The generic étaleness of the moduli space of dormant ${\mathfrak{s}\mathfrak{o}}_{2\mathrm{\ell }}$-opers

The generic étaleness is an important property on the moduli space of dormant $\mathfrak{g}$-opers (for a simple Lie algebra $\mathfrak{g}$) in the context of enumerative geometry. In the previous study, this property has been verified under the assumption that $\mathfrak{g}$ is either ${\mathfrak{s}\mathfrak{l}}_{\mathrm{\ell }}$, ${\mathfrak{s}\mathfrak{o}}_{2\mathrm{\ell }-1}$, or ${\mathfrak{s}\mathfrak{p}}_{2\mathrm{\ell }}$ for any sufficiently small positive integer $\mathrm{\ell }$. The purpose of the present paper is to prove the generic étaleness for one of the remaining cases, i.e., $\mathfrak{g}={\mathfrak{s}\mathfrak{o}}_{2\mathrm{\ell }}$. As an application of this result, we obtain a factorization formula for computing the generic degree induced from pull-back along various clutching morphisms between moduli spaces of pointed stable curves.