# A question on elliptic fibration of K3 surfaces.

Let $X$ be a K3 surface and $L$ be a non-trivial nef line bundle with self-intersection $(L,L)=0$. Then it is known that $h^0(X,L)=2$. How can one prove that the map $\phi_{|L|}\rightarrow \mathbb{P}^{h^0(X,L)-1}$ is a fibration over the image?

I know that if $\phi_{|L|}$ defines a fibration, then $(L,L)=0$ as $L$ represents a fiber class, but how can one see the converse? Are there any good way to understand this fact?

Although the question assumes that $X$ is a K$3$ surface, under what condition with $(L,L)=0$ does a similar clame hold for more general complex surface?

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People use the term "fibration" in many different ways. Does it just mean a proper, flat map here? –  Matt Nov 17 '12 at 18:45
Yes, is means a proper flat morphism. –  M. K. Nov 18 '12 at 2:29