Varieties of minimal rational tangents on double covers of projective space

Abstract

Let $\phi : X \rightarrow \mathbb{P }^n$ be a double cover branched along a smooth hypersurface of degree $2m, 2 \le m \le n-1$. 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 $\mathcal{C }_x$ 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 $X$.