Dormant opers and Gauss maps in positive characteristic

Abstract

The Gauss map of a given projective variety is the rational map that sends a smooth point to the tangent space at that point, considered as a point of the Grassmann variety. 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, as well as Frobenius-projective structures. We first prove a correspondence between dormant opers on a smooth projective variety $X$ and closed immersions from $X$ 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, the correspondence gives us a Frobenius-projective structure on a Fermat hypersurface. This example embodies an exotic phenomenon of algebraic geometry in positive characteristic.