Birational automorphisms of Fano complete intersections. (Q1432360)

The author studies smooth complete Fano intersections \(V=F_1\cap F_2\cap\dots\cap F_k\) in \(\mathbb{P}^{M+k}\) of degree \(d_1\cdot d_2\cdot \dots \cdot d_k\), where the \(F_i \) are hypersurfaces of degree \(d_i\geq 2\) and \(\sum_{i=1}^k d_i=M+k\) (i.e., \(V\) is of index 1). The main result is that for \(M\geq 2k+1\), a general \(V\) of this type is birationally superrigid. As a corollary, one obtains that \(V\) cannot be fibered by a rational mapping into uniruled lower-dimensional varieties, that \(V\) is non-rational and \(\text{Bir}(V)=\text{Aut}(V)\). The proof follows the ideas developed by the author in an earlier paper [Invent. Math. 134, No.2, 401--426 (1998; Zbl 0964.14011)].