Skip to main content
Article thumbnail
Location of Repository

The topology of terminal quartic 3-folds

By Anne-Sophie Kaloghiros

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

Topics: Algebraic Geometry, Birational Geometry
Publisher: University of Cambridge
Year: 2007
OAI identifier: oai:www.repository.cam.ac.uk:1810/214794
Provided by: Apollo

Suggested articles

Citations

  1. (1989). 3-dimensional Fano varieties with canonical singularities. doi
  2. (1968). Functors of Artin rings. doi
  3. (1976). Mixed Hodge structure on the vanishing cohomology. doi
  4. (1981). Mixed Hodge structures associated with isolated singularities. doi
  5. (1994). Nonnormal del Pezzo surfaces. doi
  6. (2001). On birational morphisms between pencils of del Pezzo surfaces.
  7. (2002). On classi of Q-Fano 3-folds of Gorenstein index 2. I, II.
  8. (1974). Projective models of K doi
  9. (1971). Rigidity of quotient singularities. doi
  10. (1983). Semicontinuity of the spectrum and an upper bound for the number of singular points of the projective hypersurface.
  11. (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.