Some elliptic fibrations arising from free rigid body dynamics

Abstract

An elliptic fibration over P₃($¥mathbb{C}$), naively arising from the Euler equation for free rigid body dynamics, is studied from the viewpoint of complex algebraic geometry. With this elliptic fibration, associated is an elliptic fibration in Weierstraß normal form, whose generic fibres are isomorphic to those of the original fibration. This normal form is desingularized in a canonical manner. It is shown that there is a four-to-one meromorphic mapping from the naive elliptic fibration to the Weierstraß mormal form. The latter fibration is also shown to be bimeromorphic to the family of spectral curves arising from the corresponding Manakov equation.