  We study canonical models of $\left(\mathbb{Z}/k\mathbb{Z}\right)^n$- covers
of 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 either
zero or does not divide $k$. We determine explicitly the structure of the
respective 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 a
graded module over the polynomial ring; in the latter case, we give generators
for its first syzygy module, which we also decompose as a direct sum of
irreducible representations.
