Non-smooth regular curves via a descent approach
Giuseppe Borelli, Camilo David Dorado Moreira, Rodrigo Salomão
TL;DR
The paper develops a descent-based framework to classify non-smooth regular curves over non-perfect fields of characteristic $3$. It introduces the category of $X$-invariant sheaves under purely inseparable base changes and proves a Cartier-type equivalence that links base-change pullbacks to coherent sheaves on the base, enabling a systematic descent of geometric properties through $K^{1/3}$. A new local invariant, the differential degree $d(C,P)$, alongside the semigroup $Gamma_{C^{ar K},P^{ar K}}$, controls regularity, singularities, and the conductor, while invertible sheaf theory is used to connect intrinsic and extrinsic geometry. In the arithmetic genus three case, the authors classify geometrically elliptic curves via explicit plane quartic models in characteristic $3$, producing three invariant families $\uC_0,\uC_1,\uC_2$ with canonical embeddings and concrete normal forms. The work thus provides both a conceptual descent strategy for non-smooth regular curves and explicit moduli-like descriptions for characteristic-$3$ genus-$3$ examples, with potential implications for broader classifications in positive characteristic.
Abstract
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 Stöhr, 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 $1$.