Optimal Descent Degrees in Reduced Hurwitz Spaces

Definitions and mathematical statement

Retain $H(C)$ for the Hurwitz space of $G$-covers with the specified inertia data $C$, $H^{\mathrm{rd}}(C)$ for its reduced Hurwitz space, and \[ \Pi:H(C)\longrightarrow H^{\mathrm{rd}}(C) \] for the source's quotient. Let $k$ be the intended field and $p^{\mathrm{rd}}\in H^{\mathrm{rd}}(C)(k)$. Write $F_{p^{\mathrm{rd}}}=\Pi^{-1}(p^{\mathrm{rd}})$ for the fiber. The invariant in the question is \[ m_k(p^{\mathrm{rd}})=\min_{p\in F_{p^{\mathrm{rd}}}}[k(p):k]. \] Under the standard finite-degree point convention, $p$ ranges over closed points with finite residue field $k(p)$ over $k$. Equivalently, when this convention applies, \[ m_k(p^{\mathrm{rd}})=\min\{[L:k]:L/k\text{ finite and }F_{p^{\mathrm{rd}}}(L)\ne\varnothing\}. \] The notation records a minimal field-extension degree needed to lift the reduced moduli point to the unreduced space. If the admissible fiber has no such point, the minimum is undefined as printed; one can use $+\infty$ for bookkeeping, but the record does not give an empty-fiber convention or hypotheses guaranteeing the relevant nonemptiness. It also does not specify separability restrictions, so full extension degree is not replaced by separable degree.

The task is to find an optimal upper bound for this invariant. The parameters allowed in the bound are not stated. For example, after fixing $k,G,C$ and the precise spaces, a uniform version would seek the smallest possible $B$ such that \[ \forall p^{\mathrm{rd}}\in H^{\mathrm{rd}}(C)(k),\qquad m_k(p^{\mathrm{rd}})\leq B. \] A bound depending on additional data of the point, or uniform over varying fields and cover data, is a different request. Finiteness of such a $B$ and attainment of its optimal value are not assumed merely from the word "optimal."

The source does not define the field hypotheses, inertia conventions, quotient equivalence, or whether these are fine moduli spaces, coarse spaces or stacks. A residue-field lift to a moduli space need not be silently equated with descent of an actual cover carrying all its chosen structure. The literature triage checked 2026-08-21 mentions a cohomological obstruction in the report but says it does not yield the requested optimal upper bound and no general resolution was located. The obstruction and its precise hypotheses are absent; the open label and notation-only verification remain supplied metadata.

Short English statement

For a rational point of the reduced Hurwitz space, bound the smallest degree of a field extension over which it lifts to the unreduced space. Optimality needs a stated set of fixed parameters, and moduli-point fields must be distinguished from fields of definition of actual covers. The source does not specify those conventions or its cohomological obstruction.

Sources

• Local UnsolvedMath registry, permanent ID 30000403, source code catalogue entry, "Optimal Descent Degrees in Reduced Hurwitz Spaces," including the quotient, fiber-degree invariant and unchanged bound question.
• The Arithmetic of Fields (2006), Oberwolfach Reports, pp. 317--378, DOI 10.4171/owr/2006/06: https://doi.org/10.4171/owr/2006/06. This is the sole supplied literature reference, labelled "original Oberwolfach report."
• Literature triage checked 2026-08-21; local record updated 2026-09-16. No external source has been visited or independently checked here.