Let $L$ be a line bundle on a smooth connected complete complex algebraic surface $X$. Assume that $L$ has enough sections i.e. that $H^0(L,X)$ has dimension $> 1$. A nonzero section $s$ of $L$ will cut out some divisor $C\_s \subset X$.


Need it be the case that, for generic $s$, the curve $C\_s$ is irreducible?


The example I have in mind is that of an elliptic fibration on a K3 surface $X$. Here, we take $L$ to be a line bundle whose $H^0$ induces a flat morphism $\pi: X \longrightarrow \mathbf{P}^1$ where a general fiber is a smooth connected curve of genus 1.


For such an $L$, we can consider its tensor square $L' = L \otimes L$. There are obvious sections $s \otimes s' \in H^0(L',X)$ coming from the product structure, whose divisors are disjoint unions of two fibers $C\_s$ and $C\_s'$ of $\pi$. But need $L'$ have other sections where the divisor is a single, connected elliptic curve?



Do line bundles with enough sections on surfaces have generic divisors which are irreducible?
Not in general. Take a high genus hyperellitpic curve $C$ with a degree 2 map $f:C\to\mathbb{P}^1$. Take your surface to be $C\times D$ for some smooth curve $D$ and let $L$ be the pull back of $\mathcal{O}\_{\mathbb{P}^1}(1)$ by the obvious map. Then $H^0(L)=2$ which is globally generated and every section vanishes along a reducible curve.


