On the Bertini regularity theorem for arithmetic varieties

TL;DR

This paper extends Bertini's theorem to arithmetic varieties, showing that most sections of high tensor powers of an ample line bundle have smooth divisors over almost all primes, with the proportion approaching a specific zeta value.

Contribution

It proves a Bertini regularity theorem for arithmetic varieties, quantifying the density of sections with smooth divisors over primes up to exponential bounds.

Findings

Proportion of sections with smooth divisors approaches a zeta value.

Most sections are regular over primes less than exponential in degree.

Asymptotic behavior of regular sections described as degree increases.

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 $\left\lVert \sigma \right\rVert_{\infty}<1$ 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 $\zeta_{\mathcal{X}}(1+\dim \mathcal{X})^{-1}$ as $d\rightarrow \infty$.