Fields of Definition under Gluing of Hurwitz Components

Definitions and mathematical statement

The supplied question concerns fields of definition of components of Hurwitz spaces, but does not specify the Hurwitz spaces, the admissible fields, or the gluing operation.

For clarity, a conventional meaning of descent can be described conditionally. Let $K$ be a field, let $\overline K$ be an algebraic closure, and suppose that the relevant ambient space $H$ is defined over $K$. A specified geometric component $C\subseteq H_{\overline K}$ is defined over $K$, in the sense of embedded descent, if there is a closed subscheme $C_0\subseteq H$ whose base change to $\overline K$ is $C$, with its specified embedding. Whether the source intends this meaning, a field of moduli, or descent of additional covering data is not specified.

Question, in schematic form. For every field $K$ and every collection of Hurwitz-space components admitted by the intended construction, if all input components are defined over $K$, must each component obtained from them by the intended gluing construction also be defined over $K$?

This restates the supplied target, subject to unresolved qualifications: the permitted fields and input collections, the moduli problem, the meaning of component, the gluing construction, and the rule selecting its output components are absent. In particular, the universal quantifiers above range only over the unspecified admissible data; they do not assert that arbitrary fields or arbitrary Hurwitz components admit this construction. A fully precise mathematical formulation requires this missing context.

Short English statement

The question asks whether gluing Hurwitz-space components defined over a field $K$ preserves definition over $K$. The supplied record omits the construction and its hypotheses, so the statement remains schematic and requires human mathematical review.

Sources

• Local UnsolvedMath registry: supplied complete record for permanent ID $30005637$, title Fields of Definition under Gluing of Hurwitz Components, exact source problem code catalogue entry. The project identifies data/unsolvedmath/problems.json and the supplementary registry catalog/problems.json; their contents were not independently inspected for this draft. No individual proposer is supplied.
• Supplied report citation: MFO--RIMS Tandem Workshop: Arithmetic Homotopy and Galois Theory (2024), Oberwolfach Reports, pp. 2377--2488, DOI 10.4171/owr/2023/42. https://doi.org/10.4171/owr/2023/42. The record gives proposed year 2023 and an owr-year:2023 tag, whereas the supplied citation title gives 2024; this discrepancy is unresolved.
• Seguin, Fields of Definition of Components of Hurwitz Spaces. https://arxiv.org/abs/2303.05903. The supplied literature assessment says that this work proves the desired statement in several positive situations while leaving the general gluing question unresolved. The record supplies no theorem numbers or hypotheses for those situations.
• Supplied literature-source link labeled "OWR report": https://doi.org/10.4171/owr/2023/42. Literature checked at: 2026-08-22. The supplied status is partially_solved; the dated assessment derives from the supplied OWR curation and calls for independent verification of time-sensitive status and cited claims. This is supplied dated curation, not an independently verified assessment.
• The record labels its statement corrected_verified and reports comparison with the official Oberwolfach report. That verification claim is supplied metadata, not verification performed for this draft. Although the original-statement field says that the relevant report passage is retained verbatim, the supplied text contains only the short question and that assertion; no detailed surrounding passage is provided.