Bertini theorem - composite with a pencil

Question

Let $X$ be a smooth projective surface over the field of complex numbers. Suppose $L$ is a base point free line bundle such that the dimension of $H^0(X,L)$ is two. The Bertini's theorem says that a general element is not necessarily irreducible. Can anyone give me some examples where the general element of the curve is reducible. I would like to know an example especially when $X$ is a K3 surface.

More generally if $|L|$ is composite with a pencil, is there a criteria as to when general element of the linear system reducible?

Answer 1

Bertini theorem says, in particular, that a general element of a pencil is smooth away from the base locus. Then a general element of a pencil without base points is reducible if it is not connected.

Choose elliptic curves $E_1$ and $E_2$ and let $S = E_1\times E_2$. We have that $E_1$ has a $2$-fold covering $\pi \colon E_1 \rightarrow \mathbb{P}^1$. Then consider the map $\sigma = \pi\circ pr_1 \colon S \rightarrow \mathbb{P}^1$ i.e. $\sigma(x,y) = \pi(x)$. Therefore you can choose $L = \sigma^\ast \mathcal{O}_{\mathbb{P}^1}(1)$. A general element of $|L|$ is given by two copies of $E_2$.

I would like to point out Dino Festi notes on elliptic $K3$ surfaces as I think the OP may be interested.