Components of moduli spaces of spin curves with the expected codimension

TL;DR

This paper proves a conjecture about the existence and codimension of certain components in the moduli space of spin curves with specified theta characteristics, confirming predicted geometric structures.

Contribution

It establishes the existence of specific components in the moduli space of spin curves with precise codimension, confirming a conjecture by Gavril Farkas.

Findings

Confirmed the existence of components with expected codimension in  spaces of spin curves.

Provided explicit conditions on g and r for these components to exist.

Validated a conjecture linking theta characteristics and moduli space geometry.

Abstract

We prove a conjecture of Gavril Farkas claiming that for all integers r \geq 2 and g \geq \binom{r+2}{2} there exists a component of the locus \mathcal{S}^r_g of spin curves with a theta characteristic L such that h^0(L) \geq r+1 and h^0(L)\equiv r+1 (mod 2) which has codimension \binom{r+1}{2} inside the moduli space \mathcal{S}_g of spin curves of genus g.