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We study the geometry and cohomology of algebraic super curves, using a new contour integral for holomorphic differentials. For a class of super curves ("generic SKP curves") we define a period matrix. We show that the odd part of the period matrix controls the cohomology of the dual curve. The Jacobian of a generic SKP curve is a smooth supermanifold; it is principally polarized, hence projective, if the even part of the period matrix is symmetric. In general symmetry is not guaranteed by the Riemann bilinear equations for our contour integration, so it remains open whether Jacobians are always projective or carry theta functions.These results on generic SKP curves are applied to the study of algebro-geometric solutions of the super KP hierarchy. The tau function is shown to be, essentially, a meromorphic section of a line bundle with trivial Chern class on the Jacobian, rationally expressible in terms of super theta functions when these exist. Also we relate the tau function and the Baker function for this hierarchy, using a generalization of Cramer's rule to the supercase.

Recent investigationby Verlinde and Verlinde has shown that the fermionic string loop amplitudes change by a total derivative term in the moduli space under a change of basis of the supermoduli. This ambiguity is addressed in the context of the heterotic string theory, and shown to be a consequence of an inherent ambiguity in defining integration over the variables of a Grassmann algebra-in this case the Grassmann valued coordinates of the supermoduli space. A resolution of this ambiguity in genus-two within this formalism is also presented.