# Is every Riemann surface a 2-sheeted covering?

Given an algebraic curve $X$ over $\mathbb{C}$, i.e. a Riemann surface and a fixed set of pairs of points $S=\{(p_1,q_1),...,(p_1,q_1)\}$ is there an algebraic curve Y, possibly singular, and a map $f: X \to Y$ such that

1. $f$ has degree 2.
2. $f$ send pairs of points $(p_i,q_i)$ to the same point $y=f(p_i)=f(q_i)$ for all $i$.

We can look at the function field of $X$, say $k(X)$ which if I'm not mistaken looks like a finite extension of $\mathbb{C}(t)$, so $k(X)=\mathbb{C}(t)[X]/f(X)$. If one finds an index two subfield this will yield a map $X\to Y$ using the smoothness of $X$ of degree 2. One idea would be to take $\mathbb{C}(t)[X^2]/f(X)$ but I'm not sure this is correct. In any case this does not allow one to describe the image of pairs of points since the map is determined birationally.

• There are many ways of seeing this is false. The quickest is to show that for a general curve $X$ of large enough genus, the group of automorphisms is trivial (though this result is non-trivial). Giving a double cover as yours would imply there is a non-trivial involution. – Mohan Nov 13 '15 at 20:56
• @Mohan What theorem are you referring to? I'd imagine you are talking about some variation of Hurwitz's automorphisms theorem for general curves. – Lee Wang Nov 14 '15 at 9:24

Here is an argument for say genus 3 curves. The dimension of the moduli spaceof genus 3 curves is 6 (=$3g-3$). If such an $X$ maps two to one onto a curve $Y$, then by Riemann-Hurwitz, $g(Y)<3$. If $g(Y)=2$, then again RH says the map has to be etale and thus determined by a genus 2 curve and a 2-torsion line bundle. Since 2-torsion line bundles are finite, the dimension of all these can at most be 4, which is the dimension of the moduli space of genus 2 curves. If $g(Y)=1$, then the map is ramified at 4 points and thus the dimension of such $X$ can be at most the dimension of moduli space of genus 1 curves plus 4, which is 5. Finally, if $g(Y)=0$, then $X$ is hyperelliptic and general curve of genus 3 is not hyperelliptic. Thus a general genus 3 curve can not map 2-1 onto a curve.