The problem of differentiation of an Abelian function over its parameters

Buchstaber, Victor and Leykin, Dmitry (2006) The problem of differentiation of an Abelian function over its parameters. [MIMS Preprint]

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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 2010, the AMS's Mathematics Subject Classification > 14 Algebraic geometry MSC 2010, the AMS's Mathematics Subject Classification > 35 Partial differential equations MSC 2010, the AMS's Mathematics Subject Classification > 53 Differential geometry
Depositing User:	Dmitry Leykin
Date Deposited:	14 Dec 2006
Last Modified:	08 Nov 2017 18:18
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/666

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• The problem of differentiation of an Abelian function over its parameters. (deposited 08 Dec 2006)
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