Singular Loci of Maximal Flat Flag Degenerations

Definitions and mathematical statement

Retain the source notation $\operatorname{Fl}^{mf}(V)$ for the maximal flat linear degeneration of the flag variety associated with $V$, and $C(A)$ for an irreducible component of that degeneration. The supplied record does not define the degeneration, the admissible indices $A$, or the dependence of the component on $A$.

For a locally Noetherian scheme $X$, define its singular set in the regular-local-ring sense by \[ \operatorname{Sing}(X)=\{x\in X:\mathcal O_{X,x}\text{ is not a regular local ring}\}. \] If $\mathfrak m_x$ is the maximal ideal of $\mathcal O_{X,x}$ and $\kappa(x)=\mathcal O_{X,x}/\mathfrak m_x$, regularity means \[ \dim_{\kappa(x)}\left(\mathfrak m_x/\mathfrak m_x^2\right)=\dim\mathcal O_{X,x}. \] This defines a set of points; it does not specify a scheme structure on the singular locus.

Problem, conditional on the source's intended construction. For every irreducible component $C(A)$ of $\operatorname{Fl}^{mf}(V)$, determine its intrinsic singular locus \[ \operatorname{Sing}\left(C(A)\right). \] The target is the singular locus of the component itself, rather than the singular locus of the entire degeneration.

Unresolved qualifications. The supplied record specifies neither the ground field and its characteristic, nor the dimension and hypotheses on $V$, nor the construction of $\operatorname{Fl}^{mf}(V)$, nor the indexing data $A$. It also does not specify the scheme structure intended on $C(A)$, whether only the set of singular points is requested, or whether singularity is intended as nonregularity or failure of smoothness over a ground field. The displayed definition makes one conventional meaning explicit without asserting that the source has fixed these choices. An explicit characterization for the intended family requires these missing definitions and hypotheses.

Short English statement

Determine the singular points of each irreducible component $C(A)$ of the maximal flat linear degeneration $\operatorname{Fl}^{mf}(V)$ of a flag variety. The supplied record does not define this family or its component indices, so the statement remains conditional on the intended source construction. Its supplied partially solved status is dated curation, not an independently verified conclusion.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004266$, source problem code catalogue entry, title Singular Loci of Maximal Flat Flag Degenerations, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2019 and names no individual proposer.
• Supplied report citation: Mini-Workshop: Degeneration Techniques in Representation Theory (2020), Oberwolfach Reports, pp. 2869--2909, DOI 10.4171/owr/2019/46. Supplied source link: https://doi.org/10.4171/owr/2019/46. The background describes the report as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source labeled OWR 2019 46: https://ems.press/content/serial-article-files/46825.
• Supplied literature source labeled Di Trani 2026: https://alco.centre-mersenne.org/articles/10.5802/alco.477/. No article title or fuller bibliographic attribution is supplied in the record.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and states that recent work gives criteria for smooth points in broad flat linear degenerations, providing partial information about singular loci, while the singular locus of every component in the extracted family is not determined. This is supplied dated curation derived from the OWR material, not independently verified here; its status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review asserting self-containment and preservation of the target. This is supplied review metadata, not independent confirmation of the missing mathematical definitions or hypotheses.