Let $X$ be a compact Riemann surface of genus $g>0$, and let $f\colon X \to \mathbb{P}^1$ be a branched covering of degree $d$. Define the ramification divisor $R_f$ on $X$ by $f$, where $\deg R_f = 2g+2d-2 :=N$. Denote $R_d = \{R_f \mid f\colon X \to \mathbb{P}^1, \deg f = d \}$.
I am interested in whether we can characterize the subset $R_d$ in the space of all effective divisors of degree $N$ on $X$, denoted as $\operatorname{Sym}^N(X)$. Specifically, is $R_d$ an algebraic subset or Zariski closed in $\operatorname{Sym}^N(X)$?
I would appreciate references on this topic, and any comments are also welcome.
Best Answer
Here is an elementary answer to your specific question. $R_d$ is not closed. For instance when $X = \mathbb P^1$ and $d = 2$, a divisor $p+q$ appears as a ramification divisor if and only if $p,q$ are distinct.
To show that $2p$ cannot occur, we may WLOG assume that $p=0$ and the map takes $0$ to $0$, so in coordinates takes the form $x \mapsto\frac{ax^2 + bx}{u x^2 + vx + w}$ for $w,a \neq 0$. Thus the Taylor expansion of the map at $0$ is at most quadratic, and so the ramification multiplicity is $\leq 1$. To show that $p +q$ can occur for any $p,q$ distinct, assume WLOG that $q = \infty$. Then if $p = c$ the map $\frac{x^2}{2} + cx$ works.
In this case, the image is open, but this cannot happen for higher genus curves for dimension reasons. By Chevalley's theorem, $R_d$ is at least constructible. I would also be interested to know more about it.