Combinatorial properties of stable spin curves

The first part of this paper describes the moduli space of stable spin curves (constructed in [Cor89]), explaining how its geometry is governed by the combinatorics of stable curves. In this context, the standard graph theoretic framework (where to every stable curve one associates its “dual graph”) is not just a book-keeping device: in Section 2, some purely combinatorial results are proved (Theorems 11 and 13), having moduli theoretic applications. More precisely, certain strata of the moduli space of stable curves are characterized by a (finite) set of integers that measures the non-reducedness of the scheme of spin curves, and which is definable in purely graph-theoretical terms (Definition 7).