Toric Orbits Meeting Subvariety Closures

Definitions and mathematical statement

Let $T=({\mathbb{C}}^*)^n$ be the given complex algebraic torus, let $Y\subseteq T$ be a subvariety, and let $X_\Sigma$ be a toric variety whose dense torus is identified with $T$. Write \[ \overline{Y}^{\,X_\Sigma} \] for the Zariski closure of $Y$ in $X_\Sigma$. For a point $x\in X_\Sigma$, its $T$-orbit is \[ T\cdot x=\{t\cdot x:t\in T\}. \] Question. For every such pair $(Y,X_\Sigma)$, characterize the set \[ \mathcal{I}(Y,X_\Sigma) = \left\{O\subseteq X_\Sigma: O\text{ is a }T\text{-orbit and }O\cap\overline{Y}^{\,X_\Sigma}\ne\varnothing\right\}. \] Thus the requested characterization should determine, for each $T$-orbit of the given toric variety, whether it meets the closure of $Y$. No completeness or projectivity hypothesis on $X_\Sigma$ is supplied.

The supplied curation describes this as a question extracted from a tropicalization passage, but states that the passage does not preserve a distinct conjecture or theorem to verify. Accordingly, this draft records the orbit-intersection question without asserting a proposed criterion or an independently verified open-problem status.

Short English statement

Given a subvariety of a complex algebraic torus and a toric variety containing that torus as its dense orbit, determine exactly which torus orbits meet the subvariety's Zariski closure. The supplied record labels the question open, while its dated assessment says that no distinct conjecture or theorem was preserved.

Sources

• Supplied local registry record: permanent UnsolvedMath ID $30002199$, title Toric Orbits Meeting Subvariety Closures, source problem code catalogue entry, collection Oberwolfach Reports Open Problems. No individual proposer is supplied; the proposed year is recorded as 2012.
• Supplied report citation: Mini-Workshop: Cohomology Rings and Fundamental Groups of Hyperplane Arrangements, Wonderful Compactifications, and Real Toric Varieties (2013), Oberwolfach Reports, pp. 2939--2983, DOI 10.4171/owr/2012/49. Supplied source URL and literature-source link, labeled "original OWR report": https://doi.org/10.4171/owr/2012/49. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature assessment, checked 2026-08-21: status "open"; the extracted tropicalization passage supplies background and a question about toric-orbit intersections, without preserving a distinct conjecture or theorem to verify. This is supplied dated curation, not independently verified here. The supplied background requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This verification is reported by the supplied record and has not been independently reproduced here.