I have to give a talk about Riemann-Roch theorem for curves. The plane of my presentation is to give first the historical motivation behind the theorem (difference in the behaviour of projective line and positive genus curves), then some words on the proof and in the end some words on Riemann surfaces and algebraic surfaces.
How can I explain the important concept (divisor, Riemann-Roch space) in every distinct context without being repetitive? The main example in my talk wold be the sphere and the rational funcition field.
I am worried on how i can describe the example in an effective way without referring to all the theory I will describe later(valuation rings, discrete valuations, etc.).
Do you have any suggestions?