Factorial threefolds and Shokurov vanishing

Abstract

We prove the factoriality of the following nodal threefolds: a complete intersection of hypersurfaces $F$ and $G\subset {\mathbb{P}}^5$ of degree $n$ and $k$ respectively, where $G$ is smooth, $|\mathrm{Sing}(F\cap G)|⩽(n+k-2)(n-1)/5$, $n⩾k$; a double cover of a smooth hypersurface $F\subset {\mathbb{P}}^4$ of degree $n$ branched over a surface that is cut out on $F$ by a hypersurface $G$ of degree $2r⩾n$, and $|\mathrm{Sing}(F\cap G)|⩽(2r+n-2)r/4$.

22 pages, extended version, to appear in Sbornik: Mathematics