Suppose I have an affine subvariety $A \subset {\mathbb C}^N$ of dimension $n \geq 3$ which has an isolated singularity at $0$ (lets say for the sake of simplicity that it is non-singular everywhere else).
Suppose that this variety $A$ is normal.
In order to study singularities it often seems like a good idea to study hyperplane sections or intersections with lower dimensional linear subspaces passing through that singularity.


Let $L$ be a generic linear subspace of ${\mathbb C}^N$ of dimension $N - n + 2$ passing through the origin. If $L$ is generic enough then $A \cap L$ has dimension $2$ and has an isolated singularity at the origin. Is it true that $A \cap L$ is normal for $L$ generic enough? 



When is the intersection of an isolated normal singularity with a generic linear subspace through that singularity  normal?
I don't think so. There are examples of isolated normal threefold singularities that are not Cohen-Macaulay. A hyperplane section is not Cohen-Macaulay, hence it can not be normal, because a normal surface is Cohen-Macaulay.


