Irreducibility of the spectral base

Determine whether the spectral base B_X^r is irreducible for a smooth variety X over an algebraically closed field of characteristic 0 and a fixed rank r.

Background

Although B_Xr is a closed subspace of the Hitchin base with an explicit construction, its global geometric and topological properties are poorly understood. The authors show B_Xr is connected via the natural scaling action but leave irreducibility unresolved.

Resolving irreducibility would have consequences for stratifications by factorization types and for understanding global behavior of spectral data.

References

Although the spectral base can be described quite explicitly as a closed subscheme of the Hitchin base, little is known about it's geometry and topology. For example, it is not known whether $\mathscr{B}r_X$ is irreducible.

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