# Is this set of curves discrete?

Let $\alpha_1, \dots, \alpha_n$ be complex numbers whose sum is zero and $u_1, \dots, u_{2g-2+n}$ be pariwise distinct nonzero complex numbers. Consider the the set of smooth genus g curves with n+1 marked points $P_0, x_1\dots, x_n$ and a marked canonical homology basis, with the following properties:

• Take the unique Abelian differential of the third kind $\omega$ on the curve having a simple pole at $x_i$ with residue $\alpha_i$, $i=1, \dots, n$ and zero a-periods. Then all b-periods of $\omega$ are zero.
• Let $P_1, \dots, P_{2g-2+n}$ be the zeroes of $\omega$. Then there exist paths $\gamma_i$ on the curve connecting $P_0$ and $P_i$ such that $\exp(\int_{\gamma_i}\omega)=u_i, i=1, \dots, 2g-2+n$.

I would like to show that this set is discrete, and possibly to find some explicit description of it. For integers values of $\alpha_i$ this is essentially Riemann's existence theorem.

Thanks

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What are a-periods and b-periods? – Will Sawin Jun 23 '12 at 2:27
Sorry, the a- and b-cycles are the canonical basis of the homology of the curve. a- and b-periods are integrals over these cycles. – user24647 Jun 23 '12 at 7:46