算术满度与一个算术Bertini定理
Arithmetic ampleness and an arithmetic Bertini theorem
摘要：
设$\mathcal X$是维数至少为$2$的射影算术簇.若$\overline{\mathcal L}$是$\mathcal X$上的充分厄米线丛，则证明了定义$\mathcal X$上不可约因子的$\overline{\mathcal L}^{\otimes n}$的有效截面的比例趋于$1$，正如$n$趋于$\infty$ .我们证明了这一说法的方案映射到这样的$\mathcal X$的变体。在得到这些结果的过程中，我们讨论了算术簇上的算术满度的一些一般性质，包括限制定理和埃尔米特线丛的有效截面数的上界。
英文摘要：
Let $\mathcal X$ be a projective arithmetic variety of dimension at least $2$. If $\overline{\mathcal L}$ is an ample hermitian line bundle on $\mathcal X$, we prove that the proportion of those effective sections of $\overline{\mathcal L}^{\otimes n}$ that define an irreducible divisor on $\mathcal X$ tends to $1$ as $n$ tends to $\infty$. We prove variants of this statement for schemes mapping to such an $\mathcal X$. On the way to these results, we discuss some general properties of arithmetic ampleness, including restriction theorems, and upper bounds for the number of effective sections of hermitian line bundles on arithmetic varieties.
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