I've been thinking about the period integrals on a hyperelliptic curve of the form $y^{2}=F(x)$, where $F(x)$ is a polynomial of degree-$2n$ with complex coefficients. Of course, a hyperelliptic curve is a double cover of $\mathbb{P}^{1}$ branched over $2n$ points. So the way I want to think of this (which I hope is correct) is that the polynomial $F(x)$ contains $2n+1$ complex pieces of data: namely it's coefficients. Using one of those pieces of data, we can choose a point on $\mathbb{P}^{1}$ to call $\infty$, and then remove this point, and un-fold the sphere into the complex plane. Our remaining $2n$ pieces of data correspond to choosing "branch points" on the complex plane, which can be connected with branch cuts. Now, to my understanding, the choice of branch cuts is irrelevant, after all, how can a polynomial contain a continuum of data to specify which branch cuts it has? Is there a more precise way to state that a hyperelliptic curve doesn't care what the choice of cuts is?

But just to make sure my reasoning is consistent, we can consider the period integrals on the curve

$S_{i}=\oint_{A_{i}} y(x) dx$

which are simply contour integrals around the i-th cut $A_{i}$. Am I correct in saying that these periods $S_{i}$ depend only on the branch points, not on the branch cuts specifically? Thanks in advance for any help!


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