Varieties of minimal rational tangents on double covers of projective space

Abstract

Let $\varphi :X\to {\mathbb{P}}^n$ 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 ${\mathbb{C}}_x$ implies a certain rigidity property of the covering morphism $\varphi$. 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 $X$.

To appear in Math. Zeit