Double cubics and double quartics

Abstract

We study a double cover $\psi :X\to V\subset {\mathbb{P}}^n$ branched over a smooth divisor $R\subset V$ such that $R$ is cut on $V$ by a hypersurface of degree $2(n-\mathrm{deg}(V))$, where $n⩾8$ and $V$ is a smooth hypersurface of degree 3 or 4. We prove that $X$ is nonrational and birationally superrigid.

9 pages