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

Abstract

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 ${\mathfrak{sl}}_{\ell }$, ${\mathfrak{so}}_{2\ell -1}$, or ${\mathfrak{sp}}_{2\ell }$ for any sufficiently small positive integer $\ell$. The purpose of the present paper is to prove the generic étaleness for one of the remaining cases, i.e., $\mathfrak{g}={\mathfrak{so}}_{2\ell }$. 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.

28 pages