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

epiga:5890 - Épijournal de Géométrie Algébrique, November 30, 2020, Volume 4 - https://doi.org/10.46298/epiga.2020.volume4.5890
Rationally connected rational double covers of primitive Fano varieties

Authors: Aleksandr V. Pukhlikov

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.

Volume: Volume 4
Published on: November 30, 2020
Submitted on: November 3, 2019
Keywords: Mathematics - Algebraic Geometry,14E05, 14E07