Geodesic
Birational geometry of Fano double spaces of index two
Izvestiya. Mathematics , Tome 74 (2010) no. 5, pp. 925-991

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We study birational geometry of Fano varieties realized as double covers $\sigma\colon V\to{\mathbb P}^M$, $M\geqslant5$, branched over generic smooth hypersurfaces $W=W_{2(M-1)}$ of degree $2(M-1)$. We prove that the only structures of a rationally connected fibre space on $V$ are pencil-subsystems of the free linear system $|{-\frac12K_V}|$. The groups of birational and biregular self-maps of $V$ coincide: $\operatorname{Bir}V=\operatorname{Aut}V$.
Keywords: Fano variety, maximal singularity, rationally connected fibre space, birational self-map.
Mots-clés : birational map 
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     author = {A. V. Pukhlikov},
     title = {Birational geometry of {Fano} double spaces of index two},
     journal = {Izvestiya. Mathematics },
     pages = {925--991},
     publisher = {mathdoc},
     volume = {74},
     number = {5},
     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2010_74_5_a1/}
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A. V. Pukhlikov. Birational geometry of Fano double spaces of index two. Izvestiya. Mathematics , Tome 74 (2010) no. 5, pp. 925-991. http://geodesic.mathdoc.fr/item/IM2_2010_74_5_a1/