The Riemann Roch Theorem (for algebraic curves)

Abstract

The Riemann-Roch theorem is a useful tool to calculate the dimension of the space of meromorphic functions with prescribed zeros and poles. There are severals versions of the theorem such as the Riemann-Roch theorem for line bundles, for (algebraic) curves, for surfaces and for higher dimensions. In this thesis, we will focus on the Riemann-Roch theorem for algebraic curves over an algebraically closed eld, which is a very important result in complex analysis and algebraic geometry. The study of the elds of rational functions on curves can be very useful in the proof. So we will recall some pre-knowledges in commutative algebra and some facts about a ne varieties. Then talk about function elds, discrete valuation rings and Weil di erentials to prove the theorem, using the methods of Andre Weil.