Representations on canonical models of generalized fermat curves and their syzygies

ABSTRACT 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.