A classification of terminal quartic 3-folds and applications to rationality questions

Details A classification of terminal quartic 3-folds and applications to rationality questions A classification of terminal quartic 3-folds and applications to rationality questions

2009-08-03

arxiv.org/abs/0908.0289v1

Mathematics

Algebraic Geometry

40 pages

Scientific paper

This paper studies the birational geometry of terminal Gorenstein Fano 3-folds. If Y is not Q-factorial, 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 Q-factorialisation of Y. In this case, Weil non-Cartier divisors are generated by "topological traces " of K-negative extremal contractions on X. One can show, as an application of these methods, that a number of families of non-factorial terminal Gorenstein Fano 3-folds 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.

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