Dormant opers and Gauss maps in positive characteristic

TL;DR

This paper explores the relationship between dormant opers, Frobenius-projective structures, and Gauss maps in positive characteristic, revealing new correspondences and geometric phenomena unique to this setting.

Contribution

It establishes a correspondence between dormant opers and purely inseparable Gauss maps, and constructs Frobenius-projective structures on Fermat hypersurfaces in positive characteristic.

Findings

Correspondence between dormant opers and purely inseparable Gauss maps.

Determination of subfields of function fields induced by Gauss maps.

Construction of Frobenius-projective structures on Fermat hypersurfaces.

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.