For a normal (you can assume smooth for this problem) quasi projective complex variety $X$ and a projective complex variety $Y$, we can endow the space of the set of morphisms $Mor(X,Y)$$\operatorname{Mor}(X,Y)$ we the topology of convergence with bounded degree in the following way: a sequence of morphisms $f_i$ is considered to converge to $f$ iff they converge as continuous maps in $Hom_{cont}(X^{an}, Y^{an})$$\operatorname{Hom}_\mathrm{cont}(X^\mathrm{an},Y^\mathrm{an})$ with compact open topology and there is a finite upper bound on the degree of closure of the graphs of $f_i$ in $\overline{X}\times Y$. Here $\overline{X}$ is a projective closure of $X$.
For this problem you may assume $X$ is affine and there is a finite group $G$ acting on $Y$. (For example you can assume $Y=(\mathbb{P^d})^{\times n}$ and $G=S_n$). Let's denote the quotient variety by $Y_{G}$. We have natural map $q: Mor(X, Y)\rightarrow Mor(X, Y_G)$$q\colon\operatorname{Mor}(X,Y)\rightarrow\operatorname{Mor}(X,Y_G)$.
- Is $q$ an open morphism?
- Is $q$ a closed morphism?
How does $Mor(X, Y)_G$$\operatorname{Mor}(X,Y)_G$ and $Mor(X, Y_G)$$\operatorname{Mor}(X,Y_G)$ compare. Is the first one an open/closed subset of the second one?
