Nondivisorial Valuations in Explicit Fano Degenerations

Definitions and mathematical statement

Under the conventional interpretation of a real valuation on the function field of an integral variety $X$ over a field $k$, a valuation is a map $v:k(X)^\times\to{\mathbb{R}}$ satisfying $v(fg)=v(f)+v(g)$, $v(f+g)\geq\min\{v(f),v(g)\}$ whenever $f+g\neq0$, and $v(a)=0$ for $a\in k^\times$. Such a valuation is divisorial if $v=c\,\operatorname{ord}_E$ for some $c>0$ and some prime divisor $E$ on a normal birational model of $X$. It is nondivisorial if no such representation exists. These definitions are conditional on this being the intended valuation domain; the supplied excerpt does not identify that domain.

The requested construction is an explicit Fano variety $X$, together with an explicit nontrivial canonical degeneration of $X$, for which the associated entropy-minimizing valuation $v$ is nondivisorial. Under the function-field interpretation above, the last requirement is \[ v\neq c\,\operatorname{ord}_E \qquad\text{for every }c>0\text{ and every prime divisor }E \text{ on every normal birational model of }X. \] The supplied record does not specify the ground field, the permitted singularities of $X$, the canonical degeneration construction, the meaning of nontriviality, the entropy functional, or its admissible and normalized valuations. Consequently, this is a qualified construction target, not a fully specified minimization problem.

The original excerpt asks for explicit nontrivial examples and expresses the additional hope of finding a nondivisorial valuation. The reconstructed statement combines these into one stronger target. The supplied dated assessment distinguishes the availability of explicit nontrivial Kähler--Ricci-soliton Fano examples, including families with nontrivial moduli, from the remaining request for an explicitly computed nondivisorial minimizing valuation.

Short English statement

Construct an explicit nontrivial canonical degeneration of a Fano variety and explicitly identify its entropy-minimizing valuation as nondivisorial. The supplied excerpt leaves the degeneration and entropy framework unspecified, so the precise hypotheses and valuation domain require human review.

Sources

• Local UnsolvedMath registry, supplied complete source record for permanent ID $30005079$, source code catalogue entry, titled "Nondivisorial Valuations in Explicit Fano Degenerations." The record belongs to the Oberwolfach Reports Open Problems collection, lists proposed year 2022, and supplies no individual proposer.
• Supplied report citation: Toric Geometry (2023), Oberwolfach Reports, DOI 10.4171/owr/2022/17, pp. 861--927. https://doi.org/10.4171/owr/2022/17. The literature-source list also labels this reference "Oberwolfach Report 2022/17." The supplied citation's 2023 designation and the report's 2022 designation are retained without reconciliation.
• Supplied literature reference: Miao--Wang, Mathematische Zeitschrift. https://doi.org/10.1007/s00208-025-03202-w. No article title, theorem number, or relevant passage is supplied; the author, venue, and DOI association is retained as given.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and reports explicit nontrivial Kähler--Ricci-soliton Fano examples, including families with nontrivial moduli, while describing an explicitly computed nondivisorial minimizing valuation as unresolved. This is supplied dated curation, not independently verified literature assessment.
• The registry labels the reconstructed statement corrected_verified and says it was checked against the official report. This is a supplied verification claim; neither that check nor the cited literature has been independently verified for this draft.