Location of Repository

Let Y be a quartic hypersurface in P^4 with terminal singularities. The Grothendieck-Lefschetz theorem states that any Cartier divisor on Y is the restriction of a Cartier divisor on P^4 . However, no such result holds for the group of Weil divisors. More generally, let Y be a terminal Gorenstein Fano 3-fold with Picard rank 1. Denote by s(Y )=h_4 (Y )-h^2 (Y ) = h_4 (Y )-1 the defect of Y. A variety is Q-factorial when every Weil divisor is Q-Cartier. The defect of Y is non-zero precisely when the Fano 3-fold Y is not Q-factorial. Very little is known about the topology of non Q-factorial terminal Gorenstein Fano 3-folds. Q-factoriality 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 3-folds and on the defect of terminal Gorenstein Fano 3-folds that do not contain a plane. Then, I state a geometric motivation of Q-factoriality. More precisely, given a non Q-factorial quartic 3-fold Y , Y contains a special surface, that is a Weil non-Cartier 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

Topics:
Algebraic Geometry, Birational Geometry

Publisher: University of Cambridge

Year: 2007

OAI identifier:
oai:www.repository.cam.ac.uk:1810/214794

Provided by:
Apollo

Downloaded from
http://www.dspace.cam.ac.uk/handle/1810/214794

- (1989). 3-dimensional Fano varieties with canonical singularities.
- (1968). Functors of Artin rings.
- (1976). Mixed Hodge structure on the vanishing cohomology.
- (1981). Mixed Hodge structures associated with isolated singularities.
- (1994). Nonnormal del Pezzo surfaces.
- (2001). On birational morphisms between pencils of del Pezzo surfaces.
- (2002). On classi of Q-Fano 3-folds of Gorenstein index 2. I, II.
- (1974). Projective models of K
- (1971). Rigidity of quotient singularities.
- (1983). Semicontinuity of the spectrum and an upper bound for the number of singular points of the projective hypersurface.
- (1989). Some birational maps of Fano 3-folds.

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.