Opers of higher types, Quot-schemes and Frobenius instability loci

In this paper we continue our study of the Frobenius instability locus in the coarse moduli space of semi-stable vector bundles of rank r and degree 0 over a smooth projective curve defined over an algebraically closed field of characteristic p > 0. In a previous paper we identified the “maximal" Frobenius instability strata with opers (more precisely as opers of type 1 in the terminology of the present paper) and related them to certain Quot-schemes of Frobenius direct images of line bundles. The main aim of this paper is to describe for any integer q ≥ 1 a conjectural generalization of this correspondence between opers of type q and Quot-schemes of Frobenius direct images of vector bundles of rank q. We also give a conjectural formula for the dimension of the Frobenius instability locus.