Integrable Systems and projective images of Kummer surfaces

Abstract

The (-1 )-involution on the Jacobian Jr of an arbitrary Riemann surfacer of genus two leads to a singular surface, the Kummer surface Icr of Jr, which,after desingularization, defines a X-3 surface Kr . We consider ample line bundleson Kr coming from the even or odd sections of [n O] with prescribed vanishingat the 2-division points of Jr (0 is the theta divisor of Jr). We use an integrablesystem to show that in the cases we study the linear system is base-point-free,to determine which curves are contracted to singular points and to compute anexplicit equation for the surface in projective space. Our explicit methods applyto the Kummer surface of any Abelian surface, given as the fiber of the momentmap of an algebraic completely integrable system.