I'm trying to study the canonical map $\phi_K$ for the algebraic curve $\mathcal{C}:y^3=x^5-1$ and to do this I need to find a basis for $\Omega^1(\tilde{\mathcal{C}})$ where $\tilde{\mathcal{C}}$ is the nonsingular model (the curve is singular at $[0:0:1]$). I have found that $div(dx)=\sum_{k=0}^4[1:\xi_5^k:0]-2[0:1:0]$ and that $div(\frac{dx}{x^5-1})=3[0:1:0]$. Now, I am stuck at this point. How can I find other differentials? Are they correct? I have tried to solve this looking on Miranda's book and there were informations about divisor on nonsingular curves or curve with nodes.
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This post might help, although your curve's singularity could cause problems. – Viktor Vaughn Nov 25 '20 at 16:38
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The problem is exactly how to overcome the singularity to use that property – cartesio Nov 25 '20 at 16:56
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Here is a paper that gives a basis of holomorphic differentials on a superelliptic curve: https://www.ams.org/journals/mcom/2019-88-316/S0025-5718-2018-03351-8/viewer/#ltxid40 – Viktor Vaughn Nov 25 '20 at 17:07
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@RichardD.James Can you post it again, please? Your link requires a password – cartesio Nov 25 '20 at 17:20
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Oh, sorry! Here's the arxiv version: https://arxiv.org/abs/1707.07249 – Viktor Vaughn Nov 25 '20 at 17:53