Birationally rigid Fano complete intersections. II

TL;DR

This paper proves that generic Fano complete intersections of certain types are birationally superrigid, ensuring unique Mori fibre space structures, automorphism groups, and non-rationality, extending previous results significantly.

Contribution

It establishes birational superrigidity for a broader class of Fano complete intersections than previously known.

Findings

Varieties are birationally superrigid under given conditions.

Unique Mori fibre space structures on these varieties.

Automorphism groups coincide with birational automorphism groups.

Abstract

We prove that a generic (in the sense of Zariski topology) Fano complete intersection $V$ of the type $(d_1,...,d_k)$ in ${\mathbb P}^{M+k}$, where $d_1+...+d_k=M+k$, is birationally superrigid if $M\geq 7$, $M\geq k+3$ and $\mathop{\rm max} \{d_i\}\geq 4$. In particular, on the variety $V$ there is exactly one structure of a Mori fibre space (or a rationally connected fibre space), the groups of birational and biregular self-maps coincide, $\mathop{\rm Bir} V= \mathop{\rm Aut} V$, and the variety $V$ is non-rational. This fact covers a considerably larger range of complete intersections than the result of [J. reine angew. Math. {\bf 541} (2001), 55-79], which required the condition $M\geq 2k+1$. The paper is dedicated to the memory of Eckart Viehweg.