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

摘要

This paper studies the birational geometry of terminal Gorenstein Fano 3-folds. If Y is not ?-factorial, in most cases, it is possible to describe explicitly the divisor class group Cl Y by running a Minimal Model Program on X, a small ?-factorialization of Y. In this case, the generators of Cl Y/ Pic Y are "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 Y _4 ? ? ~4 with rk Cl Y = 2 and show that when rk Cl Y ≥ 6, Y is always rational.