# Bertini theorem - composite with a pencil

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?

• I am not sure I understand. If you take $X=\mathbb{P}^1\times C$ where $C$ is a smooth projective curve and $L$ is the pull back of $O(1)$ from $\mathbb{P}^1$, then $H^0(X,L)$ has dimension 2, it is base point free and all members are irreducible. Aug 26, 2019 at 22:24
• @Mohan, I am sorry. My question is wrong . I will rewrite it. Aug 27, 2019 at 6:09

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.