Tropicalizing the moduli space of spin curves

Abstract

We study the tropicalization of the moduli space of algebraic spin curves, $$\overline{\mathcal {S}}_{g,n}$$. We exhibit its combinatorial stratification and prove that the strata are irreducible. We construct the moduli space of tropical spin curves $$\overline{S}_{g,n}^{{\text {trop}}}$$, prove that is naturally isomorphic to the skeleton of the analytification, $$\overline{S}_{g,n}^{{\text {an}}}$$, of $$\overline{\mathcal {S}}_{g,n}$$, and give a geometric interpretation of the retraction of $$\overline{S}_{g,n}^{{\text {an}}}$$ onto its skeleton in terms of a tropicalization map $$\overline{S}_{g,n}^{{\text {an}}}\rightarrow \overline{S}_{g,n}^{{\text {trop}}}$$.