A classification of terminal quartic 3folds and applications to rationality questions
Abstract
This paper studies the birational geometry of terminal Gorenstein Fano 3folds. If Y is not Qfactorial, in most cases, it is possible to describe explicitly the divisor class group Cl Y by running a Minimal Model Program (MMP) on X, a small Qfactorialisation of Y. In this case, Weil nonCartier divisors are generated by "topological traces " of Knegative extremal contractions on X. One can show, as an application of these methods, that a number of families of nonfactorial terminal Gorenstein Fano 3folds are rational. In particular, I give some examples of rational quartic hypersurfaces with Cl Y of rank 2, and show that when Cl Y has rank greater than 6, Y is always rational.
 Publication:

arXiv eprints
 Pub Date:
 August 2009
 DOI:
 10.48550/arXiv.0908.0289
 arXiv:
 arXiv:0908.0289
 Bibcode:
 2009arXiv0908.0289K
 Keywords:

 Mathematics  Algebraic Geometry;
 14E30
 EPrint:
 40 pages