Birational geometry of terminal quartic 3-folds, I

In this paper, we study the birational geometry of certain examples of mildly singular quartic 3-folds. A quartic 3-fold is a special case of a Fano variety, that is, a variety X with ample anticanonical sheaf [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]. Nonsingular Fano 3-folds have been studied extensively. Examples studied so far fall within two classes: either X is "close to being rational," and it then has very many biregularly distinct birational models as a Fano 3-fold, or, at the other extreme, X has a unique model. In this paper we construct examples of singular quartic 3-folds with exactly two birational models as Fano 3-folds; the other model is a complete intersection Y_3,4 ⊂ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /](1, 1, 1, 1, 2, 2) of a quartic and a cubic in weighted projective space [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /](1⁴, 2²).