Moduli and periods of supersymmetric curves

Supersymmetric curves are the analogue of Riemann surfaces in super geometry. We establish some foundational results about (Deligne-Mumford) complex superstacks, and we then prove that the moduli superstack of supersymmetric curves is a smooth Deligne-Mumford complex 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.