Kyoto Journal of Mathematics

Big arithmetic divisors on the projective spaces over $\mathbb{Z}$

Atsushi Moriwaki

Abstract

In this paper, we observe several properties of an arithmetic divisor $\overline{D}$ on ${\mathbb{P}}^{n}_{\mathbb{Z}}$ and give the exact form of the Zariski decomposition of $\overline{D}$ on ${\mathbb {P}}^{1}_{\mathbb{Z}}$. Further, we show that, if $n\geq2$ and $\overline{D}$ is big and non-nef, then for any birational morphism $f:X\to{\mathbb {P}}^{n}_{\mathbb{Z}}$ of projective, generically smooth, and normal arithmetic varieties, we cannot expect a suitable Zariski decomposition of $f^{*}(\overline{D})$. We also give a concrete construction of Fujita’s approximation of $\overline{D}$.

Article information

Source
Kyoto J. Math., Volume 51, Number 3 (2011), 503-534.

Dates
First available in Project Euclid: 1 August 2011

https://projecteuclid.org/euclid.kjm/1312205238

Digital Object Identifier
doi:10.1215/21562261-1299882

Mathematical Reviews number (MathSciNet)
MR2823999

Zentralblatt MATH identifier
1228.14023

Citation

Moriwaki, Atsushi. Big arithmetic divisors on the projective spaces over $\mathbb{Z}$. Kyoto J. Math. 51 (2011), no. 3, 503--534. doi:10.1215/21562261-1299882. https://projecteuclid.org/euclid.kjm/1312205238

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