Super elliptic curves

Abstract

A detailed study is made of super elliptic curves, namely super Riemann surfaces of genus one considered as algebraic varieties, particularly their relation with their Picard groups. This is the simplest setting in which to study the geometric consequences of the fact that certain cohomology groups of super Riemann surfaces with odd spin structure are not freely generated modules. The divisor theory of Rosly, Schwarz, and Voronov gives a map from a supertorus to its Picard group Pic, but this map is a projection, not an isomorphism as it is for ordinaty tori. The geometric realization of the addition law on Pic via intersections of the supertorus with superlines in projective space is described. The isomorphisms of Pic with the Jacobian and the divisor class group are verified. All possible isogenies, or surjective holomorphic maps between supertori, are determined and shown to induce homomorphisms of the Picard groups. Finally, the solutions to the new super Kadomtsev-Petviashvili hierarchy of Mulase-Rabin which arise from super elliptic curves via the Krichever construction are exhibited.