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.

中文翻译：

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超对称曲线的模量和周期

超对称曲线类似于超几何中的黎曼曲面。我们建立了一些关于复Deligne-Mumford 超叠堆的基础结果，然后证明了超对称曲线的模量超叠堆是一个光滑的复Deligne-Mumford 超叠堆。然后我们证明超对称曲线的超堆栈允许一个粗糙的复超空间，在这种情况下，它只是一个普通的复空间。在本文的第二部分，我们将讨论周期图。我们注意到周期域是具有对称θ因数的普通阿贝尔簇的模空间，然后我们证明周期映射的微分是满射的。换句话说，我们证明了经典雅可比的任何一阶变形都是超对称曲线的雅可比。