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Let $X$ be a normal proper variety and $D$ an $\mathbb{R}$-Cartier divisor on $X$. Then $D$ is called a semi-ample $\mathbb{R}$-divisor if there is a morphism $f:X\rightarrow Y$ and a ample $\mathbb{R}$-divisor $A$ on $Y$ such that $D\sim_{\mathbb{R}} f^*A$. I have the following two questions:

  1. Does (some multiple of)$D$ induces a morphism to some projective space?

  2. Can we always require that the morphism $f$ in the definition is a contraction morphism(i.e., $f_*\mathcal{O}_X=\mathcal{O}_Y$)?

I know the answers of the above two questions are affirmative in the $\mathbb{Q}$-cases.

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It is possible that for an ample $\mathbb{R}$-divisor no multiple is integral (hence no multiple defines a morphism to a projective space). For instance, take $X = \mathbb{P}^1 \times \mathbb{P}^1$ and $D = H_1 + \sqrt{2} H_2$, where $H_1$ and $H_2$ the classes of the rulings.

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  • $\begingroup$ Thanks. I find an argument from Fujino's book, please look it over here $\endgroup$
    – Hobo
    Commented Feb 24 at 2:43

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