Geodesic
Birationally rigid Fano fibrations
Izvestiya. Mathematics , Tome 64 (2000) no. 3, pp. 563-581

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We prove the birational superrigidity of a general Fano fibration $\pi\colon V\to\mathbf P^1$ whose fibre is a Fano hypersurface $W_M\subset\mathbf P^M$ of index 1. If the fibration is sufficiently twisted over the base $\mathbf P^1$, then $V$ has no other structure of a fibration into rationally connected varieties. We also formulate and discuss conjectures on birational rigidity for a large class of Fano varieties and Fano fibrations over a base of arbitrary dimension. 
@article{IM2_2000_64_3_a3,
     author = {A. V. Pukhlikov},
     title = {Birationally rigid {Fano} fibrations},
     journal = {Izvestiya. Mathematics },
     pages = {563--581},
     publisher = {mathdoc},
     volume = {64},
     number = {3},
     year = {2000},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2000_64_3_a3/}
}
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A. V. Pukhlikov. Birationally rigid Fano fibrations. Izvestiya. Mathematics , Tome 64 (2000) no. 3, pp. 563-581. http://geodesic.mathdoc.fr/item/IM2_2000_64_3_a3/