Birational geometry of terminal quartic 3-folds. I

Abstract

This is the unabridged web version of the paper that will be published on the American Journal of Mathematics. In this paper, we study the birational geometry of certain examples of mildly singular quartic 3-folds. A quartic 3-fold is an example of a Fano variety, that is, a variety $X$ with ample anticanonical sheaf $Ø_X(-K_X)$. From the point of view of birational geometry they basically fall within two classes: either $X$ is ``close to being rational'', and then it has very many biregularly distinct birational models as a Fano 3-fold, or, at the other extreme, $X$ has a unique model and it is often even true that every birational selfmap of $X$ is biregular. In this paper we construct examples of singular quartic 3-folds with exactly two birational models as Fano 3-folds.