On non-smooth regular curves via a descent approach - Dorado, Camilo David Moreira (2024)
This work is devoted to the study of the problem of classifying non-smooth regular curves in projective spaces. This problem has been studied to look for counterexamples to Bertini’s theorem on the variation of singular points of linear series. Such a classification has been introduced by K.-O. Stöhr, taking advantage of the fact that a non-smooth regular curve is an equivalent object to a non-conservative function field, which in turn occurs only over non-perfect fields K of characteristic p > 0. We propose here a different way to approach this problem, relying on the fact that a non-smooth regular curve in Pn K provides a singular curve when viewed in P n K1/p , after extending its base field to K1/p For this purpose, we will proceed with the following three steps. The first one is to study K-invariant sub-schemes of P n K1/p , which are those coming from base change of sub-schemes of P n K. To do this we will need two ingredients: to see P nK as a quotient of P n K1/p by a p closed foliation and to use K-invariant connections on coherent sheaves, introduced by N. Katz. The second one is to study local invariants at non-smooth regular points of algebraic curves. As an application of the theory developed in the previous two items, we classify complete, geometrically integral, non-smooth regular curves C of genus 3, over a separably closed field K, where C ×Spec K Spec K is a non-hyperelliptic curve with normalization having genus 1.