Despite the progress made in the recent years, the list open problems in characteristic p algebraic arithmetic geometry remains extensive. One of the strategies that has proven to be succesful, initially proposed by J. P. Serre in his Mexico paper, is the technique of lifting to characteristic 0: problems like the Galois module structure of (poly)differentials and Green’s syzygy conjecture are well understood in characteristic 0 but remain open in characteristic p. The above problems share a second interesting property: they involve the canonical sheaf Ω, which appears prominently in the classical theorem of M. Noether, F. Enriques and K. Petri. In this talk, following a review of the theory of lifting curves with automorphisms and the Noether-Enriques-Petri theorem, we will present joint work with H. Charalambous and A. Kontogeorgis, in which we study the relative canonical embedding of the flat family of curves obtained from lifting an Artin-Schreier curve to a Kummer curve. Combining elements of Gröbner theory with deformation-theoretic arguments we will give an explicit set of generators for the relative canonical ideal, obtaining in the process a relative version of Petri’s theorem.