  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.
