• complex tori
• fibered surfaces
• fibrations
• semi-stable curves
• Siegel upper half space

This thesis is dedicated to the article of Beauville (Le nombre minimum de fibres singulières d’une courbe stable sur P^1, Astérisque, (86):97–108, 1981) which tries to answer a classical problem first studied by Szpiro: given a fibered algebraic surfaces over the projective line which is the minimum number of singular fibres that such a fibration must have? In the first chapter we give some preliminary results about algebraic varieties, line bundles, divisors.

In the second chapter the theory of complex algebraic tori and abelian varieties is developed. In particular it is proven that the Siegel upper half space, which parametrizes the polarized abelian varieties with a given symplectic basis, is biholomorphic to a bounded domain (first proved by Siegel). This statement is fundamental in the proof of the final theorem of chapter 4.

In the third chapter, some other tools concerning complex tori are explained. In particular we define the Albanese and the Picard torus on a variety X, and we give conditions for them to be abelian varieties. At the end of the chapter there is a sketch of the proof of Torelli’s Theorem.

In the fourth chapter we give some properties concerning algebraic surfaces and curves on it. At the end of the chapter the proof of the main theorem concerning fibered surfaces is given: excluding the isotrivial case, every fibered surface has at least 3 singular fibres.

In the fifth chapter we recall the theory of abelian covers. This is very useful in the construction of examples of fibered surfaces. In particular at the end of the chapter an example of fibered surface with exactly 3 singular fibres, for every fixed genus g of the general fibre, is given.

In the last chapter we study the case of semi-stable fibrations, i.e. fibrations whose fibres have only nodal singularities. In this case the result of chapter 4 can be improved, showing that such a fibration has at least 4 singular fibres (proved by Beauville in his article). At the end of the chapter some examples of semi-stable fibrations are given, in particular a semi-stable fibration with general fibre of genus 1 with exactly 4 singular fibres.