Geodesic
Birational automorphisms of a~three-dimensional double cone
Sbornik. Mathematics, Tome 189 (1998) no. 7, pp. 991-1007

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Birational rigidity is proved for a three-dimensional double cone, that is, a double cover of a quadric cone in $\mathbb P^2$ with branching in a smooth section of a quartic, and its non-rationality and the absence of conic bundle structures are also proved. The group of birational automorphisms of this variety is computed. 
@article{SM_1998_189_7_a2,
     author = {M. M. Grinenko},
     title = {Birational automorphisms of a~three-dimensional double cone},
     journal = {Sbornik. Mathematics},
     pages = {991--1007},
     publisher = {mathdoc},
     volume = {189},
     number = {7},
     year = {1998},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1998_189_7_a2/}
}
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M. M. Grinenko. Birational automorphisms of a~three-dimensional double cone. Sbornik. Mathematics, Tome 189 (1998) no. 7, pp. 991-1007. http://geodesic.mathdoc.fr/item/SM_1998_189_7_a2/