论文标题
关于算术品种的Bertini规律定理
On the Bertini regularity theorem for arithmetic varieties
论文作者
论文摘要
令$ \ Mathcal {x} $为常规投影算术品种，配备了充足的Hermitian Line Bundle $ \ overline {\ Mathcal {l}} $。我们证明，$ \ of $ \ of $ \ of $ \ rvert \ rvert \ rvert \ rvert _ {\ infty} <1 $ of $ \ overline {\ Mathcal {\ Mathcal {l}}^{\ \ outimimes d} $的比例$ \ lvert \ lvert \ lvert \ lvert \ lvert \ rvert \ rvert _ {\ infty} <1 $ $ p <e^{\ varepsilon d} $倾向于$ζ_ {\ mathcal {x}}}（1+ \ dim \ dim \ mathcal {x}）^{ - 1} $ as $ d \ rightArrow \ rightArrow \ infty $。
Let $\mathcal{X}$ be a regular projective arithmetic variety equipped with an ample hermitian line bundle $\overline{\mathcal{L}}$. We prove that the proportion of global sections $σ$ with $\left\lVert σ\right\rVert_{\infty}<1$ of $\overline{\mathcal{L}}^{\otimes d}$ whose divisor does not have a singular point on the fiber $\mathcal{X}_p$ over any prime $p<e^{\varepsilon d}$ tends to $ζ_{\mathcal{X}}(1+\dim \mathcal{X})^{-1}$ as $d\rightarrow \infty$.
