## The topology of terminal quartic 3-folds

##### Authors

Kaloghiros, Anne-Sophie

##### Date

2007-06-20##### Awarding Institution

University of Cambridge

##### Qualification

Doctor of Philosophy (PhD)

##### Language

English

##### Type

Thesis

##### Metadata

Show full item record##### Citation

Kaloghiros, A. (2007). The topology of terminal quartic 3-folds (Doctoral thesis). https://doi.org/10.17863/CAM.16206

##### Abstract

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.

##### Keywords

Algebraic Geometry, Birational Geometry