Smooth double subvarieties on singular varieties (Q742212)

Let \(k\) be an algebraically closed field of characteristic \(0\) and \(X\subseteq \mathbb{P}^N\) an \(n\)--dimensional normal variety with canonical (resp. termial) singularities. Let \(Y\) be a generic nonsingular subvariety in \( \mathbb{P}^N\) intersecting the singular locus of \(X\). It is proved that \(X\cap Y\) is also normal with canonical (resp. terminal, log terminal) singularities. Let \(Z\) be an irreducible nonsingular \((n-1)\)--dimensional variety such that \(2Z=X\cap F\), where \(F\) is a hypersurface in \(\mathbb{P}^N\). The singularities of \(X\) which are on \(Z\) are studied.