Weak Approximation for Fano Complete Intersections in Positive Characteristic
Weak Approximation for Fano Complete Intersections in Positive Characteristic
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DOI：
10.5802/aif.3495
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发表时间：
2018-11
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影响因子：
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通讯作者：
J. Starr;Zhiyu Tian;Runhong Zong
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文献类型：
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作者：
J. Starr;Zhiyu Tian;Runhong Zong
对于代数闭域$k$上的光滑曲线$B$，对$B中的每一个$B$平坦完全交$X_B$\times_{\text{Spec}\ k} \mathbb{P}^n_k$，（d_1，\dots，d_c）$，如果Fano索引为$\geq 2$并且如果$\text{char}（k）>\max（d_1，\dots，d_c）$，我们证明了$X_B$的$\widehat{\mathcal{O}}_{B，B}$-点在（强）潜在好约化的所有地方（包括所有好约化的地方）被$k（B）$-点的弱逼近.关键的一步是证明这样的完全相交是\n {可分离地被直线无圆化}，甚至是\n {可分离地有理连通}，只要是光滑的。我们证明了这个不等式接近于sharp。我们证明了一个类似的定理的Fano流形的Picard数$1$和Fano指数$1$。
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 $1$.