Non-smooth regular curves via a descent approach

This paper aims to continue the classification of non-smooth regular curves, but over fields of characteristic three. These curves were originally introduced by Zariski as generic fibers of counterexamples to Bertini's theorem on the variation of singular points of linear series. Such a classification has been introduced by St\"ohr, taking advantage of the equivalent theory of non-conservative function fields, which in turn occurs only over non-perfect fields $K$ of characteristic $p>0$. We propose here a different way of approach, relying on the fact that a non-smooth regular curve in $\mathbb{P}^n_K$ provides a singular curve when viewed inside $\mathbb{P}^n_{K^{1/p}}$. Hence we were naturally induced to the question of characterizing singular curves in $\mathbb{P}^n_{K^{1/p}}$ coming from regular curves in $\mathbb{P}^n_K$. To understand this phenomenon we consider the notion of integrable connections with zero $p$-curvature to extend Katz's version of Cartier's theorem for purely inseparable morphisms, where we solve the above characterization for the slightly general setup of coherent sheaves. Moreover, we also had to introduce some new local invariants attached to non-smooth points, as the differential degree. As an application of the theory developed here, we classify complete, geometrically integral, non-smooth regular curves $C$ of genus $3$, over a separably closed field $K$ of characteristic $3$, whose base extension $C \times_{\operatorname{Spec} K}{\operatorname{Spec} \overline{K}}$ is non-hyperelliptic with normalization having geometric genus $1$.