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Algebraic Geometry III

Complex Algebraic Varieties Algebraic Curves and Their Jacobians

  • Book
  • © 1998

Overview

Authors:
  1. Viktor S. Kulikov

    1. Moscow State University of Transport Communications (MIIT), Moscow, Russia

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  2. P. F. Kurchanov

    1. Moscow State University of Transport Communications (MIIT), Moscow, Russia

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  3. V. V. Shokurov

    1. Department of Mathematics, The Johns Hopkins University, Baltimore, USA

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Editors:
  1. A. N. Parshin

    1. Steklov Mathematical Institute, Moscow, Russia

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  2. I. R. Shafarevich

    1. Steklov Mathematical Institute, Moscow, Russia

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Part of the book series: Encyclopaedia of Mathematical Sciences (EMS, volume 36)

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Table of contents (2 chapters)

  1. Front Matter

    Pages i-vii
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  2. Complex Algebraic Varieties: Periods of Integrals and Hodge Structures

    • Vik. S. Kulikov, P. F. Kurchanov
    Pages 1-217

  3. Algebraic Curves and Their Jacobians

    • V. V. Shokurov
    Pages 219-261

  4. Back Matter

    Pages 263-270
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Keywords

About this book

Starting with the end of the seventeenth century, one of the most interesting directions in mathematics (attracting the attention as J. Bernoulli, Euler, Jacobi, Legendre, Abel, among others) has been the study of integrals of the form r dz l Aw(T) = -, TO W where w is an algebraic function of z. Such integrals are now called abelian. Let us examine the simplest instance of an abelian integral, one where w is defined by the polynomial equation (1) where the polynomial on the right hand side has no multiple roots. In this case the function Aw is called an elliptic integral. The value of Aw is determined up to mv + nv , where v and v are complex numbers, and m and n are 1 2 1 2 integers. The set of linear combinations mv+ nv forms a lattice H C C, and 1 2 so to each elliptic integral Aw we can associate the torus C/ H. 2 On the other hand, equation (1) defines a curve in the affine plane C = 2 2 {(z,w)}. Let us complete C2 to the projective plane lP' = lP' (C) by the addition of the "line at infinity", and let us also complete the curve defined 2 by equation (1). The result will be a nonsingular closed curve E C lP' (which can also be viewed as a Riemann surface). Such a curve is called an elliptic curve.

Authors, Editors and Affiliations

  • Steklov Mathematical Institute, Moscow, Russia

    A. N. Parshin, I. R. Shafarevich

  • Moscow State University of Transport Communications (MIIT), Moscow, Russia

    Viktor S. Kulikov, P. F. Kurchanov

  • Department of Mathematics, The Johns Hopkins University, Baltimore, USA

    V. V. Shokurov

Bibliographic Information

  • Book Title: Algebraic Geometry III

  • Book Subtitle: Complex Algebraic Varieties Algebraic Curves and Their Jacobians

  • Authors: Viktor S. Kulikov, P. F. Kurchanov, V. V. Shokurov

  • Editors: A. N. Parshin, I. R. Shafarevich

  • Series Title: Encyclopaedia of Mathematical Sciences

  • DOI: https://doi.org/10.1007/978-3-662-03662-4

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 1998

  • Hardcover ISBN: 978-3-540-54681-8Published: 08 December 1997

  • Softcover ISBN: 978-3-642-08118-7Published: 01 December 2010

  • eBook ISBN: 978-3-662-03662-4Published: 17 April 2013

  • Series ISSN: 0938-0396

  • Edition Number: 1

  • Number of Pages: VIII, 270

  • Additional Information: Original Russian edition published by VINITI, Moscow, 1989

  • Topics: Algebraic Geometry, Analysis, Number Theory

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