Every birational map is an isomorphism for algebraic curves

Question

I am looking for a proof and a counter example:

• A proof of the statement: "Every birational map is an isomorphism for algebraic curves".
• A counter example for the above statement in the case of algebraic surfaces.

Many thanks for any help or references.

Answer 1

This is covered in Fulton's "Algebraic curves"

• For your first question combine 6.6 Proposition 12 and 7.1 Corollary 3 (your statement as it stands is wrong)
• For your second question you can either check out Problem 6.39 or read about blowups in Section 7.2+7.3