  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.

Algebraic Geometry