A functorial approach to the stability of vector bundles
On a normal projective variety the locus of µ-stable vector bundles that remain µ-stable on all Galois covers prime to the characteristic p (p=0 or p>0) is open in the moduli space of Gieseker semistable sheaves. On a smooth projective curve of genus at least 2 this locus is big in the moduli space of stable vector bundles.
The moduli space of µ-stable vector bundles admits a canonical stratification defined via the decomposition type of a vector bundle. We give mostly sharp dimension estimates for these strata over a smooth projective curve of genus at least 2.
As an application we obtain a mostly sharp estimate of the dimension of the closure of prime to p trivializable stable vector bundles in the moduli space of stable vector bundles over a smooth projective curve of genus at least 2. In rank 2 we give a description of its irreducible components.