# Moduli and Periods of Supersymmetric Curves

## TL;DR

This paper develops the theory of supersymmetric curves, establishing their moduli superstack as a smooth Deligne-Mumford superstack, and explores the properties of the associated period map and its surjectivity.

## Contribution

It proves that the moduli superstack of supersymmetric curves is a smooth complex Deligne-Mumford superstack and that the period map's differential is surjective, linking classical and supersymmetric Jacobians.

## Key findings

- The moduli superstack of supersymmetric curves is smooth.
- The period map's differential is surjective.
- Any first order deformation of a classical Jacobian corresponds to a supersymmetric curve.

## Abstract

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.