I'm unclear on to what extent the periods and dual-periods of a Riemann surface determine the complex structure of the surface. Perhaps to take a nice example, I'll consider a hyperelliptic curve $y^{2}=f(x)$. There's $2n$ complex pieces of data here, which correspond to choosing $2n$ branch points on the projective line.

We can connect these points pairwise with a branch cuts $A_{i}$ with $i=1,\ldots n$, and then we consider the "dual cuts" $B_{i}$ connecting the midpoint of each branch cut to $\infty \in \mathbb{P}^{1}$. This then let's us compute the periods and dual-periods of the hyperelliptic curve by contour integrals around the $A_{i}$ and $B_{i}$, like so:

$$\mathscr{A}_{i} = \oint_{A_{i}} y(x) dx$$

$$\mathscr{B}_{i} = \oint_{B_{i}} y(x) dx.$$

Now, the parameter space of the hyperelliptic curve is simply given by the placement of the $2n$ branch points (never mind quotienting by $\rm{PGL}(2,\mathbb{C})$!) and I notice above that the periods and dual-periods combine to give $2n$ pieces of data, however, isn't it really only some ratio of these quantities that we need? More specifically, I'm wondering:

(I) Since the branch points don't move, I had thought that by basic complex analysis, the $\mathscr{A}_{i}$ were independent of deforming the cuts $A_{i}$. However, is it maybe that deforming the branch cuts will lead to totally different $\mathscr{A}_{i}$ and $\mathscr{B}_{i}$, but that somehow these changes will not affect the hyperelliptic curve? Because after all, the curve only cares about the branch points, not the cuts (I believe).

(II) Secondly, in my definition of the dual-periods, I chose to start the "dual cuts" $B_{i}$ at the midpoint of the cuts $A_{i}$. This seems totally arbitrary. Could I have chosen this starting point anywhere along the cut, excluding the branch points?


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