Let $V = V(f) \subseteq \mathbb{C}^n$ be an irreducible (affine) hypersurface. The singular locus $\text{Sing}(V)$ of $V$ is the set of points in $V$ where all the partial derivatives of $f$ vanish. I am interested in examples where $\dim(\text{Sing}(V)) = n-2$, i.e. the singular locus has codimension one.
There are two classes of easy examples. One is when $V = V' \times \mathbb{C}^{n-2}$ for a non-smooth algebraic curve $V$, i.e. $f$ is a polynomial in two variables and has singularities. The other is when $f$ is a homogenous polynomial in three variables and has singularities (this feels like the "projective version" of the previous case).
- Are there other examples of irreducible complex hypersurfaces that have a codimension one singular locus?
- If there are, are there general conditions that imply that the singular locus has codimension at least two? (It is unclear, of course, what it means for a condition to be "general"; I'm hoping for one which does not involve explicitly checking the partial derivatives of the polynomial.)