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Consider the stack $Bun_{r,d}^{ss}$ of rank $r$ semi-stable vector bundle of degree $d$ over a fixed curve. There exists also a coarse moduli space $M$ built via GIT. Over the stable locus of $M$ it is well known that the natural morphism $Bun_{r,d} \to M$ is a gerbe. This is basically due to the fat that $Bun_{r,d}^{ss}$ is a $GL_n$ quotient stack of some Quot-scheme, and the moduli space below is the GIT quotient of the same scheme via $PGL_n$.

But what happens over the strictly semistable locus (which is the singular locus of the GIT moduli space)? What's the structure of the fiber? It should be more complicated.

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  • $\begingroup$ For instance I think there should be still a gerbe over some kind of mild desingularization of the GIT quotient. $\endgroup$
    – IMeasy
    Commented Feb 7, 2013 at 18:16
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    $\begingroup$ Did you already have a look at Faltings' paper "Stable G-bundles and projective connections", J. Alg. Geom. 2 (1993), 507-568? I remember something about S-equivalence classes being contracted on the strictly semistable locus... $\endgroup$
    – Christian Liedtke
    Commented Feb 7, 2013 at 18:30
  • $\begingroup$ Yes that's basically what I want to know... I take a look straight away, thank you! $\endgroup$
    – IMeasy
    Commented Feb 7, 2013 at 18:48
  • $\begingroup$ For instance, let's take trivial determinant and rank two - the easiest case. Then I expect to find just $\mathbb{P}Ext^1(L^{-1},L)$ over the S-equivalence class of $[L\oplus L^{-1}]$. $\endgroup$
    – IMeasy
    Commented Feb 7, 2013 at 19:16
  • $\begingroup$ Dear @IMeasy, I was about to ask a similar question. Is there some progress on the description of the fibers of $Bun_{r,d} \to M$? I would appreciate any hint. Thanks. $\endgroup$
    – boxdot
    Commented Oct 31, 2013 at 10:03

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