Cartan–Fubini type rigidity of double covering morphisms of quadratic manifolds

Hosung Kim

Vol. 282 (2016), No. 2, 329–339

DOI: 10.2140/pjm.2016.282.329

Abstract

Let $Z\subset {\mathbb{P}}^N$ be a Fano manifold whose Picard group is generated by the hyperplane section class. Assume that $Z$ is covered by lines and $i\left(Z\right)\ge 3$. Let $\varphi :X^Z\to Z$ be a double cover, branched along a smooth hypersurface section of degree $2m$, $1\le m\le i\left(Z\right)-2$. We describe the defining ideal of the variety of minimal rational tangents at a general point. As an application, we show that if $Z\subset {\mathbb{P}}^N$ is defined by quadratic equations and $2\le m\le i\left(Z\right)-2$, then the morphism $\varphi$ satisfies the Cartan–Fubini type rigidity property.

Keywords

double covers of Fano manifolds, varieties of minimal rational tangents, Cartan–Fubini type rigidity

Mathematical Subject Classification 2010

Primary: 14J45

Milestones

Received: 27 November 2014

Revised: 22 August 2015

Accepted: 30 September 2015

Published: 3 March 2016