The Relative Canonical Ideal of the Artin-Schreier-Kummer-Witt family of curves

TL;DR

This paper investigates the canonical models of a family of algebraic curves in mixed characteristic, providing explicit generators for their canonical ideals and a combinatorial criterion for their generation.

Contribution

It extends classical Petri's theorem to a relative setting and offers an explicit construction of the canonical ideal for Artin-Schreier-Kummer-Witt curves.

Findings

Proved a relative version of Petri's theorem.

Constructed explicit generators for the canonical ideal.

Developed a combinatorial criterion for ideal generation.

Abstract

We study the canonical model of the Artin-Schreier-Kummer-Witt flat family of curves over a ring of mixed characteristic. We first prove the relative version of a classical theorem by Petri, then use the model proposed by Bertin-M\'ezard to construct an explicit generating set for the relative canonical ideal. As a byproduct, we obtain a combinatorial criterion for a set to generate the canonical ideal, applicable to any curve satisfying the assumptions of Petri's theorem.