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

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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.

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  1. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, UK

    Anne-Sophie Kaloghiros

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  1. Anne-Sophie Kaloghiros
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Correspondence to Anne-Sophie Kaloghiros.

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A.-S. Kaloghiros was supported by Trinity Hall, Cambridge, and by the Mathematical Sciences Research Institute, Berkeley.

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Kaloghiros, AS. A classification of terminal quartic 3-folds and applications to rationality questions. Math. Ann. 354, 263–296 (2012). https://doi.org/10.1007/s00208-011-0658-z

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