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?