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Birational automorphisms of Fano hypersurfaces with singularities

1. The aim of this note is to establish birational rigidity and to compute the group of birational automorphisms of a general Fano hypersurface of index 1, V = VM ⊂ PM , M ≥ 5, with generic isolated singularities. Geometrically, birational rigidity means that there is no non-trivial rational map ρ : V S onto a variety S of positive dimension, the fibres of which are uniruled (that is, can be covered by rational curves). Moreover, birational rigidity of a variety V implies the uniqueness of the minimal model in the birational class of this variety: any Q-Fano variety V ′ with Picard number 1 and birationally equivalent to V is in fact biregularly isomorphic to V . For a precise definition of birational rigidity and an elementary derivation of the aforesaid geometric properties of the variety, see [1], [2].