Let \(k\) an algebraically closed field. Let \(char\ k=0.\) Let \(X\) be a normal variety of dimension \(n\) in \(\mathbb{P }^{N}\) with canonical (resp. terminal) singularities. Let \(Y\) be a sufficiently general nonsingular variety in \(\mathbb{P }^{N}\) such that \(Y\cap \text{ Sing} (X)\ne \emptyset ,\) (3.7). Then \(X\cap Y\) is also normal and with canonical (resp. terminal, log terminal) singularities (3.8). Let \(Z\) be an irreducible nonsingular \((n-1)\)-dimensional variety such that \(2Z=X\cap F\), where \(X\) is an \(n\)-fold and \(F\) is a \((N-1)\)-fold in \(\mathbb{P }^{N},X\) normal with canonical singularities. We study the singularities of \(X\) through which \(Z\) passes.