The structure of regular genus-one curves over imperfect fields

TL;DR

This paper classifies regular genus-one curves over imperfect fields, providing an intrinsic description using twisted forms and moduli, which advances understanding of genus-one fibrations in algebraic geometry.

Contribution

It extends the classification of genus-one curves to imperfect fields, introducing an intrinsic approach via twisted forms and moduli, beyond the known geometrically reduced case.

Findings

Classification of regular but not geometrically regular genus-one curves over imperfect fields

Description using twisted forms and infinitesimal group scheme actions

Application to genus-one fibrations over higher-dimensional bases

Abstract

Working over imperfect fields, we give a comprehensive classification of genus-one curves that are regular but not geometrically regular, extending the known case of geometrically reduced curves. The description is given intrinsically, in terms of twisted forms of standard models with respect to infinitesimal group scheme actions, and not via extrinsic equations. The main new idea is to analyze and exploit moduli for fields of representatives in Cohen's Structure Theorem. The results serve for the understanding of genus-one fibrations over higher-dimensional bases.