Let $K$ denote the function field $\mathbb C((t))$ and let $X$ be a smooth projective curve of genus $g\geq 2$ over $K$. Let $r\geq 2$ be some positive integer. Let $B$ denote the moduli of vector bundles of rank $r$ with trivial determinant. Is $B$ a very good stack(in the sense of Beilinson and Drinfeld)?
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$\begingroup$ Are you asking about "very good" in the sense of Beilinson and Drinfeld: roughly that the inertia is "small"? $\endgroup$– Jason StarrCommented Jul 4, 2023 at 11:29
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$\begingroup$ @Jason Starr Yes, whether the very good property in the sense of Beilinson and Drinfeld holds over the function field $\mathbb C((t))$? $\endgroup$– S.D.Commented Jul 4, 2023 at 12:01
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$\begingroup$ You need some inequality on the genus $g$ and the rank $r$. It is clearly false when $g$ equals $0$ and when $r$ equals $2$. $\endgroup$– Jason StarrCommented Jul 4, 2023 at 12:17
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$\begingroup$ @Jason Starr thanks, I assume $g\geq 2$. $\endgroup$– S.D.Commented Jul 4, 2023 at 12:19
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2$\begingroup$ I don't think it depends on the field. I think BD prove that the stack of bundles is very good for all curves of genus $\geq 2$. $\endgroup$– Alexander BravermanCommented Jul 5, 2023 at 0:20
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