  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.
