Let the permutation group $S\_4$ act on $\mathbb C^4$ by permuting the coordinates. Consider the categorical quotient $\mathbb P(\mathbb C^4)/S\_4$. It is a projective variety by a theorem of Mumford. Does the line bundle $\mathcal O(1)^{\otimes 12}$ on $\mathbb P(\mathbb C^4)$ descend to the quotient $\mathbb P(\mathbb C^4)/S\_4$ ?



line bundle descends?
Yes, and the good news are that there isn't anything to compute. By a result of Kempf a line bundle $L$ on $\mathbb{P}^3$ descends to the quotient if and only if the stabilizer $S\_x$ of each point $x\in \mathbb{P}^3$ acts trivially on the fiber $L\_x$. Now $S\_4$ acts on $\mathcal{O}(1)$, and $S\_x$ act on $\mathcal{O}(1)\_x$ through a character $\chi \_x:S\_x\rightarrow \mathbb{C}^\*$, and on $\mathcal{O}(1)^{\otimes k}$ through the character $\chi \_x^{k}$. The abelian subquotients of $S\_4$ have order $2,3$ or $4$, hence taking $k=12$ kills all the characters, and therefore $\mathcal{O}(1)^{\otimes 12}$ descends.


