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arXiv · 2304.02990

Representations on canonical models of generalized Fermat curves and their syzygies

Kostas Karagiannis
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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.

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Kostas Karagiannis. 2023-04-06. Representations on canonical models of generalized Fermat curves and their syzygies. https://arxiv.org/abs/2304.02990

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