MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
up vote 9 down vote accepted

Sheng-Li Tan proved that the answer is negative (see This had been conjectured by Beauville.

share|cite|improve this answer
Thank you very much. – Ariyan Javanpeykar May 6 '11 at 13:36

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.