Moduli Of Curves

Author : Joe Harris,Ian Morrison
Publisher : Springer Science & Business Media
Page : 381 pages
File Size : 47,5 Mb
Release : 2006-04-06
Category : Mathematics
ISBN : 9780387227375

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Moduli of Curves by Joe Harris,Ian Morrison Pdf

A guide to a rich and fascinating subject: algebraic curves and how they vary in families. Providing a broad but compact overview of the field, this book is accessible to readers with a modest background in algebraic geometry. It develops many techniques, including Hilbert schemes, deformation theory, stable reduction, intersection theory, and geometric invariant theory, with the focus on examples and applications arising in the study of moduli of curves. From such foundations, the book goes on to show how moduli spaces of curves are constructed, illustrates typical applications with the proofs of the Brill-Noether and Gieseker-Petri theorems via limit linear series, and surveys the most important results about their geometry ranging from irreducibility and complete subvarieties to ample divisors and Kodaira dimension. With over 180 exercises and 70 figures, the book also provides a concise introduction to the main results and open problems about important topics which are not covered in detail.

Author : Robert H. Dijkgraaf,Carel Faber,Gerard B.M. van der Geer
Publisher : Springer Science & Business Media
Page : 570 pages
File Size : 45,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461242642

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The Moduli Space of Curves by Robert H. Dijkgraaf,Carel Faber,Gerard B.M. van der Geer Pdf

The moduli space Mg of curves of fixed genus g – that is, the algebraic variety that parametrizes all curves of genus g – is one of the most intriguing objects of study in algebraic geometry these days. Its appeal results not only from its beautiful mathematical structure but also from recent developments in theoretical physics, in particular in conformal field theory.

Author : Maxim E. Kazaryan,Sergei K. Lando,Victor V. Prasolov
Publisher : Springer
Page : 231 pages
File Size : 48,7 Mb
Release : 2019-01-21
Category : Mathematics
ISBN : 9783030029432

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Algebraic Curves by Maxim E. Kazaryan,Sergei K. Lando,Victor V. Prasolov Pdf

This book offers a concise yet thorough introduction to the notion of moduli spaces of complex algebraic curves. Over the last few decades, this notion has become central not only in algebraic geometry, but in mathematical physics, including string theory, as well. The book begins by studying individual smooth algebraic curves, including the most beautiful ones, before addressing families of curves. Studying families of algebraic curves often proves to be more efficient than studying individual curves: these families and their total spaces can still be smooth, even if there are singular curves among their members. A major discovery of the 20th century, attributed to P. Deligne and D. Mumford, was that curves with only mild singularities form smooth compact moduli spaces. An unexpected byproduct of this discovery was the realization that the analysis of more complex curve singularities is not a necessary step in understanding the geometry of the moduli spaces. The book does not use the sophisticated machinery of modern algebraic geometry, and most classical objects related to curves – such as Jacobian, space of holomorphic differentials, the Riemann-Roch theorem, and Weierstrass points – are treated at a basic level that does not require a profound command of algebraic geometry, but which is sufficient for extending them to vector bundles and other geometric objects associated to moduli spaces. Nevertheless, it offers clear information on the construction of the moduli spaces, and provides readers with tools for practical operations with this notion. Based on several lecture courses given by the authors at the Independent University of Moscow and Higher School of Economics, the book also includes a wealth of problems, making it suitable not only for individual research, but also as a textbook for undergraduate and graduate coursework

Author : Alexander Vasil'ev
Publisher : Springer
Page : 214 pages
File Size : 50,8 Mb
Release : 2004-10-20
Category : Mathematics
ISBN : 9783540454373

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Moduli of Families of Curves for Conformal and Quasiconformal Mappings by Alexander Vasil'ev Pdf

The monograph is concerned with the modulus of families of curves on Riemann surfaces and its applications to extremal problems for conformal, quasiconformal mappings, and the extension of the modulus onto Teichmüller spaces. The main part of the monograph deals with extremal problems for compact classes of univalent conformal and quasiconformal mappings. Many of them are grouped around two-point distortion theorems. Montel's functions and functions with fixed angular derivatives are also considered. The last portion of problems is directed to the extension of the modulus varying the complex structure of the underlying Riemann surface that sheds some new light on the metric problems of Teichmüller spaces.

Author : Benson Farb,Richard Hain,Eduard Looijenga
Publisher : American Mathematical Soc.
Page : 371 pages
File Size : 40,6 Mb
Release : 2013-08-16
Category : Mathematics
ISBN : 9780821898871

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Moduli Spaces of Riemann Surfaces by Benson Farb,Richard Hain,Eduard Looijenga Pdf

Mapping class groups and moduli spaces of Riemann surfaces were the topics of the Graduate Summer School at the 2011 IAS/Park City Mathematics Institute. This book presents the nine different lecture series comprising the summer school, covering a selection of topics of current interest. The introductory courses treat mapping class groups and Teichmüller theory. The more advanced courses cover intersection theory on moduli spaces, the dynamics of polygonal billiards and moduli spaces, the stable cohomology of mapping class groups, the structure of Torelli groups, and arithmetic mapping class groups. The courses consist of a set of intensive short lectures offered by leaders in the field, designed to introduce students to exciting, current research in mathematics. These lectures do not duplicate standard courses available elsewhere. The book should be a valuable resource for graduate students and researchers interested in the topology, geometry and dynamics of moduli spaces of Riemann surfaces and related topics. Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.

Author : Martin Schlichenmaier
Publisher : Springer
Page : 149 pages
File Size : 54,8 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 : Enrico Arbarello,Maurizio Cornalba,Phillip Griffiths,Joseph Daniel Harris
Publisher : Springer
Page : 387 pages
File Size : 49,6 Mb
Release : 2013-08-30
Category : Mathematics
ISBN : 1475753241

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Geometry of Algebraic Curves by Enrico Arbarello,Maurizio Cornalba,Phillip Griffiths,Joseph Daniel Harris Pdf

In recent years there has been enormous activity in the theory of algebraic curves. Many long-standing problems have been solved using the general techniques developed in algebraic geometry during the 1950's and 1960's. Additionally, unexpected and deep connections between algebraic curves and differential equations have been uncovered, and these in turn shed light on other classical problems in curve theory. It seems fair to say that the theory of algebraic curves looks completely different now from how it appeared 15 years ago; in particular, our current state of knowledge repre sents a significant advance beyond the legacy left by the classical geometers such as Noether, Castelnuovo, Enriques, and Severi. These books give a presentation of one of the central areas of this recent activity; namely, the study of linear series on both a fixed curve (Volume I) and on a variable curve (Volume II). Our goal is to give a comprehensive and self-contained account of the extrinsic geometry of algebraic curves, which in our opinion constitutes the main geometric core of the recent advances in curve theory. Along the way we shall, of course, discuss appli cations of the theory of linear series to a number of classical topics (e.g., the geometry of the Riemann theta divisor) as well as to some of the current research (e.g., the Kodaira dimension of the moduli space of curves).

Author : Daniel Huybrechts,Manfred Lehn
Publisher : Cambridge University Press
Page : 345 pages
File Size : 40,9 Mb
Release : 2010-05-27
Category : Mathematics
ISBN : 9781139485821

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The Geometry of Moduli Spaces of Sheaves by Daniel Huybrechts,Manfred Lehn Pdf

This edition has been updated to reflect recent advances in the theory of semistable coherent sheaves and their moduli spaces. The authors review changes in the field and point the reader towards further literature. An ideal text for graduate students or mathematicians with a background in algebraic geometry.

Author : Joachim Kock,Israel Vainsencher
Publisher : Springer Science & Business Media
Page : 162 pages
File Size : 42,6 Mb
Release : 2007-12-27
Category : Mathematics
ISBN : 9780817644956

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An Invitation to Quantum Cohomology by Joachim Kock,Israel Vainsencher Pdf

Elementary introduction to stable maps and quantum cohomology presents the problem of counting rational plane curves Viewpoint is mostly that of enumerative geometry Emphasis is on examples, heuristic discussions, and simple applications to best convey the intuition behind the subject Ideal for self-study, for a mini-course in quantum cohomology, or as a special topics text in a standard course in intersection theory

Author : Edward Frenkel,David Ben-Zvi
Publisher : American Mathematical Soc.
Page : 418 pages
File Size : 46,9 Mb
Release : 2004-08-25
Category : Mathematics
ISBN : 9780821836743

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Vertex Algebras and Algebraic Curves by Edward Frenkel,David Ben-Zvi Pdf

Vertex algebras are algebraic objects that encapsulate the concept of operator product expansion from two-dimensional conformal field theory. Vertex algebras are fast becoming ubiquitous in many areas of modern mathematics, with applications to representation theory, algebraic geometry, the theory of finite groups, modular functions, topology, integrable systems, and combinatorics. This book is an introduction to the theory of vertex algebras with a particular emphasis on the relationship with the geometry of algebraic curves. The notion of a vertex algebra is introduced in a coordinate-independent way, so that vertex operators become well defined on arbitrary smooth algebraic curves, possibly equipped with additional data, such as a vector bundle. Vertex algebras then appear as the algebraic objects encoding the geometric structure of various moduli spaces associated with algebraic curves. Therefore they may be used to give a geometric interpretation of various questions of representation theory. The book contains many original results, introduces important new concepts, and brings new insights into the theory of vertex algebras. The authors have made a great effort to make the book self-contained and accessible to readers of all backgrounds. Reviewers of the first edition anticipated that it would have a long-lasting influence on this exciting field of mathematics and would be very useful for graduate students and researchers interested in the subject. This second edition, substantially improved and expanded, includes several new topics, in particular an introduction to the Beilinson-Drinfeld theory of factorization algebras and the geometric Langlands correspondence.

Author : Steve Bradlow
Publisher : Cambridge University Press
Page : 516 pages
File Size : 40,5 Mb
Release : 2009-05-21
Category : Mathematics
ISBN : 9780521734714

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Moduli Spaces and Vector Bundles by Steve Bradlow Pdf

Coverage includes foundational material as well as current research, authored by top specialists within their fields.

Author : Shigeru Mukai
Publisher : Cambridge University Press
Page : 528 pages
File Size : 41,8 Mb
Release : 2003-09-08
Category : Mathematics
ISBN : 0521809061

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An Introduction to Invariants and Moduli by Shigeru Mukai Pdf

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Author : Nicholas M. Katz,Barry Mazur
Publisher : Princeton University Press
Page : 528 pages
File Size : 53,6 Mb
Release : 2016-03-02
Category : Mathematics
ISBN : 9781400881710

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Arithmetic Moduli of Elliptic Curves. (AM-108), Volume 108 by Nicholas M. Katz,Barry Mazur Pdf

This work is a comprehensive treatment of recent developments in the study of elliptic curves and their moduli spaces. The arithmetic study of the moduli spaces began with Jacobi's "Fundamenta Nova" in 1829, and the modern theory was erected by Eichler-Shimura, Igusa, and Deligne-Rapoport. In the past decade mathematicians have made further substantial progress in the field. This book gives a complete account of that progress, including not only the work of the authors, but also that of Deligne and Drinfeld.

Author : Michael Artin,John Tate
Publisher : Springer Science & Business Media
Page : 485 pages
File Size : 41,7 Mb
Release : 2013-11-11
Category : Mathematics
ISBN : 9781475792867

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Arithmetic and Geometry by Michael Artin,John Tate Pdf

Author : Gary Cornell,Joseph H. Silverman,Glenn Stevens
Publisher : Springer Science & Business Media
Page : 592 pages
File Size : 52,6 Mb
Release : 2013-12-01
Category : Mathematics
ISBN : 9781461219743

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Modular Forms and Fermat’s Last Theorem by Gary Cornell,Joseph H. Silverman,Glenn Stevens Pdf

This volume contains the expanded lectures given at a conference on number theory and arithmetic geometry held at Boston University. It introduces and explains the many ideas and techniques used by Wiles, and to explain how his result can be combined with Ribets theorem and ideas of Frey and Serre to prove Fermats Last Theorem. The book begins with an overview of the complete proof, followed by several introductory chapters surveying the basic theory of elliptic curves, modular functions and curves, Galois cohomology, and finite group schemes. Representation theory, which lies at the core of the proof, is dealt with in a chapter on automorphic representations and the Langlands-Tunnell theorem, and this is followed by in-depth discussions of Serres conjectures, Galois deformations, universal deformation rings, Hecke algebras, and complete intersections. The book concludes by looking both forward and backward, reflecting on the history of the problem, while placing Wiles'theorem into a more general Diophantine context suggesting future applications. Students and professional mathematicians alike will find this an indispensable resource.