Rationally connected rational double covers of primitive Fano varieties
Aleksandr V. Pukhlikov

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 of degree in for there are no Galois rational covers of degree with an abelian Galois group, where is a rationally connected variety. In particular, there are no rational maps of degree 2 with 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.
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