Optimal Pluricanonical Bounds for Stable Log Surfaces

Definitions and mathematical statement

The supplied record concerns stable log surfaces $(X,\Delta)$ and the Gorenstein locus \[ U=\{x\in X:\mathcal O_{X,x}\text{ is Gorenstein}\}. \] In the usual terminology, a stable log surface is a projective surface pair with semi-log-canonical singularities for which $K_X+\Delta$ is an ample ${\mathbb{Q}}$-Cartier divisor. The record does not specify the ground field, the allowed coefficients of the effective boundary $\Delta$, or additional conventions defining the class of surfaces. These choices must be fixed before a uniform bound can be stated unconditionally. On $U$, write $\omega_U$ for the invertible dualizing sheaf and interpret $rK_U$ as the line bundle $\omega_U^{\otimes r}$.

The first supplied question asks for the optimal positive integer $r$ such that $rK_U$ induces a birational map or an embedding. Under the interpretation that the bound is uniform, it asks for the least positive integers $r_{\mathrm{bir}}$ and $r_{\mathrm{emb}}$, separately, such that for every stable log surface in the intended class the map induced by $r_{\mathrm{bir}}K_U$ is birational onto its image, respectively the map induced by $r_{\mathrm{emb}}K_U$ is an embedding. Existence of such uniform integers is part of the question.

This formulation remains conditional: the record does not specify whether sections are taken on $U$ or obtained by restriction from $X$, whether the maps are required only on $U$ or on $X$, or whether an embedding means a locally closed immersion or a closed immersion. Since $U$ need not be proper, using all sections on $U$ does not automatically specify a finite-dimensional projective linear system. Moreover, the first question explicitly uses $K_U$, whereas the second uses $K_X+\Delta$; no identification or replacement of these divisors is justified by the supplied record.

For the second question, if $I$ denotes the global Cartier index of $K_X+\Delta$, define \[ I=I(X,\Delta):=\min\{m\in{\mathbb{Z}}_{>0}:m(K_X+\Delta)\text{ is Cartier}\}. \] With this interpretation, the requested existence statement is \[ \exists\,(X,\Delta)\text{ a stable log surface in the intended class} \quad\text{such that}\quad \mathcal O_X\left(5I(X,\Delta)(K_X+\Delta)\right) \text{ is not very ample}. \] Here a line bundle is very ample if its complete linear system defines a closed immersion into projective space. The meaning of $I$ is not explicitly supplied, so this precise interpretation requires confirmation from the original source.

Short English statement

Determine the smallest uniform pluricanonical multiples giving birational maps and embeddings for the stated stable log surfaces, and ask whether the index-normalized log canonical multiple $5I(K_X+\Delta)$ can fail to be very ample. The ground field, boundary conventions, section spaces, embedding target, and meaning of $I$ require clarification. This is a statement-only draft requiring human mathematical review.

Sources

• Local registry record: permanent UnsolvedMath ID $30002343$, source problem code catalogue entry; project registry data/unsolvedmath/problems.json, with catalog/problems.json designated as the supplementary registry. The supplied record is titled "Optimal Pluricanonical Bounds for Stable Log Surfaces."
• Supplied report citation: Complex Algebraic Geometry (2014), Oberwolfach Reports, DOI 10.4171/owr/2013/27, pp. 1563--1627. https://doi.org/10.4171/owr/2013/27. The record gives proposed year $2013$ and names no individual proposer. The report-year discrepancy is unresolved.
• Supplied literature reference: Liu--Rollenske, Pluricanonical maps of stable log surfaces. https://arxiv.org/abs/1211.1291.
• Supplied publication reference: Advances in Mathematics publication. https://doi.org/10.1016/j.aim.2014.03.009.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports that the index multiples $4I$ and $8I$ are respectively base-point free and very ample, with stronger results for semi-canonical surfaces. It states that the optimal uniform $r$ and the proposed $5I$ counterexample question remain unresolved. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its hypotheses, terminology, bounds, and time-sensitive status require independent checking.
• The supplied statement status is corrected_verified, accompanied by a claim of manual semantic, typographic, and LaTeX review. This records the supplied editorial assessment and does not establish independent source verification or mathematical correctness of this draft.