The generic étaleness of the moduli space of dormant so2ℓ-opers

Abstract

The generic étaleness is an important property on the moduli space of dormant $g$-opers (for a simple Lie algebra $g$) in the context of enumerative geometry. In the previous study, this property has been verified under the assumption that $g$ is either ${\mathrm{sl}}_{\ell }$, ${\mathrm{so}}_{2\ell -1}$, or ${\mathrm{sp}}_{2\ell }$ for any sufficiently small positive integer ℓ. The purpose of the present paper is to prove the generic étaleness for one of the remaining cases, i.e., $g={\mathrm{so}}_{2\ell }$. 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.