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I have a forgetful map between moduli spaces, I want to prove that it is an embedding, In fact, I have a reductive algebric group (which is not constant) over a curve $X$ whose geniric fiber is semisimple. And I have a forgetful map from $Bun_\mathcal G$ to the algebraic stuck of vector bundle $\mathscr {SL}(r)$

Is there any usuful critiria in this situation?

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  • $\begingroup$ Does this forgetful morphism arise from a faithful morphism of $X$-group schemes, $\mathcal{G} \to \textbf{SL}_{r,X}$? If so, for a fixed $\textbf{SL}_{r,X}$-torsor $E$, the "reductions of structure group" to $\mathcal{G}$ are naturally in bijection with sections of the associated affine morphism $E/\mathcal{G} \to X$. $\endgroup$ Sep 15, 2015 at 15:19
  • $\begingroup$ I am not sure if that morphism is faithfull or not, but even we suppose so, I don't see your response, could you please explain a litte bit? $\endgroup$
    – Gest2015
    Sep 15, 2015 at 15:47

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