Weak approximation for Fano complete intersections in positive characteristic
| Autor: | Starr, Jason Michael, Tian, Zhiyu, Zong, Runhong |
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| Rok vydání: | 2018 |
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| Druh dokumentu: | Working Paper |
| Popis: | For a smooth curve $B$ over an algebraically closed field $k$, for every $B$-flat complete intersection $X_B$ in $B\times_{\text{Spec}\ k} \mathbb{P}^n_k$ of type $(d_1,\dots,d_c)$, if the Fano index is $\geq 2$ and if $\text{char}(k)>\max(d_1,\dots,d_c)$, we prove weak approximation of $\widehat{\mathcal{O}}_{B,b}$-points of $X_B$ by $k(B)$-points at all places of (strong) potentially good reduction, including all places of good reduction. The key step is the proof that such complete intersections are \emph{separably uniruled by lines}, and even \emph{separably rationally connected}, whenever smooth. We prove that the inequality is close to sharp. We prove a similar theorem for Fano manifolds of Picard number $1$ and Fano index $1$. Comment: 22 pages. Following discussions with Shizhang Li and Bhargav Bhatt, added hypothesis about torsion in crystalline cohomology for specializations of complex Fano manifolds of Picard number $1$ and Fano index $1$. Also proved a version eliminating this hypothesis and the hypothesis that the mixed characteristic DVR is unramified |
| Databáze: | arXiv |
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