Geodesic
Birationally rigid varieties. I. Fano varieties
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 62 (2007) no. 5, pp. 857-942

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The theory of birational rigidity of rationally connected varieties generalises the classical rationality problem. This paper gives a survey of the current state of this theory and traces its history from Noether's theorem and the Lüroth problem to the latest results on the birational superrigidity of higher-dimensional Fano varieties. The main components of the method of maximal singularities are considered. 
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     author = {A. V. Pukhlikov},
     title = {Birationally rigid varieties. {I.} {Fano} varieties},
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A. V. Pukhlikov. Birationally rigid varieties. I. Fano varieties. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 62 (2007) no. 5, pp. 857-942. http://geodesic.mathdoc.fr/item/RM_2007_62_5_a1/