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This question is based on Beauville's article in Szpiro's asterisque Seminaire sur les pinceaux de courbes de genre au moins deux from 1986.

We will work over the complex numbers $\mathbf{C}$.

Let $f:X\longrightarrow \mathbf{P}^1_{\mathbf{C}}$ be a semi-stable curve. This means that $f$ is a projective flat morphism of relative dimension 1 such that the complex algebraic surface $X$ is nonsingular, the fibres of $f$ are connected curves of positive genus which have only ordinary double points as singularities and there does not exist an exceptional curve on $X$ contained in a fibre of $f$.

Theorem. Assume that $f$ is non-trivial. Then $f$ has at least $4$ singular fibres.

Question. Does there exist a non-trivial semi-stable curve of genus $>1$ which has precisely 4 singular fibres?

Example. There exists a non-trivial semi-stable curve of genus $1$ which has precisely 4 fibres. Namely, the so-called Hesse cubic $C_t: (X^3+Y^3+Z^3)-3tXYZ =0$, where $t\in \mathbf{P}^1$.

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up vote 8 down vote accepted

Sheng-Li Tan proved that the answer is negative (see http://arxiv.org/pdf/alg-geom/9411002). This had been conjectured by Beauville.

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Thank you very much. –  Ari May 6 '11 at 13:36

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