## The topology of terminal quartic 3-folds

dc.contributor.author | Kaloghiros, Anne-Sophie | |

dc.date.accessioned | 2009-03-02T09:37:57Z | |

dc.date.available | 2009-03-02T09:37:57Z | |

dc.date.issued | 2007-06-20 | |

dc.identifier.other | PhD.30695 | |

dc.identifier.uri | http://www.dspace.cam.ac.uk/handle/1810/214794 | |

dc.identifier.uri | https://www.repository.cam.ac.uk/handle/1810/214794 | |

dc.description.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. | en |

dc.language.iso | en | en |

dc.subject | Algebraic Geometry | en |

dc.subject | Birational Geometry | en |

dc.title | The topology of terminal quartic 3-folds | en |

dc.type | Thesis | en |

dc.type.qualificationlevel | Doctoral | |

dc.type.qualificationname | Doctor of Philosophy (PhD) | |

dc.publisher.institution | University of Cambridge | |

dc.identifier.doi | 10.17863/CAM.16206 |