Question

Background: Generalizing the notion of upper half plane to compact Riemann surfaces:

Suppose $p(x,y) \in \mathbb{R}[x,y]$ is a polynomial in 2 variables with real coefficients, defining a smooth complex plane algebraic curve $C_0 = {(x,y) \in \mathbb{C}^2:p(x,y)=0}$. Let $C$ be the projective closure of $C_0$ in $P^2\mathbb{C}$, and assume that $C$ is also smooth. Since $C$ is defined over the real numbers, it comes equipped with an involution $\sigma:C\rightarrow C$, $\sigma(x,y) = (\overline{x},\overline{y})$. Denote by $X$ the compact Riemann surface associated to $C$, and let $X_\mathbb{R}$ be the set of fixed points of $\sigma$.

If the space $X - X_\mathbb{R}$ has exactly two connected components, then $X$ is called a real compact Riemann surface of dividing type, and the two connected components are denoted by $X_+$ and $X_{-}$ (the decision between "the positive half plane" and "the negative half plane" arbitrarily).

And finally, to the question:

I am given a real compact Riemann surface of dividing type $X$, and interested in interpolation problems of meromorphic functions with conditions such as "all the poles of $f$ lie in the upper half plane". Does anybody knows of any previous work in the area? Any known techniques to relate these topological and algebraic constructions?

Answer 1

Let me give a version of the question in the comment:

Let $X$ be a curve of genus $g$ with a real separting involution, and conisder the map $Sym^n(X_+)\to Jac^n(X)$.

For wich $n$ is this map surjective? Or, in other words, what is the minimal number of poles of a meromorphic function with poles in $X_+$ that garanties that zeros can happen at any collection of points?

This sounlds like a very nice question. In the case $g=1$ you can always take $n=2$. Also for any $g$ you sould take $n>g$ because $Sym^g(X)$ maps to $Jac^g(X)$ with degree $1$.

Added. The notation $Sym^n(X)$ means the symmetric power of $X$. Let me explain also why what is above is a reformulation of the original question. Indeed, a divisor $\sum_i x_i-\sum_i y_i$ on $X$ is a divisor of a meromorfic function iff it represent zero in $Jac^0(X)$. So if we want to chose arbitraly zeros $x_i$ of a meromorphic function $f$ keeping the poles $y_i$ in $X_+$ it is enouth to know that $\sum_i y_i$ can take any value in $Jac^n(X)$ (to cancel the point $\sum_i x_i$). This is eactly the condition that $Sym^n(X_+)\to Jac^n(X)$ is surjective.