Big arithmetic divisors on the projective spaces over Z

TL;DR

This paper studies properties and explicit Zariski decompositions of certain arithmetic divisors on projective spaces over Z, revealing limitations and providing constructions related to big and non-nef divisors.

Contribution

It provides explicit Zariski decompositions for divisors on P^1_Z and discusses the non-existence of such decompositions under certain conditions for higher dimensions.

Findings

Explicit Zariski decomposition on P^1_Z

Limitations of Zariski decompositions for big, non-nef divisors in higher dimensions

Construction of Fujita's approximation for divisors

Abstract

This paper is an enhancement of the previous note "Explicit computations of Zariski decompositions on P_Z^1". In this paper, we observe several properties of a certain kind of an arithmetic divisor D on the n-dimensional projective space over Z and give the exact form of the Zariski decomposition of D on the projective line over Z. Further, we show that, if n>=2 and D is big and non-nef, then, for any birational morphism f: X --> P^n_Z of projective, generically smooth and normal arithmetic varieties, we can not expect a suitable Zariski decomposition of f^*(D). We also give a concrete construction of Fujita's approximation of D.