About the semi-ampleness for $\mathbb{R}$-divisor

Question

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.

Answer 1

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.