Here i assume to work on $\mathbb{C}$. Let $X$ be a projective K3 surface. On Reid's "Chapters on algebraic surfaces" there is a theorem (page 69) whose point (b) says: If $D>0$ is nef and $D^2=0$ then $D=aE$, where |E| is a free pencil. At first, he shows that $|D|$ is base point free (and it is ok).
Then he claims that "Every element of $|D|$ is made up of components of fibres of a morphism $f \colon X \to \mathbb{P}^1$":
(1) Does this mean that the irreducible components of an element of $|D|$ are made up of fibres of $f$?
Now He says that, since $D^2=0$ then the image of the morphism $\phi_D$ is a curve:
(2) Why is $\mathrm{Im}(\phi_D)$ a curve?
He goes on by saying that $\tilde{C}$ is isomorphic to $\mathbb{P}^1$, because $h^1(\mathcal{O}_X)=0$ (using the Stein factorization $X \to \tilde{C} \to C$).
(3) Why is $\tilde{C}$ a curve? Moreover, if $\tilde{C}$ is smooth, then $h^1(\mathcal{O}_X)=0$ implies that $\tilde{C}$ cannot have $1$-forms and so $\tilde{C} \cong \mathbb{P}^1$. But how do i know that with that factorization $\tilde{C}$ is smooth?
Now, the morphism $\tilde{C} \to \mathbb{P}^{h^0(D)-1}$ is an embedding, because it is defined by a complete linear: i guess the one defined by the pull-back of $\mathcal{O}_{{\mathbb{P}}^{h^0(D)-1}}(1)$.
(4) He says that $E$ is reduced and i don't undersand why.
(5) Eventually he concludes by saying that $D \equiv aE$ with $h^0(D)=a+1$, by iterating this argument to $D-E$, $D-2E$ ecc. Even this step is not clear to me.
It seems that we construct an element of the linear system $|D|$ step by step, but i cannot see well what's going on.
Thanks to everyone
1) is correct.
2) Since $D$ is base point free, it gives a morphism to an irreducible variety. Since $D>0$, Riemann-Roch says the $h^0(D)\geq 2$ and thus the image is not a point. If it is not a curve, then the dimension of the image $Y$ is at least 2 and then it must be 2. Also, for an ample divisor $L$ on $Y$, $\phi_D^*(L)=O_X(D)$. This says, $D^2=(\phi_D(L))^2=mL^2>0$ for some positive number $m$. So, $Y$ must be a curve, since $D^2=0$.
3) $\bar{C}$ is just the normalization of $C$ in $K(X)$, the function field of $X$ and thus a curve. $\phi_D$ factors through $\bar{C}$ since $X$ is normal. All normal curves are smooth.
4) If the degree of $\bar{C}\to C$ is $a>0$, and a general fiber of $X\to\bar{C}$ is $E$, it follows that $D=aE$, since $\bar{C}$ is just $\mathbb{P}^1$. Since $\bar{C}$ is the Stein factorization of $\phi_D$, $E$ is reduced irreducible.
5) From the above, you have $h^0(E)=2$, since $E$ is the pull back of a point in $\mathbb{P}^1$. And similarly, one gets $h^0(D)=h^0(aE)=a+1$, since it is the pull back of $a$ points in $\mathbb{P}^1$.