Let $X$ be a projective variety and $G$ a finite group acting on $X$. We consider the quotient $\pi:X\rightarrow Y :=X/G$. 


I'm interested in the relation between $Eff(X)$ and $Eff(Y)$. In particular, is it true that if $Eff(X)$ has infinitely many extremal rays then $Eff(Y)$ has infinitely many extremal rays as well?



A question on the effective cone
Here is an example with $Pic(X)$ finitely generated. Take $X = \overline{M}\_{1,n}$ the moduli space of genus one curves with $n$ marked points. Then $Pic(\overline{M}\_{1,n})$ is finitely generated.


Now, by Theorem $1.1$ of this paper:


<https://www2.bc.edu/dawei-chen/Extremal.pdf>


we have that $Eff(\overline{M}\_{1,n})$ is not finitely generated. Now, consider the action of the symmetric group $S\_n$ on $\overline{M}\_{1,n}$ permuting the markings, and the quotient $\widetilde{M}\_{1,n}:=\overline{M}\_{1,n}/S\_n$. Then, by Theorem $5.1$ of the same paper we get that $Eff(\widetilde{M}\_{1,n})$ is the closed, simplicial cone generated by the boundary divisors.


Therefore, $Eff(\overline{M}\_{1,n}/S\_n)$ is finitely generated while $Eff(\overline{M}\_{1,n})$ is not. 


