Rationally connected rational double covers of primitive Fano varieties

TL;DR

The paper proves that for a general hypersurface of degree M+1 in projective space, there are no Galois rational covers of degree at least 2 with abelian Galois groups from rationally connected varieties, supporting a conjecture on their rigidity.

Contribution

It establishes the non-existence of certain Galois rational covers of primitive Fano varieties, highlighting their rigidity and motivating a broader conjecture.

Findings

No Galois rational covers of degree ≥ 2 with abelian Galois group exist for general hypersurfaces in projective space.

The result extends to many primitive Fano varieties, indicating their absolute rigidity.

Supports conjecture on the rigidity of primitive Fano varieties.

Abstract

We show that for a Zariski general hypersurface $V$ of degree $M+1$ in ${\mathbb P}^{M+1}$ for $M\geqslant 5$ there are no Galois rational covers $X\dashrightarrow V$ of degree $d\geqslant 2$ with an abelian Galois group, where $X$ is a rationally connected variety. In particular, there are no rational maps $X\dashrightarrow V$ of degree 2 with $X$ rationally connected. This fact is true for many other families of primitive Fano varieties as well and motivates a conjecture on absolute rigidity of primitive Fano varieties.