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Abstract: Topologically, a compact Riemann surface $X$ of genus $g$ is a $g$-holedtorus a sphere with $g$ handles. This paper is an introduction to the theoryof compact Riemann surfaces and algebraic curves. It presents the basic ideasand properties as an expository essay, explores some of their numerousconsequences and gives a concise account of the elementary aspects of differentviewpoints in curve theory. We discuss and prove most intuitively somegeometric-topological aspects of the algebraic functions and the associatedRiemann surfaces. Abelian and normalized differentials, Riemann-s bilinearrelations and the period matrix for $X$ are defined and some consequencesdrawn. The space of holomorphic 1-forms on $X$ has dimension $g$ as a complexvector space. Fundamental results on divisors on compact Riemann surfaces arestated and proved. The Riemann-Roch theorem is of utmost importance in thealgebraic geometric theory of compact Riemann surfaces. It tells us how manylinearly independent meromorphic functions there are having certainrestrictions on their poles. We present a simple direct proof of this theoremand explore some of its numerous consequences. We also give an analytic proofof the Riemann-Hurwitz formula. As an application, we compute the genus of someinteresting algebraic curves. Abel-s theorem classifies divisors by theirimages in the jacobian. The Jacobi inversion problem askes whether we can finda divisor that is the preimage for an arbitrary point in the jacobian. In thefirst appendix, we introduced intuitively and explicitly elliptic andhyperelliptic Riemann surfaces. In the second appendix, we study some resultsof resultant and discriminant as needed in the paper.



Author: A. Lesfari

Source: https://arxiv.org/



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