Title:

The topology of terminal quartic 3folds

Let Y be a quartic hypersurface in P⁴ with terminal singularities. The GrothendieckLefschetz theorem states that any Cartier divisor on Y is the restriction of a Cartier divisor on P⁴. However, no such result holds for the group of Weil divisors. More generally, let Y be a terminal Gorenstein Fano 3fold with Picard rank 1. Denote by s(Y )=h_4 (Y )h² (Y ) = h_4 (Y )1 the defect of Y. A variety is Qfactorial when every Weil divisor is QCartier. The defect of Y is nonzero precisely when the Fano 3fold Y is not Qfactorial. Very little is known about the topology of non Qfactorial terminal Gorenstein Fano 3folds. Qfactoriality is a subtle topological property: it depends both on the analytic type and on the position of the singularities of Y . In this thesis, I endeavour to answer some basic questions related to this global topolgical property. First, I determine a bound on the defect of terminal quartic 3folds and on the defect of terminal Gorenstein Fano 3folds that do not contain a plane. Then, I state a geometric motivation of Qfactoriality. More precisely, given a non Qfactorial quartic 3fold Y , Y contains a special surface, that is a Weil nonCartier divisor on Y . I show that the degree of this special surface is bounded, and give a precise list of the possible surfaces. This question has traditionally been studied in the context of Mixed Hodge Theory. I have tackled it from the point of view of Mori theory. I use birational geometric methods to obtain these results.
