A formal local model at a type-2b stable numerical quintic

Definitions and mathematical statement

Understand the geometry of moduli of stable surfaces by identifying tractable components and studying their singularities and boundary. A stable surface in the usual characteristic-zero KSBA setting is projective, semi-log-canonical, with ample ${\mathbb{Q}}$-Cartier canonical divisor; a particular moduli problem fixes its numerical invariants and compatible families. The original fixes no invariants or component. It warns that, even after fixing differentiable type in the smooth locus, moduli can be disconnected, connected components reducible, and irreducible components everywhere nonreduced; the boundary need not be a divisor or normal crossings. It also attributes arbitrarily complicated singularities of canonically polarized surface moduli to Vakil, without giving the precise singularity-type statement.

The attached unreviewed aid proposes one formal local example. Let $W$ be a general Rana type-2b stable numerical quintic, retaining the classification label because its definition is not supplied. Its distinguished Wahl point has type $\frac14(1,1)$, the cyclic quotient by $\mu_4$ acting on two local coordinates with weights $(1,1)$. The aid claims that the completed miniversal ${\mathbb{Q}}$-Gorenstein deformation ring and the distinguished-singularity-preserving locus are \[ R={\mathbb{C}}[[a,b,z_1,\ldots,z_{38},\tau]]/(ab),\qquad \tau=0. \] The two ambient branches are smooth of dimension $40$ and meet in dimension $39$. The reduced Cartier locus $\tau=0$ has two smooth $39$-dimensional branches meeting in the $38$-dimensional type-2b stratum. The proposed normalization and conductor are \[ \overline R={\mathbb{C}}[[a,z_1,\ldots,z_{38},\tau]]\oplus {\mathbb{C}}[[b,z_1,\ldots,z_{38},\tau]],\qquad \mathfrak c=(a,b)\subset R. \] Thus the conductor image inside the distinguished boundary is $V(a,b,\tau)$, while the ambient conductor locus is $V(a,b)$; these have different dimensions.

This ring is asserted for a miniversal/stack chart. With $G=\operatorname{Aut}(W)$, the aid states the formal stack germ as $[\operatorname{Spf}R/G]$ and the coarse completed local ring as $R^G$. Its $G$-action is uncomputed, so the displayed hypersurface equation cannot simply be called the coarse moduli germ. The aid describes Rana's transverse equisingular branches and a product with one Wahl-smoothing parameter as inputs, but does not provide their full hypotheses or citations. This local example is only one proposed step toward the broader moduli-geometric problem.

Short English statement

Study useful components and boundary geometry of stable-surface moduli. The aid gives a proposed normal-crossing formal deformation chart at one classified surface, while keeping that chart separate from the coarse quotient by automorphisms.

Sources

• Problem 4.2, AIM Compact moduli spaces and birational geometry: https://aimath.org/WWN/birational/birational.pdf. Canonical location: aim-algebraic-geometry-notes.json, notes[335]. The supplied text includes an appended page header and following "4.1. Intersection theory" heading, which have no mathematical role in this question. No PDF or source TeX was consulted here.
• The original attributes arbitrary moduli singularities to Vakil without a complete reference. The aid invokes Rana's type-2b results and Wahl smoothing without a complete citation or classification definition, and proposes the formal model with low novelty confidence. It is machine generated and not peer reviewed; original remarks and literature are empty.
• Local registry: data/unsolvedmath/problems.json, permanent catalogue entry, exact source code catalogue entry.