Algebraic Geometry III

Complex Algebraic Varieties Algebraic Curves and Their Jacobians

A.N. Parshin, I.R. Shafarevich (Herausgeber)

Buch | Hardcover

VIII, 270 Seiten

1997 | 1998
Springer Berlin (Verlag)
978-3-540-54681-8 (ISBN)

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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.

Complex Algebraic Varieties: Periods of Integrals and Hodge Structures by Viktor S.Kulikov and P.V.Kurchanov; Algebraic Curves and Their Jacobians by V.V.Shokurov

Erscheint lt. Verlag	8.12.1997
Reihe/Serie	Encyclopaedia of Mathematical Sciences
Co-Autor	V.S. Kulikov, P.F. Kurchanov, V.V. Shokurov
Übersetzer	I. Rivin
Zusatzinfo	VIII, 270 p.
Verlagsort	Berlin
Sprache	englisch
Maße	155 x 235 mm
Gewicht	526 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
	Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
Schlagworte	algebraic curves • Algebraic Varieties • Algebraische Geometrie • algebraische Kurven • complex algebraic varieties • differential equation • Hodgesche Struktur • Hodge structures • Jacobian varieties • Jacobischen Varietät • komplexe algebraische Varietät • Kurven • Mathematical Physics • Sätze von • Torelli theorems
ISBN-10	3-540-54681-2 / 3540546812
ISBN-13	978-3-540-54681-8 / 9783540546818
Zustand	Neuware