论文标题
群集的家庭和对朗型猜想的申请
Clustered families and applications to Lang-type conjectures
论文作者
论文摘要
我们在Grassmannian $ \ Mathbb {g}（k-1，n）$中介绍并分类了1个线性空间的群集家庭，并向lang-type猜想提供了申请。令$ x \ subset \ mathbb {p}^n $为$ d $的非常通用的高度表面。令$ z_l $为$ x $的行中包含的点的轨迹。让$ z_2 $是$ x $上的积分源，这些线被$ x $以$ 2 $ $ 2 $的点扫除。我们证明1）如果$ d \ geq \ frac {3n+2} {2} $，则$ x $在$ z_l $之外为代数夸张。 2）如果$ d \ geq \ frac {3n} {2} $，$ x $包含行，但没有其他理性曲线3）如果$ d \ geq \ frac {3n+3} {2} $，那么$ x $上$ x $的唯一点上的点与本身比本身的$ x $相等，而不是z______2 $。 4）如果$ d \ geq \ frac {3n+2} {2} $和一个相对的绿色 - 格里奇斯 - 兰格构构保持，则$ x $的特殊位置包含在$ z_2 $中。
We introduce and classify 1-clustered families of linear spaces in the Grassmannian $\mathbb{G}(k-1,n)$ and give applications to Lang-type conjectures. Let $X \subset \mathbb{P}^n$ be a very general hypersurface of degree $d$. Let $Z_L$ be the locus of points contained in a line of $X$. Let $Z_2$ be the locus of points on $X$ that are swept out by lines that meet $X$ in at most $2$ points. We prove that 1) If $d \geq \frac{3n+2}{2}$, then $X$ is algebraically hyperbolic outside $Z_L$. 2) If $d \geq \frac{3n}{2}$, $X$ contains lines but no other rational curves 3) If $d \geq \frac{3n+3}{2}$, then the only points on $X$ that are rationally Chow zero equivalent to points other than themselves are contained in $Z_2$. 4) If $d \geq \frac{3n+2}{2}$ and a relative Green-Griffiths-Lang Conjecture holds, then the exceptional locus for $X$ is contained in $Z_2$.
