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 $\mathbb P(\mathbb C^4)/S_4$ ?
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.

$\begingroup$ The statement "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 $L_x$." is not very precise. You probably mean : an equivariant line bundle, else the action on the fiber is not even defined. Could you give a reference ? $\endgroup$ – Niels Jan 27 '15 at 13:37

2$\begingroup$ Yes, of course, I meant an equivariant line bundle. A good reference is Theorem 2.3 in the paper of Drezet and Narasimhan, Groupe de Picard des variétés de modules de fibrés semistables sur les courbes algébriques, Invent. math. 97 (1989), 5394. $\endgroup$ – abx Jan 27 '15 at 13:49

$\begingroup$ I think the same proof will work for any $k$ a multiple of $2,3,...,n$ with the action of $S_n$ on $\mathbb P^{n1}$. $\endgroup$ – Ram Jan 28 '15 at 9:47

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1$\begingroup$ I gave the precise condition in my answer, please read it. $\endgroup$ – abx Jan 21 '16 at 9:53