Components of moduli spaces of spin curves with the expected codimension II

TL;DR

This paper proves the existence of specific components within the moduli space of spin curves, characterized by certain cohomological properties, for all sufficiently large genus relative to the parameter r.

Contribution

It establishes the existence of components of the spin moduli space with expected codimension for all r ≥ 2 and large enough genus g, extending prior results.

Findings

Existence of components with specified properties for all r ≥ 2

Components have expected codimension in the moduli space

Results hold for genus g above a certain bound

Abstract

We prove that for all integers $r \geq 2$ and $g \geq \lfloor \frac{r^2+10r+1}{4} \rfloor$ 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 $h^0(L)\equiv r+1 (\text{mod} 2)$ which has expected codimension $\binom{r+1}{2}$ inside the moduli space $\mathcal{S}_g$ of spin curves of genus $g$.