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We study canonical models of $\left(\mathbb{Z}/k\mathbb{Z}\right)^n$- coversof the projective line, tamely ramified at exactly $n+1$ points each of index$k$, when $k,n\geq 2$ and the characteristic of the ground field $K$ is eitherzero or does not divide $k$. We determine explicitly the structure of therespective homogeneous coordinate ring first as a graded $K$-algebra, next as a$\left(\mathbb{Z}/k\mathbb{Z}\right)^n$- representation over $K$, and then as agraded module over the polynomial ring; in the latter case, we give generatorsfor its first syzygy module, which we also decompose as a direct sum ofirreducible representations.

Representations on canonical models of generalized Fermat curves and their syzygies