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 $t\in \mathbf{P}^1$.



Does there exist a non-trivial semi-stable curve of genus >1 with only 4 singular fibres
Sheng-Li Tan proved that the answer is negative (see <http://arxiv.org/pdf/alg-geom/9411002>). This had been conjectured by Beauville.


