- \( g_{X} \geq g_{Y} \)
- If \( g_{X} = g_{Y} > 1 \), then \( f \) is biholomorphic.
- If \( g_{X} = g_{Y} = 1 \), then \( f \) is unramified.
- If \( g_{X} = g_{Y} = 0 \), then \( f \) has ramification points.
- If \( g_{Y} = 0 \) and \( g_{X} > 0 \), then \( f \) has ramification points.
Riemann Surfaces and their Genus
-
- Community Team
- Posts: 314
- Joined: Tue Nov 10, 2015 8:25 pm
Riemann Surfaces and their Genus
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:
Create an account or sign in to join the discussion
You need to be a member in order to post a reply
Create an account
Not a member? register to join our community
Members can start their own topics & subscribe to topics
It’s free and only takes a minute
Sign in
Who is online
Users browsing this forum: No registered users and 16 guests