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Let us start with a multiple cover $C$ of the $x$-plane branched at $z$ and $-z$, and so described by an equation $y^N = x^2 - z^2$.

For $N=2$, it is known that there are globally-defined holomorphic vector fields on $C$ that are of the form $V_n = u^{n+1} \frac{d}{du} $ where $u= x \pm y$ for any $n\in \mathbb{Z}$.

Can we also construct such well-defined vector fields on $C$ for any $N$?

Naively, they can be generalized to $V_n = u^{n+1} \frac{d}{du} $ where $u= \left(x^{\frac 2N} - w^i y\right)^{\frac N2}$ for $i=0,1,..,N-1$ and $w$ is the N-th root of unity. Am I missing something here?

I am a theoretical physicist, and not good at math. Need your help.

Thank you very much in advance.

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  • $\begingroup$ Is your initial equation written correctly? For $N=2$ this does not look like a hyperelliptic curve to me. $\endgroup$
    – Lazzaro Campeotti
    May 5, 2016 at 13:59
  • $\begingroup$ Oh you're right. I shouldn't call the curve with N=2 as a hyper elliptic curve because the polynomial of x is of order two. Thank you very much for your comment. However it is still true that the holomorphic vector field V_n is well-defined on $y^2=x^2-e^2$. $\endgroup$
    – sealiy
    May 5, 2016 at 14:15

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