Modular Compactifications from Curve Singularities

Definitions and mathematical statement

Let $\mathcal S$ be a deformation-open collection of isolated curve singularities. A customary interpretation of deformation-openness is that, in an admissible flat family of curves, the locus of fibers whose singularities all belong to the allowed collection is open. The source does not specify the category of singularities, permitted families or base field, so this interpretation needs confirmation.

The question is whether one can associate to $\mathcal S$ a stability condition on curves such that its moduli object is a proper compactification of the relevant moduli of smooth curves. For fixed genus and any intended marking data, denote the latter moduli object by $\mathcal M_{\mathrm{sm}}$. A precise target would require a moduli object $\overline{\mathcal M}_{\mathcal S}^{\mathrm{st}}$ with a dense open immersion \[ \mathcal M_{\mathrm{sm}}\hookrightarrow\overline{\mathcal M}_{\mathcal S}^{\mathrm{st}}, \] where the right-hand object is proper over the specified base and parametrizes curves satisfying the proposed stability condition, with singularities allowed as prescribed by $\mathcal S$. The stability rule should thus be genuinely related to the given singularity collection, not merely an unrelated proper compactification.

The source leaves "corresponding" unspecified: it may mean allowing only singularities in $\mathcal S$, realizing particular singularities, or imposing more precise local and global conditions. These alternatives are not equated, and no requirement that every allowed singularity occur at every fixed genus is inserted. Likewise, stack properness, properness of a coarse moduli space, automorphism restrictions and separatedness requirements must be distinguished and specified. No particular GIT or other pre-existing notion of stability is chosen.

The source asks about arbitrary deformation-open collections; its assessment uses the additional word "reasonable" without defining admissibility conditions. Whether deformation-openness alone suffices, and what additional restrictions are necessary, are therefore part of the unresolved formulation.

The supplied triage checked 2026-08-21 labels the question partially solved. It reports proper modular compactifications for important classes such as cusps and elliptic $m$-fold points, but no answer for an arbitrary reasonable collection. The registry also reports manual review finding the extracted statement coherent and exact. Those source-review labels do not supply the missing moduli and stability conventions.

Short English statement

Given deformation-open allowed curve singularities, ask for a stability condition producing a proper modular compactification of smooth-curve moduli. The intended genus, markings, moduli category and relation between stability and the allowed singularities must be specified.

Sources

• Local UnsolvedMath registry, permanent ID 30000972, source code catalogue entry; supplied complete record, including its exact-statement manual-review note and dated partial-progress assessment.
• Classical Algebraic Geometry (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/26, pp. 1425--1474: https://doi.org/10.4171/owr/2008/26. Proposed-year metadata gives 2008.
• Smyth, modular compactifications of pointed elliptic curves: https://openresearch-repository.anu.edu.au/items/4cdbb338-d369-464b-b747-0a89ca28f3b7.