Flatness of the spectral cover over the non-étale locus

Ascertain whether the spectral cover morphism π_s: X_s → X associated to a spectral datum s ∈ B_X^r is flat over the closed subset X \ X^∘, where X^∘ is the open subset of X that is mapped into the multiplicity-free locus [B^{r,∘}/GL_d] by s (equivalently, the locus over which π_s is étale).

Background

Given s ∈ B_Xr, the spectral cover X_s → X is finite and étale over the open locus X∘ where the corresponding cycle has no multiplicities; outside X∘, X_s need not be Cohen–Macaulay, so flatness and local freeness of the pushforward are subtle.

Understanding flatness over X \ X∘ is important for constructing vector bundles (via pushforward) and for comparing spectral data with Higgs bundles in higher dimensions.

References

More precisely, the morphism $\pi_s$ is étale over the open subset $X\circ$ of $X$ that is mapped into $[B{r,\circ}/_d]$ by $s$. It is unclear whether $\pi_s$ is flat over the locus $X\setminus X\circ$.

— The Hitchin morphism for K-trivial varieties  (2604.03217 - Patel et al., 3 Apr 2026) in Subsection "The spectral base", Section 2 (Spectral covers)