On the Slope and Geography of Fibred Surfaces and Threefolds.

Abstract

In this tesis we study numerical propieties of surfaces and threefolds, mainly fibred over curves, the so called "slope" of the fibration. We prove partially a conjecture of Fujita on the semiampleness of the direct image of the relative dualizing sheaf of a fibration.<br/>We give new lower bounds of the slope of a fibred surface depending on data of the general fibre (existence of involutions) and on data of the hole surface (the fibration not being the Albanese morphism, for example). <br/>We study the case of threefolds over curves. We prove that, in general, the relative algebraic Euler characteristic is nonnegative and give lower bound for the slope. We classify the lowest cases of the invariants.