Birational geometry of quartic 3-folds. II: The importance of being \(\mathbb{Q}\)-factorial
The author studies the birational geometry of a \(\mathbb Q\)-factorial quartic 3-fold \(X\subset \mathbb P^4_k\) with only ordinary double points as singularities, where \(k\) is either \(\mathbb C\) or a (not necessarily algebraically closed) field of characteristic 0. He proves that such a 3-fold is birationally rigid, i.e., any birational map \(X\to V\) to a Mori fibre space is a self map. The main tools used in the proof are: 1) the maximal singularities with center of positive dimension: in this case the centers are shown to be lines through some singular point: 2) the Sarkisov links, which give the required self maps \(X\to V\) to a Mori fibre space.
scientific article; zbMATH DE number 2104999