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Bumped in to this problem while trying to understand hyper-elliptic Riemann surfaces, and being a bit new to the subject of Riemann surfaces, was not that much confident on a couple of things.

Consider the hyper-elliptic Riemann surface $\mathcal{R}$ defined as the affine curve

$y^2=z(z-1)(z-a)(z-b)(z-c)(z-d)=P(z)$,

where $1<a<b<c<d\in \mathbb{R}$ and compactified at $z=\infty$ with two points and with local coordinate $\zeta=\frac{1}{z}$.

  1. Find the genus.

  2. Find a basis for the holomorphic differentials of $\mathcal{R}$.

  3. Consider the divisor of points $D=(z_1,y_1)+(z_2,y_2)$, where $y_j$ are one of the solutions of the equation $y_j^2=P(z_j)$. Find $i(D)$ (the dimension of the space of holomorphic differentials vanishing at those two points) assuming that $z_1\neq z_2$.

  4. Find $i(D)$ assuming that $z_1=z_2$ and $y_1=-y_2$.

Obviously, (1) is straightforward. Any help/suggestions on the other three parts will be very helpful. I believe I have to use Riemann-Roch for (3) and (4) but not sure how.

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  • $\begingroup$ Perhaps this will help. $\endgroup$ Commented Nov 5, 2018 at 1:26

1 Answer 1

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A suitable basis is $\frac{z^k}{y}dz$, $k=0,\dots,g-1$. You need to verify that these differentials are holomorphic. (3) and (4) can be solved using this basis.

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