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

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