Let $f=[F,G,H]:\mathbb{P}^2\to\mathbb{P}^2$ be a morphism of degree $d\ge2$. The critical locus $C_f$ of $f$ is the zero-locus of the Jacobian determinant: $$ C_f = \left\{ [x,y,z]\in\mathbb{P}^2 : \det\begin{pmatrix} F_X&F_Y&F_Z\\ G_X&G_Y&G_Z\\ H_X&H_Y&H_Z\\ \end{pmatrix} = 0 \right\}. $$ In the parameter space of such $f$, it is clear that $$ \{ f : \text{the critical locus $C_f$ is smooth} \} $$ is Zariski open. Our question is whether this set is non-empty, i.e., for each $d\ge2$, does there exist at least one degree $d$ morphism $f:\mathbb{P}^2\to\mathbb{P}^2$ such that $C_f$ is smooth?

We suspect that for every $d\ge2$ the map $$ f=[X^d+X^2Y^{d-1}+XZ^d,Y^d+Y^2Z^{d-1}+YX^d,Z^d+Z^2X^{d-1}+ZY^d] $$ has smooth $C_f$, and have verified this for all $d\le 10$ by explicitly computing resultants modulo an appropriate prime.

More generally, we would be interested in the same question for morphisms $\mathbb{P}^n\to\mathbb{P}^n$ on projective space of higher dimension.

Let $f=[F,G,H]:\mathbb{P}^2\to\mathbb{P}^2$ be a morphism of degree $d\ge2$. The critical locus $C_f$ of $f$ is the zero-locus of the Jacobian determinant: $$ C_f = \left\{ [x,y,z]\in\mathbb{P}^2 : \det\begin{pmatrix} F_X&F_Y&F_Z\\ G_X&G_Y&G_Z\\ H_X&H_Y&H_Z\\ \end{pmatrix} = 0 \right\}. $$ In the parameter space of such $f$, it is clear that $$ \{ f : \text{the critical locus $C_f$ is smooth} \} $$ is Zariski open. Our question is whether this set is non-empty, i.e., for each $d\ge2$, does there exist at least one degree $d$ morphism $f:\mathbb{P}^2\to\mathbb{P}^2$ such that $C_f$ is smooth?

We suspect that for every $d\ge2$ the map $$ f=[X^d+X^2Y^{d-2}+XZ^{d-1},Y^d+Y^2Z^{d-2}+YX^{d-1},Z^d+Z^2X^{d-2}+ZY^{d-1}] $$ has smooth $C_f$, and have verified this for all $d\le 10$ by explicitly computing resultants modulo an appropriate prime.

More generally, we would be interested in the same question for morphisms $\mathbb{P}^n\to\mathbb{P}^n$ on projective space of higher dimension.