Let $ X$ be a smooth curve of genus $ g$ . Let $ Q_n$ be the quotient scheme of quotients of trivial bundle $ E$ of rank $ n$ (say) of Hilbert polynomial $ P$ . Let $ R_n$ be the open subset which consists of locally free sheaves $ F$ of rank $ r$ and degree $ d$ , and $ H^1(F)$ vanishes and $ H^0(E)=H^0(F)$ . We know that $ GL(n)$ acts on $ R_n$ .

Now we know that we have a smooth morphism $ \coprod R_n \rightarrow \mathcal{M}_{VB}$ , where $ \mathcal{M}_{VB}$ is the moduli stack of vector bundles of rank $ r$ and degree $ d$ . This is an atlas. Now this morphism factors through $ \coprod R_n \rightarrow \coprod [\frac{R_n}{GL(n)}]\rightarrow \mathcal{M}_{VB}$ , where $ [\frac{R_n}{GL(n)}]$ denotes the quotient stack. It follows that $ \coprod [\frac{R_n}{GL(n)}]\rightarrow \mathcal{M}_{VB}$ is smooth. It is basically a “smooth gluing” of quotient stacks.

Now to do GIT on $ R_n$ we choose suitable $ GL(n)$ -equivariant embedding into projective space (let us denote it by $ H_n$ ). Consider $ \coprod H_n$ . We have $ \coprod R_n\subset \coprod H_n$ . Also $ \coprod [\frac{R_n}{GL_n}]\subset \coprod [\frac{H_n}{GL(n)}]$ . Now we have this “smooth gluing” $ \coprod [\frac{R_n}{GL(n)}]\rightarrow \mathcal{M}_{VB}$ as mentioned earlier.

Question: Does this “smooth gluing” on $ \coprod [\frac{R_n}{GL_n}]$ induce a gluing on $ \coprod [\frac{H_n}{GL(n)}]$ to produce a new proper stack containing $ \mathcal{M}_{VB}$ ??