Inseparable Gauss maps and dormant opers

The present paper aims to generalize a result by H. Kaji on Gauss maps in positive characteristic and establish an interaction with the study of dormant opers and Frobenius-projective structures. We prove a correspondence between dormant opers on a smooth projective variety and closed immersions into a projective space with purely inseparable Gauss map. By using this, we determine the subfields of the function field of a smooth curve in positive characteristic induced by Gauss maps. Moreover, this correspondence gives us a Frobenius-projective structure on a Fermat hypersurface.