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:

  1. 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 $D$.
  2. All other fibres are integral.
  • $\begingroup$ Could please say what "integral fibre" means? Is this the same as to say that $\pi$ has non-vanishing differential at any point of the fibre (so that the fibre is a smooth curve in $S$)? $\endgroup$ – aglearner Oct 16 '17 at 17:59
  • $\begingroup$ @DanielLoughran As you probably already know, such a morphism $\pi$ is flat projective, and its generic fibre is a smooth projective geometrically connected curve of genus > 0 with no $\mathbb{C}(t)$-rational point. $\endgroup$ – Ariyan Javanpeykar Oct 16 '17 at 19:44
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    $\begingroup$ @aglearner: By "integral" I mean its standard usage in algebraic geometry: the scheme theoretic fibres $\pi^{-1}(x)$ are reduced ad irreducible. $\endgroup$ – Daniel Loughran Oct 17 '17 at 8:16
  • $\begingroup$ @Ariyan: Indeed! (For the interested reader: As $S$ is regular, any section must meet each fibre in a smooth point.Thus there can be no multiple fibre in this case. If the generic fibre has genus $0$, then there is always a section by Tsen's theorem). $\endgroup$ – Daniel Loughran Oct 17 '17 at 8:36

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.

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    $\begingroup$ I know this must be in the article, but I could not quickly find it: are Fujimoto's examples projective? Some logarithmic transforms do not preserve projectivity. $\endgroup$ – Jason Starr Oct 17 '17 at 5:58
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    $\begingroup$ @JasonStarr: In Step 3 of the Proof of Proposition 1.1 Fujimoto proves that his fibrations can be obtained as nine-points blowing-up of $\mathbb{P}^2$, so they are projective. . $\endgroup$ – Francesco Polizzi Oct 17 '17 at 6:07
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    $\begingroup$ After chasing references, it seems that the key word for the surfaces in this paper is "Halphen surface of index $m$". A useful reference for me was Section 2 of arxiv.org/pdf/1106.0930.pdf. $\endgroup$ – Daniel Loughran Oct 17 '17 at 11:10

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