Birational rigidity of Fano complete intersections (Q352228)

In the paper under review, the author presents, without proofs, some results on the superrigidity of Fano complete intersections. These theorems improve considerably previous results obtained by the author [J. Reine Angew. Math. 541, 55--79 (2001; Zbl 1037.14004). Let \((d_{i})_{i=1}^{k}\) be a non-decreasing sequence of integers such that \(d_{i}\geq 2\), \(d_{k}=\max\left\{d_{i}\right\}\geq 4\) and \(d_{1}+d_{2}+\dots +d_{k}\geq \max\left\{2k+3,k+7\right\}\). Let \(M=d_{1}+d_{2}+\dots +d_{k}-k\). Consider the smooth complete intersection \(V=F_{1}\cap F_{2}\cap \dots \cap F_{k}\subseteq\mathbb{P}^{M+k}\), where \(F_{i}\) has degree \(d_{i}\). In particular, \(V\) is a Fano variety of dimension \(M\) and it has Picard number one. The main result presented in the paper is that a generic complete intersection \(V\), as above, is birationally superrigid. In particular, it implies a number of interesting corollaries: there exists no rational dominant mapping \(\gamma:V\rightarrow S\) onto a variety \(S\) of positive dimension whose general fiber \(\gamma^{-1}(s)\) has negative Kodaira dimension, any birational mapping \(\chi:V\rightarrow V'\) onto a Fano variety \(F'\) with \(\mathbb{Q}\)-factorial terminal singularities and Picard number one is a biregular isomorphism and the group \(\mathrm{Bir}(V)\) of birational automorphisms coincides with the group \(\mathrm{Aut}(V)\) of biregular projective automorphism and, therefore, is trivial. Furthermore, the author presents other results concerning the birationally superrigidity of complete intersections of quadrics and cubics.