Mathematische Annalen

, Volume 354, Issue 1, pp 263–296 | Cite as

A classification of terminal quartic 3-folds and applications to rationality questions



This paper studies the birational geometry of terminal Gorenstein Fano 3-folds. If Y is not \({\mathbb{Q}}\) -factorial, in most cases, it is possible to describe explicitly the divisor class group Cl Y by running a Minimal Model Program on X, a small \({\mathbb{Q}}\) -factorialization of Y. In this case, the generators of Cl Y/ Pic Y are “topological traces” of K-negative extremal contractions on X. One can show, as an application of these methods, that a number of families of non-factorial terminal Gorenstein Fano 3-folds are rational. In particular, I give some examples of rational quartic hypersurfaces \({Y_4 \subset \mathbb{P}^4}\) with rk Cl Y = 2 and show that when rk Cl Y ≥ 6, Y is always rational.


Exceptional Divisor Pezzo Surface Hyperplane Section Cartier Divisor Fano Variety 


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© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of Pure Mathematics and Mathematical StatisticsUniversity of CambridgeCambridgeUK

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