricting to \(\mathrm{Spec}(\mathcal{O}_{B,b_i}/\mathfrak{m}_{b_i}^{N})\). \textit{B. Hassett} and \textit{Y. Tschinkel} [Invent. Math. 163, No. 1, 171--190 (2006; Zbl 1095.14049), Conjecture 2] conjectured that if \(\pi\) is a family of rationally connected varieties, then weak approximation holds at all places. They proved this conjecture in [loc.~cit.] assuming \(\pi^{-1}(b_i)\) smooth. \par We say \(\pi\) has ``strong potentially good reduction'' at a point \(b \in B\), if \begin{itemize} \item the fraction field \(K_b\) of \(\widehat{\mathcal{O}}_{B,b}\) admits a finite Galois extension \(K'_b\) such that \(X \times_B \mathrm{Spec}(K'_{b})\) extends to a smooth proper morphism \(X' \to \mathrm{Spec}(\mathcal{O}')\) where \(\mathcal{O}'\) is the ring of integers of \(K'_b\); and \item \(X'\) admits an action of \(\mathrm{Gal}(K'_b/K_b)\) such that the morphism \(X' \to \mathrm{Spec}(\mathcal{O}')\) is equivariant with respect to the action. \end{itemize} In the article under review, the authors prove that if \(X\) has strong potentially good reduction at \(b_i\), then weak approximation holds at these points. For example, in an isotrivial family, all places are of strong potentially good reduction. \par When \(N=1\), one needs to find a section passing through a prescribed point \(x\) on a fiber \(\pi^{-1}(b)\). A standard method for producing such a section is: \begin{itemize} \item[(1)] choose an arbitrary section \(\sigma\), \item[(2)] choose a ``good'' rational curve \(R\) joining \(\sigma(b)\) and \(x\), and \item[(3)] perturb the nodal curve \(\sigma(B) \cup R\) (with \(x\) fixed) to produce the desired section. \end{itemize} The subtlety is that the singularity of \(\pi^{-1}(b)\) may prevent one from applying deformation theory naively. Thus, the authors work with \(X'\) instead of \(X\). This bypasses the complication caused by the singularity. But the authors have to carry out the above procedure in an equivariant fashion. The first and the third steps are relatively straightforward. The equivariant version of Step 2 is nontrivial. It entails finding, on a smooth rationally connected variety with a cyclic group action, an equivariant rational curve passing through a pair of fix points of the group action. This fact is proved in Section 3 of the paper. Finally, the authors use induction and iterated blow-ups to deal with an arbitrary \(N\). Due to the presence of the group action, this last step too is more complicated than the situation considered by Hassett and Tschinkel.