Frobenius-Projective Structures on Curves in Positive Characteristic

In the present paper we study Frobenius-projective structures on projective smooth curves in positive characteristic. The notion of Frobenius-projective structures may be regarded as an analogue, in positive characteristic, of the notion of complex projective structures in the classical theory of Riemann surfaces. By means of the notion of Frobenius-projective structures we obtain a relationship between a certain rational function, i.e., a pseudocoordinate, and a certain collection of data which may be regarded as an analogue, in positive characteristic, of the notion of indigenous bundles in the classical theory of Riemann surfaces, i.e., a Frobenius-indigenous structure. As an application of this relationship, we also prove the existence of certain Frobenius-destabilized locally free coherent sheaves of rank two.