Geodesic
Birationally rigid Fano varieties
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 60 (2005) no. 5, pp. 875-965

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The birational superrigidity and, in particular, the non-rationality of a smooth three-dimensional quartic was proved by V. Iskovskikh and Yu. Manin in 1971, and this led immediately to a counterexample to the three-dimensional Lüroth problem. Since then, birational rigidity and superrigidity have been proved for a broad class of higher-dimensional varieties, among which the Fano varieties occupy the central place. The present paper is a survey of the theory of birationally rigid Fano varieties. 
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     author = {I. A. Cheltsov},
     title = {Birationally rigid {Fano} varieties},
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I. A. Cheltsov. Birationally rigid Fano varieties. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 60 (2005) no. 5, pp. 875-965. http://geodesic.mathdoc.fr/item/RM_2005_60_5_a2/