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

TL;DR

This paper proves the generic e9taleness of the moduli space of dormant f6_{2e9} f6pers, extending previous results to a new Lie algebra case, and derives a formula for computing degrees in related moduli spaces.

Contribution

It establishes the generic e9taleness for f6_{2e9} f6pers, a case not previously verified, and applies this to a factorization formula for degrees in moduli spaces.

Findings

Proves generic e9taleness for f6_{2e9} f6pers.

Derives a factorization formula for degrees in moduli spaces.

Extends previous results to a new Lie algebra case.

Abstract

The generic \'{e}taleness 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 \'{e}taleness 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.