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Rationally connected rational double covers of primitive Fano varieties

Aleksandr V. Pukhlikov

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.

Rationally connected rational double covers 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.

Paper Structure

This paper contains 7 sections, 7 theorems, 51 equations.

Table of Contents

  1. Statement of the main results
  2. Divisorially canonical varieties
  3. Proof of Theorem \ref{['th1']}
  4. Resolution of a family of curves
  5. Construction of the cyclic cover
  6. Generalizations
  7. Historical remarks and acknowledgements

Key Result

Theorem \oldthetheorem

Assume that the Fano variety $V$, introduced above, is divisorially canonical and satisfies the conditions $(\star1)$ and $(\star2)$. Then there are no rational Galois covers $X\stackrel{d:1}{\dashrightarrow} V$ with an abelian Galois group of order $d\geqslant 2$, where $X$ is a rationally connecte

Theorems & Definitions (17)

  • Definition \oldthetheorem: cf. Pukh05
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Corollary \oldthetheorem
  • Conjecture \oldthetheorem: on absolute rigidity
  • Corollary \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • ...and 7 more