Resolutions of Zero-Mutable Log Structures

Definitions and mathematical statement

The supplied record does not define a zero-mutable log structure, a log resolution, or the additional condition log-crepant. These terms must be interpreted in the precise setting of the cited sources; their hypotheses are not reproduced in the record.

For purposes of recording the intended quantifiers, let $\mathcal Z$ denote the source-defined class of zero-mutable log structures, and let $\mathcal R_{\mathrm{lc}}(X)$ denote the collection of log-crepant log resolutions of $X$ in that setting. These are placeholders for source-specific definitions, not independent definitions of those notions.

Conjecture (normalized formulation supplied by the registry). \[ \forall X\in\mathcal Z,\qquad \mathcal R_{\mathrm{lc}}(X)\ne\varnothing. \] In words, every zero-mutable log structure admits a log-crepant log resolution.

Unresolved qualification. The supplied original passage says only: "We conjecture that zero-mutable log structures always have log resolutions." It does not explicitly impose log-crepancy. The record does not establish whether log-crepancy is part of the source's definition of log resolution or is an additional requirement. Accordingly, the normalized formulation above and the original formulation cannot be identified without checking the source definitions. The ambient category, base, and precise resolution and crepancy conditions also remain unspecified here. This is a statement-only draft requiring human mathematical review.

Short English statement

The intended conjecture is that every zero-mutable log structure has a log-crepant log resolution. The supplied original passage mentions log resolutions without explicitly requiring log-crepancy, and the technical definitions and ambient hypotheses are absent. The record labels the problem open; this status is supplied curation, not an independently verified conclusion.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006302$, title Resolutions of Zero-Mutable Log Structures, exact source problem code catalogue entry. The proposed year is recorded as 2025; no proposer is supplied.
• Toric Geometry (2025), Oberwolfach Reports, pp. 883--940, DOI 10.4171/owr/2025/19. Supplied workshop-report provenance: https://doi.org/10.4171/owr/2025/19.
• Corti, Graefnitz, and Ruddat, Singular Log Structures and Log Crepant Log Resolutions I. Supplied literature reference: https://arxiv.org/abs/2503.11610.
• Graefnitz, Smoothings from zero mutable Laurent polynomials. Supplied literature reference: https://arxiv.org/abs/2503.18661.
• Supplied literature assessment dated 2026-08-22: the primary paper and workshop report formulate the assertion as a conjecture, while subsequent constructions establish examples and infinite families rather than the general statement. This is supplied dated curation derived from the Oberwolfach Reports collection, not independently verified literature assessment; time-sensitive status and cited claims require independent checking.
• The registry assigns statement status corrected_verified and reports that the official workshop report was checked, the question isolated from surrounding exposition, and notation repaired for publication. These are supplied verification claims; no source consultation or independent verification is asserted in this draft.