# Does there exist a non-trivial semi-stable curve of genus >1 with only 4 singular fibres

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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