Orbit Closures in Degenerate Flag Varieties

Definitions and mathematical statement

The supplied record concerns a degenerate flag variety $X=\operatorname{Fl}^{a}(V)$, an action of a group denoted $G_a$, and a parametrization of its orbits by tuples of coordinate subsets. The record does not define the degeneration, the group or its action, the ground field, the dimension of $V$, the chosen coordinates, or the admissibility conditions on these tuples. Consequently, the following formulation retains these objects as source-dependent data rather than selecting a particular construction.

Let $\Lambda$ denote the source's unspecified set of admissible tuples. For $\lambda\in\Lambda$, write $\mathcal O_\lambda\subseteq X$ for the corresponding orbit and $Z_\lambda=\overline{\mathcal O_\lambda}$ for its closure, with closure understood in the Zariski topology of the intended algebraic variety. The requested tasks are:

1. Determine, for every $\lambda,\mu\in\Lambda$, a criterion in terms of their coordinate subsets for \[ \mathcal O_\mu\subseteq Z_\lambda, \qquad\text{equivalently}\qquad Z_\mu\subseteq Z_\lambda. \]
2. Classify the minimal singularities of the varieties $Z_\lambda$, for all $\lambda\in\Lambda$. The source does not specify the meaning of minimality, the relevant singular strata, or the equivalence relation under which singularities are to be classified; this part therefore remains an incompletely specified classification request.
3. Describe the intersection cohomology complexes $\operatorname{IC}(Z_\lambda)$ for all $\lambda\in\Lambda$. The coefficient system, sheaf-theoretic setting, normalization, and desired form of the description are not specified in the supplied record.

These are the mathematical targets recorded by the source. A fully specified problem requires the missing geometric and sheaf-theoretic data; no choice of these data is asserted here.

Short English statement

For the intended group action on the intended degenerate flag variety, determine which orbits lie in the closures of other orbits, classify the minimal singularities of those closures, and describe their intersection cohomology complexes. The source leaves essential definitions and conventions unspecified, so this is a conditional statement-only draft requiring mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004263$, title Orbit Closures in Degenerate Flag Varieties, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year $2019$ and names no proposer.
• Supplied report citation: Mini-Workshop: Degeneration Techniques in Representation Theory (2020), Oberwolfach Reports, pp. 2869--2909, DOI 10.4171/owr/2019/46. https://doi.org/10.4171/owr/2019/46. The supplied background describes Oberwolfach Reports 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 further bibliographic details are supplied.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and states that recent work gives combinatorial descriptions and smooth-locus criteria for broad classes of flat linear degenerations, while a general classification of minimal singularities and intersection cohomology complexes on all orbit closures remains unresolved. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The source itself requests independent verification of time-sensitive status and cited claims.