Algebraic Curves And Riemann Surfaces

Author : Rick Miranda
Publisher : American Mathematical Soc.
Page : 390 pages
File Size : 52,8 Mb
Release : 1995
Category : Mathematics
ISBN : 9780821802687

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Algebraic Curves and Riemann Surfaces by Rick Miranda Pdf

The book was easy to understand, with many examples. The exercises were well chosen, and served to give further examples and developments of the theory. --William Goldman, University of Maryland 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. Therefore, many examples of algebraic curves are presented in the first chapters. In this way, the book begins as a primer on Riemann surfaces, with complex charts and meromorphic functions taking center stage. But the main examples come from projective curves, and slowly but surely the text moves toward the algebraic category. Proofs of the Riemann-Roch and Serre Duality Theorems are presented in an algebraic manner, via an adaptation of the adelic proof, expressed completely in terms of solving a Mittag-Leffler problem. Sheaves and cohomology are introduced as a unifying device in the latter chapters, so that their utility and naturalness are immediately obvious. 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.

Author : Rick Miranda
Publisher : American Mathematical Soc.
Page : 418 pages
File Size : 43,6 Mb
Release : 1995
Category : Mathematics
ISBN : 0821802682

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Algebraic Curves and Riemann Surfaces by Rick Miranda Pdf

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. Therefore, many examples of algebraic curves are presented in the first chapters. In this way, the book begins as a primer on Riemann surfaces, with complex charts and meromorphic functions taking centre stage. But the main examples come fromprojective curves, and slowly but surely the text moves toward the algebraic category. Proofs of the Riemann-Roch and Serre Dualtiy Theorems are presented in an algebraic manner, via an adaptation of the adelic proof, expressed completely in terms of solving a Mittag-Leffler problem. Sheaves andcohomology are introduced as a unifying device in the later chapters, so that their utility and naturalness are immediately obvious. Requiring a background of one term of complex variable theory and a year of abstract algebra, this is an excellent graduate textbook for a second-term course in complex variables or a year-long course in algebraic geometry.

Author : Rick Miranda
Publisher : American Mathematical Soc.
Page : 390 pages
File Size : 42,8 Mb
Release : 1995
Category : Mathematics
ISBN : 1470411407

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Algebraic Curves and Riemann Surfaces by Rick Miranda Pdf

The author of this monograph argues 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. Therefore, many examples of algebraic curves are presented in the first chapters. In this way, the book begins as a primer on Riemann surfaces, with complex charts and meromorphic functions taking centre stage. But the main examples come from projective curves, and slowly but surely the text moves toward the algebraic category. Proofs of the Riemann-Roch and Serre Dualtiy Theorems are presented in an algebraic manner, via an adaptation of the adelic proof, expressed completely in terms of solving a Mittag-Leffler problem. Sheaves and cohomology are introduced as a unifying device in the later chapters, so that their utility and naturalness are immediately obvious. Requiring a background of one term of complex variable theory and a year of abstract algebra, this is a graduate textbook for a second-term course in complex variables or a year-long course in algebraic geometry.

Author : Anil Nerode,Noam Greenberg
Publisher : Springer Nature
Page : 453 pages
File Size : 47,6 Mb
Release : 2023-01-16
Category : Mathematics
ISBN : 9783031116162

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Algebraic Curves and Riemann Surfaces for Undergraduates by Anil Nerode,Noam Greenberg Pdf

The theory relating algebraic curves and Riemann surfaces exhibits the unity of mathematics: topology, complex analysis, algebra and geometry all interact in a deep way. This textbook offers an elementary introduction to this beautiful theory for an undergraduate audience. At the heart of the subject is the theory of elliptic functions and elliptic curves. A complex torus (or “donut”) is both an abelian group and a Riemann surface. It is obtained by identifying points on the complex plane. At the same time, it can be viewed as a complex algebraic curve, with addition of points given by a geometric “chord-and-tangent” method. This book carefully develops all of the tools necessary to make sense of this isomorphism. The exposition is kept as elementary as possible and frequently draws on familiar notions in calculus and algebra to motivate new concepts. Based on a capstone course given to senior undergraduates, this book is intended as a textbook for courses at this level and includes a large number of class-tested exercises. The prerequisites for using the book are familiarity with abstract algebra, calculus and analysis, as covered in standard undergraduate courses.

Author : Renzo Cavalieri,Eric Miles
Publisher : Cambridge University Press
Page : 197 pages
File Size : 42,8 Mb
Release : 2016-09-26
Category : Mathematics
ISBN : 9781107149243

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Riemann Surfaces and Algebraic Curves by Renzo Cavalieri,Eric Miles Pdf

Classroom-tested and featuring over 100 exercises, this text introduces the key algebraic geometry field of Hurwitz theory.

Author : Frances Clare Kirwan
Publisher : Cambridge University Press
Page : 278 pages
File Size : 52,5 Mb
Release : 1992-02-20
Category : Mathematics
ISBN : 0521423538

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Complex Algebraic Curves by Frances Clare Kirwan Pdf

This development of the theory of complex algebraic curves was one of the peaks of nineteenth century mathematics. They have many fascinating properties and arise in various areas of mathematics, from number theory to theoretical physics, and are the subject of much research. By using only the basic techniques acquired in most undergraduate courses in mathematics, Dr. Kirwan introduces the theory, observes the algebraic and topological properties of complex algebraic curves, and shows how they are related to complex analysis.

Author : Kichoon Yang
Publisher : World Scientific
Page : 184 pages
File Size : 50,6 Mb
Release : 1988-11-01
Category : Mathematics
ISBN : 9789814520034

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Compact Riemann Surfaces and Algebraic Curves by Kichoon Yang Pdf

This volume is an introduction to the theory of Compact Riemann Surfaces and algebraic curves. It gives a concise account of the elementary aspects of different viewpoints in curve theory. Foundational results on divisors and compact Riemann surfaces are also stated and proved. Contents:Topological Preliminaries —Singular Homology and Relative HomologyCellular HomologyDe Rham CohomologyCommutative Algebra — An Introduction —Closed Ideals and VarietiesCoordinate RingsDimension TheoryIntersection NumbersSingular Plane Curves —The Classical Plücker FormulaeDivisors on a Compact Complex Manifold —Divisors and Holomorphic Line BundlesLinear Systems on a Compact Riemann Surface and Holomorphic MapsCompact Riemann Surfaces —The Jacobian Variety and Abel's TheoremThe Riemann-Roch Theorem and the Canonical EmbeddingHyperelliptic Riemann Surfaces and the Weierstrass PointsGeometry of Projective Curves — The Complex Flag ManifoldMetric Geometry of Projective CurvesPlücker Formulae for Projective Algebraic CurvesHarmonic Maps from a Compact Riemann SurfaceA Brief Look at Algebraic Surfaces —The Intersection FormBlow-Ups and Rational MapsThe Kodaira Dimension of an Algebraic Surface Readership: Mathematicians. Keywords:Compact Riemann Surfaces;Algebraic Curves;Compact Complex Manifold;Algebraic Surfaces

Author : Martin Schlichenmaier
Publisher : Springer
Page : 149 pages
File Size : 47,8 Mb
Release : 2014-10-09
Category : Mathematics
ISBN : 3662137283

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An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces by Martin Schlichenmaier Pdf

This lecture is intended as an introduction to the mathematical concepts of algebraic and analytic geometry. It is addressed primarily to theoretical physicists, in particular those working in string theories. The author gives a very clear exposition of the main theorems, introducing the necessary concepts by lucid examples, and shows how to work with the methods of algebraic geometry. As an example he presents the Krichever-Novikov construction of algebras of Virasaro type. The book will be welcomed by many researchers as an overview of an important branch of mathematics, a collection of useful formulae and an excellent guide to the more extensive mathematical literature.

Author : V.I. Danilov,V.V. Shokurov
Publisher : Springer Science & Business Media
Page : 314 pages
File Size : 44,9 Mb
Release : 2013-12-01
Category : Mathematics
ISBN : 9783642578786

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Algebraic Geometry I by V.I. Danilov,V.V. Shokurov Pdf

"... To sum up, this book helps to learn algebraic geometry in a short time, its concrete style is enjoyable for students and reveals the beauty of mathematics." --Acta Scientiarum Mathematicarum

Author : Emilio Bujalance,Emilio Bujalance García,Francisco Javier Cirre,José Manuel Gamboa,Grzegorz Gromadzki
Publisher : Springer Science & Business Media
Page : 181 pages
File Size : 50,6 Mb
Release : 2010-10-06
Category : Mathematics
ISBN : 9783642148279

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Symmetries of Compact Riemann Surfaces by Emilio Bujalance,Emilio Bujalance García,Francisco Javier Cirre,José Manuel Gamboa,Grzegorz Gromadzki Pdf

This monograph deals with symmetries of compact Riemann surfaces. A symmetry of a compact Riemann surface S is an antianalytic involution of S. It is well known that Riemann surfaces exhibiting symmetry correspond to algebraic curves which can be defined over the field of real numbers. In this monograph we consider three topics related to the topology of symmetries, namely the number of conjugacy classes of symmetries, the numbers of ovals of symmetries and the symmetry types of Riemann surfaces.

Author : Otto Forster
Publisher : Springer Science & Business Media
Page : 262 pages
File Size : 47,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461259619

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Lectures on Riemann Surfaces by Otto Forster Pdf

This book grew out of lectures on Riemann surfaces given by Otto Forster at the universities of Munich, Regensburg, and Münster. It provides a concise modern introduction to this rewarding subject, as well as presenting methods used in the study of complex manifolds in the special case of complex dimension one. From the reviews: "This book deserves very serious consideration as a text for anyone contemplating giving a course on Riemann surfaces."—-MATHEMATICAL REVIEWS

Author : Gareth A. Jones,Jürgen Wolfart
Publisher : Springer
Page : 259 pages
File Size : 51,6 Mb
Release : 2016-03-23
Category : Mathematics
ISBN : 9783319247113

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Dessins d'Enfants on Riemann Surfaces by Gareth A. Jones,Jürgen Wolfart Pdf

This volume provides an introduction to dessins d'enfants and embeddings of bipartite graphs in compact Riemann surfaces. The first part of the book presents basic material, guiding the reader through the current field of research. A key point of the second part is the interplay between the automorphism groups of dessins and their Riemann surfaces, and the action of the absolute Galois group on dessins and their algebraic curves. It concludes by showing the links between the theory of dessins and other areas of arithmetic and geometry, such as the abc conjecture, complex multiplication and Beauville surfaces. Dessins d'Enfants on Riemann Surfaces will appeal to graduate students and all mathematicians interested in maps, hypermaps, Riemann surfaces, geometric group actions, and arithmetic.

Author : Norbert A'Campo
Publisher : Springer Nature
Page : 282 pages
File Size : 51,8 Mb
Release : 2021-10-27
Category : Mathematics
ISBN : 9783030890322

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Topological, Differential and Conformal Geometry of Surfaces by Norbert A'Campo Pdf

This book provides an introduction to the main geometric structures that are carried by compact surfaces, with an emphasis on the classical theory of Riemann surfaces. It first covers the prerequisites, including the basics of differential forms, the Poincaré Lemma, the Morse Lemma, the classification of compact connected oriented surfaces, Stokes’ Theorem, fixed point theorems and rigidity theorems. There is also a novel presentation of planar hyperbolic geometry. Moving on to more advanced concepts, it covers topics such as Riemannian metrics, the isometric torsion-free connection on vector fields, the Ansatz of Koszul, the Gauss–Bonnet Theorem, and integrability. These concepts are then used for the study of Riemann surfaces. One of the focal points is the Uniformization Theorem for compact surfaces, an elementary proof of which is given via a property of the energy functional. Among numerous other results, there is also a proof of Chow’s Theorem on compact holomorphic submanifolds in complex projective spaces. Based on lecture courses given by the author, the book will be accessible to undergraduates and graduates interested in the analytic theory of Riemann surfaces.

Author : Martin Schlichenmaier
Publisher : Springer Science & Business Media
Page : 228 pages
File Size : 42,9 Mb
Release : 2010-02-11
Category : Science
ISBN : 9783540711759

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An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces by Martin Schlichenmaier Pdf

This book gives an introduction to modern geometry. Starting from an elementary level, the author develops deep geometrical concepts that play an important role in contemporary theoretical physics, presenting various techniques and viewpoints along the way. This second edition contains two additional, more advanced geometric techniques: the modern language and modern view of Algebraic Geometry and Mirror Symmetry.

Author : Maurizio Cornalba,Xavier Gomez-mont,Alberto Sola Verjovsky
Publisher : World Scientific
Page : 716 pages
File Size : 50,9 Mb
Release : 1989-06-01
Category : Mathematics
ISBN : 9789814590877

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Lectures On Riemann Surfaces - Proceedings Of The College On Riemann Surfaces by Maurizio Cornalba,Xavier Gomez-mont,Alberto Sola Verjovsky Pdf