Algebraic Geometry Seminar
Anne-Sophie Kaloghiros
Cambridge University
The defect of Fano 3-folds
Abstract: Let X be a quartic hypersurface in P^4 with no worse than terminal
singularities. The Grothendieck-Lefschetz theorem states that the Picard rank
of X is 1, i.e. that every Cartier divisor on X is a hyperplane section of X.
However, no such result holds for the group of Weil divisors of X if X is not
factorial. I will bound the rank of the group of Weil divisors of X when X is a Fano
3-fold with terminal Gorenstein singularities. This bound is optimal in the
case of the quartic 3-fold. I will show how to use birational geometry in order
to understand some aspects of the topology of Fano 3-folds with mild
singularities. If time permits, I will show that these methods yield an
"explicit" description of the lattice of Weil divisors and provide some
additional information on the geometry of X .
Thursday March 5, 2009 at 4:00 PM in SEO 636