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2006.414: The problem of differentiation of an Abelian function over its parameters

2006.414: Victor Buchstaber and Dmitry Leykin (2006) The problem of differentiation of an Abelian function over its parameters.

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Abstract

The present work is devoted to the problem of differentiation of an Abelian function, defined by a family of plane algebraic curves, over the parameters of the family.

A precise formulation of the problem involves the language of Differential Geometry.

We give an effective solution, which is based on our theory of multivariate sigma-function. We obtain explicit expressions for the generators of the module of differentiations of a ring of Abelian functions. This result is equivalent, as we show, to an explicit construction of a Gauss-Manin connection and a Koszul connection in the appropriate vector bundles.

In the course of exposition we outline the key classic results relevant to the problem.

Item Type:MIMS Preprint
Uncontrolled Keywords:Abelian fuctions on a Jacobian, plane algebraic curves, Gauss-Manin connection, Coszul connection
Subjects:MSC 2000 > 14 Algebraic geometry
MSC 2000 > 35 Partial differential equations
MSC 2000 > 53 Differential geometry
MIMS number:2006.414
Deposited By:Dmitry Leykin
Deposited On:14 December 2006

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