Factoriality of nodal three-dimensional varieties and connectedness of the locus of log canonical singularities

Shokurov's vanishing theorem is used for the proof of the $\mathbb Q$-factoriality of the following nodal threefolds: a complete intersection of hypersurfaces $F$ and $G$ in $\mathbb P^5$ of degrees $n$ and $k$, $n\geqslant k$, such that $G$ is smooth and $|{\operatorname{Sing}(F\cap G)}|\leqslant(n+k-2)(n-1)/5$; a double cover of a smooth hypersurface $F\subset\mathbb P^4$ of degree $n$ branched over the surface cut on $F$ by a hypersurface $G\subset\mathbb P^4$ of degree $2r\geqslant n$, provided that $|{\operatorname{Sing}(F\cap G)}|\leqslant(2r+n-2)r/4$.
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