Big arithmetic divisors on the projective spaces over Z

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This paper is an enhancement of the previous note "Explicit computations of Zariski decompositions on PZ¹". 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 --> PⁿZ 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.