大域或有限域上曲线在函数域上的弱逼近
Weak Approximation over Function Fields of Curves over Large or Finite Fields
摘要：
设$K=k(C)$是域$k$上曲线的函数域，$X$是光滑的、射影的、可分有理连通的$K$ -簇，且具有$X(K)\neq\emptyset$ .在假定$X$是光滑投射模型$π: \mathcal{X}\to C$的条件下，证明了如下弱逼近结果：（1）若$k$是一个大域，则$X(K)$是Zerkki稠密的;（2）若$k$是一个有限域的无限代数扩张，则$X$在好约化的地方满足弱逼近;（3）若$k$是非阿基米德局部域，且$R$ -等价在纤维$\mathcal{X}_p$上的好约化点上是平凡的，则存在一个Zebraki稠密子集$W\subseteq C(k)$，使得$X$在$W\subseteq C(k)$中的某些地方满足弱逼近.作为这些方法的应用，我们还得到了有限域$k$上的下列结果：（4）若$|k|>10$，则对于光滑三次超曲面$X/K$，在任意给定的好约化点处的特化映射$X(K)\longrightarrow \prod_{p\in P}\mathcal{X}_p(κ(p))$是满射的;（5）若$\mathrm{char} k\neq 2, 3$和$|k|>47$，则$K$上的光滑三次曲面$X$在任意给定的好约化点处满足弱逼近.
英文摘要：
Let $K=k(C)$ be the function field of a curve over a field $k$ and let $X$ be a smooth, projective, separably rationally connected $K$-variety with $X(K)\neq\emptyset$. Under the assumption that $X$ admits a smooth projective model $π: \mathcal{X}\to C$, we prove the following weak approximation results: (1) if $k$ is a large field, then $X(K)$ is Zariski dense; (2) if $k$ is an infinite algebraic extension of a finite field, then $X$ satisfies weak approximation at places of good reduction; (3) if $k$ is a nonarchimedean local field and $R$-equivalence is trivial on one of the fibers $\mathcal{X}_p$ over points of good reduction, then there is a Zariski dense subset $W\subseteq C(k)$ such that $X$ satisfies weak approximation at places in $W$. As applications of the methods, we also obtain the following results over a finite field $k$: (4) if $|k|>10$, then for a smooth cubic hypersurface $X/K$, the specialization map $X(K)\longrightarrow \prod_{p\in P}\mathcal{X}_p(κ(p))$ at finitely many points of good reduction is surjective; (5) if $\mathrm{char} k\neq 2, 3$ and $|k|>47$, then a smooth cubic surface $X$ over $K$ satisfies weak approximation at any given place of good reduction.
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