Surfaces de Riemann compactes, courbes alg\'{e}briques complexes et leurs Jacobiennes

TL;DR

This paper provides an introductory overview of compact Riemann surfaces and algebraic curves, discussing their properties, key theorems, and applications in complex geometry and algebraic geometry.

Contribution

It offers a clear, expository account of fundamental concepts, proofs, and consequences in the theory of Riemann surfaces and algebraic curves, including new simplified proofs.

Findings

Dimension of holomorphic 1-forms equals genus g

Proof of Riemann-Roch theorem and Riemann-Hurwitz formula

Explicit computation of genus for specific algebraic curves

Abstract

Topologically, a compact Riemann surface $X$ of genus $g$ is a $g$-holed torus (a sphere with $g$ handles). This paper is an introduction to the theory of compact Riemann surfaces and algebraic curves. It presents the basic ideas and properties as an expository essay, explores some of their numerous consequences and gives a concise account of the elementary aspects of different viewpoints in curve theory. We discuss and prove most intuitively some geometric-topological aspects of the algebraic functions and the associated Riemann surfaces. Abelian and normalized differentials, Riemann's bilinear relations and the period matrix for $X$ are defined and some consequences drawn. The space of holomorphic 1-forms on $X$ has dimension $g$ as a complex vector space. Fundamental results on divisors on compact Riemann surfaces are stated and proved. 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. We 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. Abel's theorem classifies divisors by their images in the jacobian. The Jacobi inversion problem askes whether we can find a divisor that is the preimage for an arbitrary point in the jacobian. In the first appendix, we introduced intuitively and explicitly elliptic and hyperelliptic Riemann surfaces. In the second appendix, we study some results of resultant and discriminant as needed in the paper.