Minimal Rational Curves on Almost Homogeneous Varieties

Definitions and mathematical statement

The supplied problem calls a smooth projective variety $X$ almost homogeneous if it is equipped with an algebraic action $G\times X\to X$ of a connected linear algebraic group $G$ having an open orbit. The base field, its characteristic, and whether the open-orbit condition is geometric are not specified.

For clarity, one possible interpretation of the remaining terminology is the following, conditional on working over an algebraically closed field $k$. A rational curve on $X$ is an integral curve whose normalization is isomorphic to ${\mathbb{P}}^1_k$. Write $\operatorname{RatCurves}^{n}(X)$ for the normalized parameter space of rational curves on $X$. A family is an irreducible component $\mathcal K$ of this parameter space. It is covering if the union of its curves is dense in $X$. Under a common convention, such a covering family is minimal if, for a general point $x\in X$, the normalized parameter space $\mathcal K_x$ of members passing through $x$ is nonempty and proper over $k$. Here "for a general point" means for every point in some dense open subset of $X$. This convention does not require $\mathcal K$ itself to be proper.

Requested classification. For every smooth projective almost homogeneous variety $X$, describe all its minimal families of rational curves.

The preceding definition of minimality is an explicit conditional interpretation, not a definition established by the supplied source record. The record does not specify whether this covering-family convention, a degree-minimizing convention, or another convention is intended. It also does not specify the form of the requested description. Consequently, the field assumptions and the intended meaning of minimality must be resolved before this can be treated as an unqualified precise classification problem. No positivity, Fano, or rational-connectedness hypothesis is supplied.

Short English statement

Describe all minimal families of rational curves on every smooth projective variety admitting an action of a connected linear algebraic group with an open orbit. The field assumptions and the meaning of "minimal family" remain unresolved; the mathematical formulation above supplies only a conditional standard interpretation.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30003100$, title "Minimal Rational Curves on Almost Homogeneous Varieties," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied proposed year is 2016; no individual proposer is identified.
• Supplied report citation: Toric Geometry (2017), Oberwolfach Reports, pp. 963--1026, DOI 10.4171/owr/2016/19: https://doi.org/10.4171/owr/2016/19. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The literature-source list labels this reference "OWR 2016 report." The differing year labels are retained without reconciliation.
• Supplied literature reference, labeled "Brion--Perrin--Kim": https://arxiv.org/abs/2303.03013. No title or more detailed attribution is supplied for this reference.
• Supplied literature assessment dated 2026-08-21: status "partially solved." The assessment states that toric and complete symmetric or wonderful cases have been described, while a description for every smooth projective almost homogeneous variety remains incomplete. This is supplied dated curation derived from the OWR record, not an independently verified assessment. The record expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says that mathematical content was preserved while notation, typography, or omitted definitions were repaired for self-contained publication. These are supplied editorial assertions, not independent verification of this draft or its conditional definitions.