TITLE: A pencil with exactly one multiple fibre
QUESTION [10 upvotes]: Does there exist a smooth projective surface $S / \mathbb{C}$ equipped with a dominant morphism $\pi: S \to \mathbb{P}^1$ which has the following properties:

The fibre at infinity is "multiple", i.e. as divisors on $S$ one has $\pi^{-1}(\infty) = mD$ for some $m > 1$ and some divisor $D$.
All other fibres are integral.

REPLY [5 votes]: The answer is yes. 
In fact, it is possible to construct a rational elliptic fibration $f \colon S \to \mathbb{P}^1$ with exactly one multiple fibre of multiplicity $m \geq 2$, by starting from the blow-up of $\mathbb{P}^2$ at nine points that are the base locus of a pencil $\mathscr{P}$ of elliptic curves and then performing a logarithmic transformation centered at one point of $\mathbb{P}^1$.
Since the logarithmic transformation does not change the fibres outside the center, choosing a sufficiently general pencil $\mathscr{P}$ the reduced fibres of $f$ will be all irreducible.
For more details and examples, see 
Y. Fujimoto, On rational elliptic surfaces with multiple fibers, Publ. Res. Inst. Math. Sci. 26, No.1, 1-13 (1990). ZBL0729.14027,
in particular Proposition 1.1.