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 $J=\{p\in \mathbb{CP}^n\ :\ \det\mathrm{Jac}(f)=0\}$:


1. can $J$ be a smooth submanifold of $\mathbb{CP}^n$?
2. can $J$ be an irreducible subvariety of $\mathbb{CP}^n$?
3. 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.



Geometry of critical points of holomorphic maps in projective space
For $n=2$, the locus $J$ is smooth and irreducible for a general $f$; i.e., these $f$ form a Zariski dense subset of the parameter space of such $f$. For $n\ge3$ and for general $f$, the locus $J$ will be (mildly) singular, irreducible, and of general type. [See Theorems 14 and 15 in the linked paper.](https://arxiv.org/abs/1910.11290) (The proof of Theorem 14, which proves general type, was shown to the authors by Jason Starr.)


