Generic singular hypersurface

Question

I've heard in informal conversations before the claim that:

"a generic singular hypersurface has a single singularity of type $\mathbf{A}_1$".

What is the precise statement of this result? Where can I find a proof?

Answer 1

Consider the space of all hypersurfaces of degree $d$ in $\mathbb P^n$, say, with $n \geq 1$ and $d \geq 2$. Consider the closed subspace of singular hypersurfaces. Then there is a dense open set of this closed subspace over which all hypersurfaces have a single singularity of type $\mathbf A_1$.

The proof is to check that a sufficient condition for a singularity to have type $\mathbf A_1$ is for the Hessian to be nondegenerate, then to check that the locus of singular hypersurfaces has codimension $1$, while the locus of hypersurfaces with two singularities and with a singularity with degenerate Hessian each have codimension at least $2$. These can all be proved by viewing the moduli space of hypersurfaces with a certain type of singularity as a bundle over the space of possible singular points and calculating its dimension as the dimension of the base plus the dimension of the fiber.