Say I have compact Riemann surfaces $X$, $Y$. Is there necessarily a Riemann surface $Z$ which maps holomorphically onto both $X$, $Y$?



Existence of Riemann surface, holomorphic maps
Here is some construction. $X$ and $Y$ are smooth projective algebraic complex curves. So the complex surface $X \times Y$ is projective (e.g. by Segre embedding) and so admits a very ample line bundle $L$. By Bertini theorem, the vanishing locus of a general section of $L$ is a smooth projective complex curve $Z$ (so a compact Riemann surface) in $X \times Y$. The composition of the inclusion of $Z$ in $X \times Y$ with the projection on $X$ (or $Y$) is not constant: if it were constant, $Z$ would be contained in some $Y$ fiber, having zero intersection with a generic $Y$ fiber, and contradicting ampleness of $L$.


