Most texts and university courses on curve theory generally conclude with the Riemann-Roch theorem, despite the fact that this theorem is the gateway to some of the most fascinating results in the theory of algebraic curves. From this embedding one can find equations for the image of the curve X. Therefore, many examples of algebraic curves are presented in the first chapters. We describe this form and give an algorithm for computing it. This is a surprising theorem: Riemann surfaces are given by locally patching charts. This particular surface is called the because it can be interpreted as wrapping the complex plane around the sphere. For example, hyperbolic Riemann surfaces are of the sphere they have non-constant meromorphic functions , but the sphere does not cover or otherwise map to higher genus surfaces, except as a constant.
You can also draw a picture of Riemann surface covering a sphere by projection onto a coordinate. Any one of these has a finite number of moduli and so a finite-dimensional Teichmüller space. Use MathJax to format equations. Containing numerous exercises and two exams, this book would make an excellent introductory text. The basic idea underlying the method is to reduce the problem to the known case of genus 1.
This next step consists of the black-box computation of the Abel map from a Riemann surface to its Jacobian. Keywords: Complex plane, open set, analytic or holomorphic function, Cauchy-Riemann equations, complex differentiable, convergent power series, Taylor expansion, Taylor coefficients, open map, biholomorphic map or holomorphic isomorphism, homeomorphism or topological isomorphism, complex coordinate chart, compatibility of charts, transition functions, Riemann surface structure. Two conformally equivalent Riemann surfaces are for all practical purposes identical. According to the uniformization theorem above, all hyperbolic surfaces are quotients of the unit disk. Riemann surfaces on which all bounded holomorphic functions are constant, or on which all bounded harmonic functions are constant, or on which all positive harmonic functions are constant, etc. Every compact Riemann surface admits a holomorphic embedding into P3. Reviews: The text grew out of lecture notes for courses which the author has taught several times during the last ten years.
In this book, Miranda takes the approach that algebraic curves are best encountered for the first time over the complex numbers, where the reader's classical intuition about surfaces, integration, and other concepts can be brought into play. In this alternative classification scheme, a Riemann surface is called parabolic if there are no nonconstant negative subharmonic functions on the surface and is otherwise called hyperbolic. The general numerical uniformization problem may be stated as to find numerical algorithms which permits to find, given explicitly one of the representations of a closed Riemann surface, some of the other ones. And a nonsingular variety in complex projective space has nonconstant globally meromorphic functions on it; just use ratios of homogenous coordinates. In this way, the book begins as a primer on Riemann surfaces, with complex charts and meromorphic functions taking center stage. In this way, the book begins as a primer on Riemann surfaces, with complex charts and meromorphic functions taking centre stage. For this function, it is also the height after rotating the plot 180° around the vertical axis.
Readership Graduate students studying complex variables and algebraic geometry. The authors propose a numerical method for the uniformization of Riemann surfaces and algebraic curves in genus two with highly accurate results. However, the majority of books written on the subject discuss algebraic curves and compact Riemann surfaces separately, as parts of distinct general theories. Choosing an equivalence class of metrics on X is the additional datum of the conformal structure. We use the term algebraic variety ambiguously to mean either projective algebraic variety or affine algebraic variety. Requiring a background of one semester of complex variable theory and a year of abstract algebra, this is an excellent graduate textbook for a second-semester course in complex variables or a year-long course in algebraic geometry. For example, a nonhyperelliptic curve of genus 3 is given by the vanishing of a quartic polynomial in P 2, a nonhyperelliptic curve of genus 4 is defined by the vanishing of a quadratic and a cubic polynomial in P 3.
So the sphere and torus admit complex structures, but the , and do not. More precisely, the of X is a finite of C t , the function field in one variable, i. In fact, the marked moduli space of the torus is biholomorphic to the upper half-plane or equivalently the open unit disk. There are lots of gems in this short statement. Kirwan introduces the theory, observes the algebraic and topological properties of complex algebraic curves, and shows how they are related to complex analysis.
To learn more, see our. For real curves of genus 2 and 3, however, such a form does exist up to a finite, but explicit, ambiguity for genus 3. . The lectures began with minimal technical requirements a working knowledge of elementary complex function theory and algebra together with some exposure to topology of compact surfaces and proceeded directly to the Riemann-Roch and Abel theorems. This book is based on a six-week series of lectures presented by the author to third- and fourth-year undergraduates and graduate students at Beijing University in 1982.
At the level of uniformizations, in general if we start with some explicit uniformization for a closed Riemann surface S, then it is not easy to get explicitly all or some of the other uniformizations of it. The aim of this paper is to present theoretical basis for computing a representation of a compact Riemann surface as an algebraic plane curve and to compute a numerical approximation for its period matrix. Now, in its evolved and fully ripe form, the text impressively reflects his apparently outstanding teaching skills as well as his admirable ability for combining great expertise in the field with masterly aptitude for representation and didactical sensibility. In this note we discuss a theoretical algorithm which relates a Schottky group from a given Fuchsian group both uniformizing the same closed Riemann surface. The ideas underlying the algorithm are based on the geometry of the moduli space of such curves. Thanks for contributing an answer to Mathematics Stack Exchange! Complete and sign the license agreement. The existence of these three types parallels the several.
With two punctures, it is the punctured plane or alternatively annulus or cylinder, which is parabolic. But the main examples come from projective curves, and slowly but surely the text moves toward the algebraic category. Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially such as the and other , or the. Namely it was Riemann's introduction of the topological and analytic points of view, showing that path integrals and differential forms could be profitably used to study projective algebraic curves, that deepened and revolutionized algebraic geometry forever. This feature of Riemann surfaces allows one to study them with either the means of or.
Consequently, the complex atlas is an oriented atlas. There is a different classification for Riemann surfaces which is typically used by complex analysts. For example, as I said there is a way to count the algebraic forms in terms of inner geometry of algebraic curve. A classification is possible for surfaces of finite type: those isomorphic to a compact surface with a finite number of points removed. But the beautiful fact is that those are two sides of the same medal. Cars allows one also to perform the Fenchel--Nielsen twist and other deformations on Riemann surfaces. Readership Graduate students studying complex variables and algebraic geometry.