论文标题
在Bombieri-lang上，猜想有限生成的领域
On the Bombieri-Lang Conjecture over finitely generated fields
论文作者
论文摘要
强大的Bombieri-lang猜想假设，对于$ \ m m mathbb {q} $有限生成的每个品种$ x $的通用类型，存在一个开放子集$ u \ subset x $，因此$ u（k）$都是有限生成的扩展$ k/k/k $的有限的。 Bombieri-lang的弱猜想假设，对于$ \ m athbb {q} $，在field $ k $上的每个正数$ x $的通用类型中，理性点$ x（k）$都不密度。此外，Lang猜想的是，在特征$ 0 $的字段上，每种类型的$ x $都包含一个开放子集$ u \ subset x $，因此$ u $的每个子变量都是通用类型的，此语句通常称为“几何” lang costixture。 我们将强大的Bombieri-lang猜想减少到案例$ k = \ mathbb {q} $。假设Lang的猜想假设，我们也将弱的Bombieri-Lang猜想减少到$ K = \ Mathbb {q} $。
The strong Bombieri-Lang conjecture postulates that, for every variety $X$ of general type over a field $k$ finitely generated over $\mathbb{Q}$, there exists an open subset $U\subset X$ such that $U(K)$ is finite for every finitely generated extension $K/k$. The weak Bombieri-Lang conjecture postulates that, for every positive dimensional variety $X$ of general type over a field $k$ finitely generated over $\mathbb{Q}$, the rational points $X(k)$ are not dense. Furthermore, Lang conjectured that every variety of general type $X$ over a field of characteristic $0$ contains an open subset $U\subset X$ such that every subvariety of $U$ is of general type, this statement is usually called geometric Lang conjecture. We reduce the strong Bombieri-Lang conjecture to the case $k=\mathbb{Q}$. Assuming the geometric Lang conjecture, we reduce the weak Bombieri-Lang conjecture to $k=\mathbb{Q}$, too.
