The authors in this article study the geometry of the moduli space of spin curves by using tropical geometric techniques.
A spin curve \(X\) is a projective curve of genus \(g\) along with a choice of a theta characteristic, that is, is a square root of the dualizing sheaf of \(X\). The moduli space of such curves, \(\overline{S}_g\), was constructed by M. Cornalba [in: Proceedings of the first college on Riemann surfaces held in Trieste, Italy, November 9-December 18, 1987. Teaneck, NJ: World Scientific Publishing Co.. 560–589 (1989; Zbl 0800.14011)]. It comes together with a natural map \[ \pi:\overline{S}_g \longrightarrow \overline{M}_g \] to the moduli space of stable curves, whose fiber over a curve \(X\) parametrizes spin structures on \(X\).
The authors define combinatorial analogues of spin curves, as spin tropical curves and construct the tropical analogue \(\overline{S}_g ^{\mathrm{Trop}}\) of the moduli space \(\overline{S}_g \). One of the main results of the article shows that there is a natural map \[ \pi^{\mathrm{Trop}}:\overline{S}_g ^{\mathrm{Trop}} \longrightarrow \overline{M}_g^{\mathrm{Trop}} \] to the moduli space of tropical curves, which arises as a morphism of extended generalized cone complexes. The fiber of this map is descibed in terms the set of spin structures on a tropical curve associated to a spin curve.
The authors then investigate the moduli space of \(n\)-pointed stable spin curves \( \overline{S}_{g,n} \) and describe a stratification of it using spin graphs. They also describe the skeleton of the Berkovich analityfication of \( \overline{S}_{g,n} \) and prove that that there is an isomorphism between this skeleton and the moduli space of spin tropical curves.