Functional Analysis and Its Applications

, Volume 42, Issue 4, pp 268–278

Solution of the problem of differentiation of Abelian functions over parameters for families of (n, s)-curves


DOI: 10.1007/s10688-008-0040-4

Cite this article as:
Buchstaber, V.M. & Leykin, D.V. Funct Anal Its Appl (2008) 42: 268. doi:10.1007/s10688-008-0040-4


We consider a wide class of models of plane algebraic curves, so-called (n, s)-curves. The case (2, 3) is the classical Weierstrass model of an elliptic curve. On the basis of the theory of multivariate sigma functions, for every pair of coprime n and s we obtain an effective description of the Lie algebra of derivations of the field of fiberwise Abelian functions defined on the total space of the bundle whose base is the parameter space of the family of nondegenerate (n, s)-curves and whose fibers are the Jacobi varieties of these curves. The essence of the method is demonstrated by the example of Weierstrass elliptic functions. Details are given for the case of a family of genus 2 curves.

Key words

sigma functiondifferentiation with respect to parametersuniversal bundle of Jacobi varieties(n, s)-curvevector field tangent to the discriminant of a singularity

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteSteklovRussia
  2. 2.Institute of MagnetismKievUkraine