Algebraic Geometry Algebraic Geometry III : Complex Algebraic Varieties Algebraic Curves and Their Jacobians

A. N. Parshin

ISBN 10: 3540546812 / ISBN 13: 9783540546818

Editore: Springer, Berlin, Springer Berlin Heidelberg, Springer, 1997

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Neuware - 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.

Codice articolo 9783540546818

Dati bibliografici

• Titolo: Algebraic Geometry Algebraic Geometry III : Complex Algebraic Varieties Algebraic Curves and Their Jacobians
• Editore: Springer, Berlin, Springer Berlin Heidelberg, Springer
• Data di pubblicazione: 1997
• Lingua: Inglese
• ISBN 10: 3540546812
• ISBN 13: 9783540546818
• Rilegatura: Buch
• Condizione: Neu
• Dimensioni: 244 L x 157 A x 18 P (millimetri) 244 millimetri di larghezza per 157 millimetri di altezza per 18 millimetri di profondità
• Peso: 616 grammi

Descrizione articolo

Riassunto

This two-part EMS volume provides a succinct summary of complex algebraic geometry, coupled with a lucid introduction to the recent work on the interactions between the classical area of the geometry of complex algebraic curves and their Jacobian varieties. An excellent companion to the older classics on the subject.

Dalla quarta di copertina

The first contribution of this EMS volume on the subject of complex algebraic geometry touches upon many of the central problems in this vast and very active area of current research. While it is much too short to provide complete coverage of this subject, it provides a succinct summary of the areas it covers, while providing in-depth coverage of certain very important fields - some examples of the fields treated in greater detail are theorems of Torelli type, K3 surfaces, variation of Hodge structures and degenerations of algebraic varieties.
The second part provides a brief and lucid introduction to the recent work on the interactions between the classical area of the geometry of complex algebraic curves and their Jacobian varieties, and partial differential equations of mathematical physics. The paper discusses the work of Mumford, Novikov, Krichever, and Shiota, and would be an excellent companion to the older classics on the subject by Mumford.