Smooth double subvarieties on singular varieties. II

Abstract

Let k be an algebraically closed field of characteristic 0. We give a brief survey on multiplicity-2 structures on varieties. Let $Z$ be a reduced irreducible nonsingular $(n-1)$-dimensional variety such that $2Z = X \cap F$, where $X$ is a normal $n$-fold with canonical singularities, $F$ is an $(N-1)$-fold in $\mathbb{P}^N$, such that $Z \cap \mathrm{Sing}(X) \neq \emptyset$. Assume that $\mathrm{Sing}(X)$ is equidimensional and $\mathrm{codim}_X(\mathrm{Sing}(X)) = 3$. We study the singularities of $X$ through which $Z$ passes. We also consider Fano cones. We discuss the construction of some vector bundles and the resolution property of a variety.