Let $f:\mathbb{CP}^n\to\mathbb{CP}^n$ be a holomorphic map; I am interested in what the subvariety of critical points could be.
More specifically, let $J=\{p\in \mathbb{CP}^n\ :\ \det\mathrm{Jac}(f)=0\}$:
- can $J$ be a smooth submanifold of $\mathbb{CP}^n$?
- can $J$ be an irreducible subvariety of $\mathbb{CP}^n$?
- is there a way to describe the subvarieties $J$ that can be obtained this way?
Obviously, for $n=1$, the answer to 1,2 is easy, because $J$ is a union of isolated points. Regarding 3, if we want a constructive answer (i.e., given $J$, we want to produce $f$) it is not immediate to answer to 3, already for $n=1$.
On a further level, I am also wondering if it can happen that $J$ is, e.g., smooth but $(\det\mathrm{Jac}(f))$ is not reduced (i.e., locally $\det\mathrm{Jac}(f)=(g)^k$ for some $g$ holomorphic and $k>1$).
I tried to write down the equations explicitly fixing the degree of $f$ (as a homogeneous polynomial on $\mathbb{C}^{n+1}$), but the result is not easily manageable, at least for me; I also tried to look at it from and algebraic viewpoint, noticing a link with the concept of socle of a Gorenstein algebra, but, as much as it was interesting, it did not give me any hint on how to exlpicitly describe $J$ (or $f$ given $J$, or given the equation I want for $J$).
I believe this kind of question could already appear somewhere in the literature, but it evaded my googleing efforts, up to now.