I'm reading the proof of Proposition IX.2 in Beauville's "Complex algebraic surfaces", and I do not understand a part of it:
Suppose we have a minimal surface $S$ of Kodaira dimension $\kappa(S)=1$. Then we have that $K_S^2=0$, so we can apply Lemma IX.1 and get that we can write $$rK_S\sim Z+M$$ where $Z$ is the fixed part of the system $|rK_S|$ and $M$ is the mobile part which satisfies $M^2=K_S\cdot M=0$. Then $|M|$ defines a morphism to $\mathbb{P}^N$ for some $N$ the image of which is a curve.
I do not understand how does it follow that the image of this morphism is a curve? Is this some quality of $|M|$ being a mobile part?
I will appreciate any help!