Frobenius Factorization Loci for Gauss Maps of Smooth Plane Curves

We construct canonical closed subschemes representing relative Frobenius factorization of a morphism from a smooth projective curve family, including over nonreduced bases.

For Gauss maps of smooth plane curves of degree \(d\), we identify these subschemes with explicit linear coefficient loci. For \(q=p^e>2\), the \(e\)th locus is empty unless \(q\mid d-1\); otherwise it is the smooth open in the space of equations \(\sum_{i=0}^2 X_iR_i(X_0^q,X_1^q,X_2^q)\), of dimension \(3\binom{(d-1)/q+2}{2}-1\).

The classification on geometric points is due to Pardini and Homma. The scheme-level argument uses the pulled-back Euler sequence, Serre duality and a regular-sequence calculation to rule out infinitesimal thickenings. We determine exact-height dimensions and geometric irreducibility, treat characteristic two separately, and give explicit smooth families with height jumps.