Nonrational Nodal Quartic Threefolds

The Q-factoriality of a nodal quartic 3-fold implies its non-rationality. We prove that a nodal quartic 3-fold with at most 8 nodes is Q-factorial, and we show that a nodal quartic 3-fold with 9 nodes is not Q-factorial if and only if it contains a plane. However, there are non-rational non-Q-factorial nodal quartic 3-folds in P. In particular, we prove the non-rationality of a general non-Q-factorial nodal quartic 3-fold that contains either a plane or a smooth del Pezzo surface of degree 4.