Riemann Surfaces and their Genus

[Tsakanikas Nickos]

Let \( X \) and \( Y \) be two compact Riemann surfaces of genus \( g_{X} \) and \( g_{Y} \), respectively, and let \( \displaystyle f \ \colon X \longrightarrow Y\) be a non-constant holomorphic map. Show the following:

1. \( g_{X} \geq g_{Y} \)
2. If \( g_{X} = g_{Y} > 1 \), then \( f \) is biholomorphic.
3. If \( g_{X} = g_{Y} = 1 \), then \( f \) is unramified.
4. If \( g_{X} = g_{Y} = 0 \), then \( f \) has ramification points.
5. If \( g_{Y} = 0 \) and \( g_{X} > 0 \), then \( f \) has ramification points.