  Supersymmetric curves are the analogue of Riemann surfaces in super geometry.
We establish some foundational results about complex Deligne-Mumford
superstacks, and we then prove that the moduli superstack of supersymmetric
curves is a smooth complex Deligne-Mumford superstack. We then show that the
superstack of supersymmetric curves admits a coarse complex superspace, which,
in this case, is just an ordinary complex space. In the second part of this
paper we discuss the period map. We remark that the period domain is the moduli
space of ordinary abelian varieties endowed with a symmetric theta divisor, and
we then show that the differential of the period map is surjective. In other
words, we prove that any first order deformation of a classical Jacobian is the
Jacobian of a supersymmetric curve.
