Birational rigidity of complete intersections

TL;DR

This paper establishes conditions under which smooth and singular complete intersections are birationally superrigid, implying they are non-rational and have trivial birational automorphism groups, extending previous results in algebraic geometry.

Contribution

It provides new numerical criteria for birational superrigidity of complete intersections, including singular cases, generalizing earlier work by de Fernex.

Findings

Smooth complete intersections are birationally superrigid under specified conditions.

Singular complete intersections also exhibit birational superrigidity with similar criteria.

Complete intersections are proven to be non-rational and have trivial birational automorphism groups.

Abstract

We prove that every smooth complete intersection X defined by s hypersurfaces of degree d_1, ... , d_s in a projective space of dimension d_1 + ... + d_s is birationally superrigid if 5s +1 is at most 2(d_1 + ... + d_s + 1)/sqrt{d_1...d_s}. In particular, X is non-rational and Bir(X)=Aut(X). We also prove birational superrigidity of singular complete intersections with similar numerical condition. These extend the results proved by Tommaso de Fernex.