Le th\'eor\`eme de Riemann-Roch et ses applications

TL;DR

This paper provides a straightforward proof of the Riemann-Roch theorem, explores its consequences, and applies it to compute the genus of algebraic curves, highlighting its significance in algebraic geometry.

Contribution

It offers a simple direct proof of the Riemann-Roch theorem and demonstrates its applications in algebraic geometry and the computation of curve genus.

Findings

Presented a simple proof of the Riemann-Roch theorem.

Derived the Riemann-Hurwitz formula analytically.

Computed the genus of specific algebraic curves.

Abstract

The Riemann-Roch theorem is of utmost importance in the algebraic geometric theory of compact Riemann surfaces. It tells us how many linearly independent meromorphic functions there are having certain restrictions on their poles. The aim of this article is to present a simple direct proof of this theorem and explore some of its numerous consequences. We also give an analytic proof of the Riemann-Hurwitz formula. As an application, we compute the genus of some interesting algebraic curves.