Arithmetic ampleness and an arithmetic Bertini theorem

Let $\mathcal{X}$ be a projective arithmetic variety of dimension at least $2$. If $\bar{\mathcal{L}}$ is an ample hermitian line bundle on $\mathcal X$,weprovethattheproportionofthoseeffectivesectionsof$\overline{\mathcal L}^{\otimes n}$thatdefineanirreducibledivisoron$\mathcal X$tendsto$1$as$n$tendsto$\infty$. 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.