Let $G=PGL(n)$ act on a smooth projective scheme $X$ over $\mathbb{C}$ with nontrivial finite stabilizers ($\cong \mathbb{Z}/2\mathbb{Z}$) only along a divisor $D\subset X$. Furthermore there a is a good quotient $Y=X//G$, with $Y$ also smooth and projective. This is constructed via GIT with a $G$-linearized ample line bundle $\mathcal{L}$ on $X$ and we have the morphism $\pi: X \rightarrow Y$.
There is a $G$-linearized locally free sheaf $\mathcal{E}$ of rank two on $X$, then we have the projective bundle $p: \mathbb{P}(\mathcal{E}^{*})\rightarrow X$ associated to the dual sheaf.
$\bf{Question\,1:}$ is there a $G$-linearized ample line bundle $\mathcal{L}'$ on $\mathbb{P}(\mathcal{E})$ (probably some combination of the pullback of $\mathcal{L}$ and $\mathcal{O}(1)$?) so that we can construct a GIT quotient $f: \mathbb{P}(\mathcal{E}^{*})\rightarrow \mathbb{P}(\mathcal{E}^{*})//G$ such that $p: \mathbb{P}(\mathcal{E}^{*})\rightarrow X$ induces a morphism of quotients $\hat{p}: \mathbb{P}(\mathcal{E}^{*})//G \rightarrow X//G$ with an evident commutative diagram?
If Question 1 has a positive answer:
$\bf{Question\,2:}$ Can we describe $\hat{p}:\mathbb{P}(\mathcal{E}^{*})//G\rightarrow X//G$? I doubt that it is again a projective bundle. My guess is that it is maybe the conic bundle (Brauer Severi variety) associated to the sheaf of algebras associated to the invariant direct image of $\mathcal{E}nd(\mathcal{E})$ on $X//G$. This is because $\mathbb{P}(\mathcal{E}^{*})$ is isomorphic to the Brauer Severi variety associated to $\mathcal{E}nd(\mathcal{E})$ on $X$ and the morphism $\pi: X \rightarrow Y$ is a principal $PGL(n)$-bundle away from $D$. That is the conic bundle $\mathbb{P}(\mathcal{E})//G\rightarrow X//G$ degenerates along the divisor $\pi(D)$. Or is there another description of this space?
