# Stack of vector bundles (on a curve) over a strictly semi-stable point of the moduli space

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.

• For instance I think there should be still a gerbe over some kind of mild desingularization of the GIT quotient. – IMeasy Feb 7 '13 at 18:16
• 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... – Christian Liedtke Feb 7 '13 at 18:30
• Yes that's basically what I want to know... I take a look straight away, thank you! – IMeasy Feb 7 '13 at 18:48
• 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 $[L\oplus L^{-1}]$. – IMeasy Feb 7 '13 at 19:16
• 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. – boxdot Oct 31 '13 at 10:03