Symmetries of Compact Riemann Surfaces (Lecture Notes in Mathematics, Vol. 2007)

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9783642148279: Symmetries of Compact Riemann Surfaces (Lecture Notes in Mathematics, Vol. 2007)
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This monograph covers symmetries of compact Riemann surfaces. It examines the number of conjugacy classes of symmetries, the numbers of ovals of symmetries and the symmetry types of Riemann surfaces.

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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.

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“The monograph under review is primarily a survey of recent advances in the theory of symmetries of compact Riemann surfaces. It also provides a number of new interesting developments and different methods of proof for some of the recent and classical results in this area as well as a number of illustrative and detailed examples highlighting these results. With its informative and well-written introduction and a substantial preliminaries section, this monograph is ideal for both beginners to the area and current researchers.” (Aaron D. Wootton, Mathematical Reviews, Issue 2011 h)

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Book Description Condition: New. Publisher/Verlag: Springer, Berlin | The content of this monograph is situated in the intersection of important branches of mathematics like the theory of one complex variable, algebraic geometry, low dimensional topology and, from the point of view of the techniques used, com- natorial group theory. The main tool comes from the Uniformization Theorem for Riemannsurfaces,whichrelatesthetopologyofRiemannsurfacesandhol omorphic or antiholomorphic actions on them to the algebra of classical cocompact Fuchsian groups or, more generally, non-euclidean crystallographic groups. Foundations of this relationship were established by A. M. Macbeath in the early sixties and dev- oped later by, among others, D. Singerman. Another important result in Riemann surface theory is the connection between Riemannsurfacesandtheir symmetrieswith complexalgebraiccurvesandtheirreal forms. Namely, there is a well known functorial bijective correspondence between compact Riemann surfaces and smooth, irreducible complex projective curves. The fact that a Riemann surface has a symmetry means, under this equivalence, that the corresponding complex algebraic curve has a real form, that is, it is the complex- cation of a real algebraic curve. Moreover, symmetries which are non-conjugate in the full group of automorphisms of the Riemann surface, correspond to real forms which are birationally non-isomorphic over the reals. Furthermore, the set of points xedbyasymmetryishomeomorphictoaprojectivesmoothmodeloftherealform. | Preliminaries.- On the Number of Conjugacy Classes of Symmetries of Riemann Surfaces.- Counting Ovals of Symmetries of Riemann Surfaces.- Symmetry Types of Some Families of Riemann Surfaces.- Symmetry Types of Riemann Surfaces with a Large Group of Automorphisms. | Format: Paperback | Language/Sprache: english | 290 gr | 164 pp. Seller Inventory # K9783642148279

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Book Description Springer-Verlag Gmbh Okt 2010, 2010. Taschenbuch. Condition: Neu. Neuware - The content of this monograph is situated in the intersection of important branches of mathematics like the theory of one complex variable, algebraic geometry, low dimensional topology and, from the point of view of the techniques used, com- natorial group theory. The main tool comes from the Uniformization Theorem for Riemannsurfaces,whichrelatesthetopologyofRiemannsurfacesandholomorphic or antiholomorphic actions on them to the algebra of classical cocompact Fuchsian groups or, more generally, non-euclidean crystallographic groups. Foundations of this relationship were established by A. M. Macbeath in the early sixties and dev- oped later by, among others, D. Singerman. Another important result in Riemann surface theory is the connection between Riemannsurfacesandtheir symmetrieswith complexalgebraiccurvesandtheirreal forms. Namely, there is a well known functorial bijective correspondence between compact Riemann surfaces and smooth, irreducible complex projective curves. The fact that a Riemann surface has a symmetry means, under this equivalence, that the corresponding complex algebraic curve has a real form, that is, it is the complex- cation of a real algebraic curve. Moreover, symmetries which are non-conjugate in the full group of automorphisms of the Riemann surface, correspond to real forms which are birationally non-isomorphic over the reals. Furthermore, the set of points xedbyasymmetryishomeomorphictoaprojectivesmoothmodeloftherealform. 158 pp. Englisch. Seller Inventory # 9783642148279

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