On the Bertini regularity theorem for arithmetic varieties

Abstract

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 $\sigma$ with ${\Vert \sigma \Vert }_{\infty }<1$ of ${\overline{\mathcal{L}}}^{\otimes d}$ whose divisor does not have a singular point on the fiber ${\mathcal{X}}_p$ over any prime $p<e^{\epsilon d}$ tends to ${\zeta }_{\mathcal{X}}(1+\dim \mathcal{X}{)}^{-1}$ as $d\to \infty$.

55 pages, comments welcome

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