Big arithmetic divisors on the projective spaces over Z

Abstract

In this paper, we observe several properties of an arithmetic divisor $\overline{D}$ on ${\mathbb{P}}_{\mathbb{Z}}^n$ and give the exact form of the Zariski decomposition of $\overline{D}$ on ${\mathbb{P}}_{\mathbb{Z}}^1$. Further, we show that, if $n\ge 2$ and $\overline{D}$ is big and non-nef, then for any birational morphism $f:X\to {\mathbb{P}}_{\mathbb{Z}}^n$ of projective, generically smooth, and normal arithmetic varieties, we cannot expect a suitable Zariski decomposition of $f^{\ast }\left(\overline{D}\right)$. We also give a concrete construction of Fujita’s approximation of $\overline{D}$.