Birational geometry of quartic 3-folds II: The importance of being -factorial

The paper explores the birational geometry of terminal quartic 3-folds. In doing this I develop a new approach to study maximal singularities with positive dimensional centers. This allows to determine the pliability of a -factorial quartic with ordinary double points, and it shows the importance of -factoriality in the context of birational geometry of uniruled 3-folds.Mathematics Subject Classification (1991): 14E07, 14J30, 14E30Partially supported by EAGER and Geometria sulle Varietà Algebriche (MIUR).Revised version: 2 January 2004