Combinatorics of Bondal-Thomsen Polytopes

Definitions and mathematical statement

For terminology only, a polytope in a finite-dimensional real vector space $V$ is the convex hull of a finite nonempty subset of $V$. Its nonempty faces are the sets on which some linear functional attains its maximum; the empty face is also included. If an ambient lattice $M\subset V$ is specified, its lattice points are $P\cap M$. Two vertices are adjacent when they are the endpoints of an edge. These conventions do not specify the Bondal--Thomsen construction or resolve what the source intends by adjacency.

Requested research task. Determine the combinatorial properties and significance of the Bondal--Thomsen polytopes, including their face structure, lattice points, adjacency, and their relationships with toric-arrangement strata and derived categories.

Unresolved formulation. The supplied record does not define the Bondal--Thomsen polytopes, their ambient spaces or lattices, their indexing data, or the hypotheses on the associated toric geometry. It also does not specify the relevant arrangements, strata, categories, or comparison maps. In particular, adjacency might concern vertices, faces, or distinct polytopes. Consequently, the requested task is a research programme rather than a uniquely specified proposition: its objects, quantifiers, and intended conclusions cannot be made precise from the supplied evidence alone. No integrality, smoothness, completeness, projectivity, or other geometric hypothesis is imposed by this draft.

Short English statement

The source asks for a combinatorial understanding of Bondal--Thomsen polytopes and their connections with toric arrangements and derived categories. The defining construction and intended comparison statements are absent, so a precise theorem or conjecture cannot yet be stated. The supplied partially solved label is dated curation, not an independently verified conclusion.

Sources

• Local UnsolvedMath problem registry, permanent ID $30006303$, source code catalogue entry, title Combinatorics of Bondal--Thomsen Polytopes. The supplied record attributes the question to the 2025 Oberwolfach workshop context; no individual proposer is identified.
• Toric Geometry (2025), Oberwolfach Reports, pp. 883--940, DOI 10.4171/owr/2025/19. Supplied report reference: https://doi.org/10.4171/owr/2025/19. No precise page or passage for this question is supplied.
• Bauermeister et al., Strata of toric hyperplane arrangements. Supplied literature link: https://arxiv.org/abs/2507.14356.
• Ballard et al., Realizing King's Conjecture via Cox Category. Supplied literature link: https://arxiv.org/abs/2501.00130.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and says it is not settled in full. It reports that the 2025 report gives a definition and a combinatorial description through the Cox category, and that Bauermeister et al. prove a bijection between relevant toric-arrangement strata and zonotope lattice points and relate dimensions. These are supplied dated curation claims, not independently verified mathematical or literature assessments; their hypotheses and precise scope require checking in the cited sources.