Varieties of minimal rational tangents on double covers of projective space

TL;DR

This paper investigates the varieties of minimal rational tangents on double covers of projective space, describing their structure, variation, and rigidity properties, and applies these findings to morphism classification and extension properties.

Contribution

It provides a detailed description of the minimal rational tangent varieties and establishes their maximal variation and rigidity, leading to new results on morphism isomorphisms and extension properties.

Findings

The ideal of the variety of minimal rational tangents is explicitly described.

The projective isomorphism type of these varieties varies maximally across general points.

Any finite morphism between certain double covers is an isomorphism.

Abstract

Let $\phi: X \to \mathbb P^n$ be a double cover branched along a smooth hypersurface of degree $2m, 2 \leq m \leq n-1$. We study the varieties of minimal rational tangents $\mathcal C_x \subset \mathbb P T_x(X)$ at a general point $x$ of $X$. We describe the homogeneous ideal of $\mathcal C_x$ and show that the projective isomorphism type of $\mathcal C_x$ varies in a maximal way as $x$ varies over general points of $X$. Our description of the ideal of $\mathbb C_x$ implies a certain rigidity property of the covering morphism $\phi$. As an application of this rigidity, we show that any finite morphism between such double covers with $m=n-1$ must be an isomorphism. We also prove that Liouville-type extension property holds with respect to minimal rational curves on $X$.