Moduli of log twisted $\mathcal{N} =1$ SUSY curves

TL;DR

This paper constructs a smooth, compact moduli superstack for pointed $ ext{SUSY}_1$ curves using log superscheme theory, extending classical and spin curve compactifications to supersymmetric settings.

Contribution

It introduces a framework of log superstacks and defines pointed log twisted $ ext{SUSY}_1$ curves, providing a new smooth compactification of the moduli superstack.

Findings

The moduli superstack is represented by a log superstack that is superproper and supersmooth.

The construction offers a different compactification than Deligne's.

The framework extends classical notions to supersymmetric and logarithmic contexts.

Abstract

The goal of the present paper is to construct a smooth compactification of the moduli superstack classifying pointed $\mathcal{N} =1$ SUSY (= $\text{SUSY}_1$) curves. This construction is based on the Abramovich-Jarvis-Chiodo compactification of the moduli stack classifying spin curves. First, we give a general framework of a theory of log superschemes (or more generally, log superstacks). Then, we introduce the notion of a pointed (stable) log twisted $\text{SUSY}_1$ curve; it may be thought of as a logarithmic and twisted generalization of the classical notion of a pointed $\text{SUSY}_1$ curve, as well as a supersymmetric analogue of the notion of a pointed (log) twisted curve. The main result of the present paper asserts that the moduli superstack classifying pointed stable log twisted $\text{SUSY}_1$ curves may be represented by a log superstack whose underlying superstack is a superproper and supersmooth Deligne-Mumford superstack. Consequently, this moduli superstack forms a smooth compactification different from the compactification proposed by P. Deligne.