Harder-Narasimhan filtration for rank 2 tensors and stable coverings

Abstract: We construct a Harder-Narasimhan filtration for rank $2$ tensors, where there does not exist any such notion a priori, as coming from a GIT notion of maximal unstability. The filtration associated to the 1-parameter subgroup of Kempf giving the maximal way to destabilize, in the GIT sense, a point in the parameter space of the construction of the moduli space of rank $2$ tensors over a smooth projective complex variety, does not depend on certain integer used in the construction of the moduli space, for large values of the integer. Hence, this filtration is unique and we define the Harder-Narasimhan filtration for rank $2$ tensors as this unique filtration coming from GIT. Symmetric rank $2$ tensors over smooth projective complex curves define curve coverings lying on a ruled surface, hence we can translate the stability condition to define stable coverings and characterize the Harder-Narasimhan filtration in terms of intersection theory.