Algebraic geometry of the eigenvector mapping for a free rigid body
This article deals with the algebro-geometric aspects of the eigenvector mapping for a dynamical system of a free rigid body on SO(3), which is governed by the Manakov equation. The authors investigate the eigenvector mapping which is regarded as a rational mapping to the complex projective plane \(P_2(\mathbb C)\) from the product of the elliptic curves, one of which is the curve given by the first integrals of the dynamical system and the other the spectral curve of the Manakov equation. The authors show that the eigenvector mapping admits a factorisation through a Kummer surface, which is a double covering of the projective plane branched along a sextic curve associated with the dynamics. It turns out that their essentials are determined by the Cremona transformation of the projective plane and some elliptic fibrations of the Kummer surface.
scientific article; zbMATH DE number 5957585