Normal directions and forced nodes for two explicit stable limits

Definitions and mathematical statement

Give explicit moduli descriptions and stable degenerations of surfaces of general type, especially octic double planes, quintic hypersurfaces, bidouble covers and other abelian covers, following the work named in the original. An octic double plane is a double cover of ${\mathbb{P}}^2$ branched over a degree-eight curve; a quintic surface is a degree-five hypersurface in ${\mathbb{P}}^3$; a bidouble cover has group $({\mathbb{Z}}/2)^2$. Stable degenerations are understood in a specified KSBA family, with semi-log-canonical central surface and ample ${\mathbb{Q}}$-Cartier canonical divisor. The source does not fix invariants, cover data or a classification target beyond these examples.

The attached unreviewed aid studies two complex simple-normal-crossing limits. First, $X_0=V(\ell q_4)\subset{\mathbb{P}}^3$ is a plane and a smooth quartic meeting transversely in a smooth plane quartic $C$. With $V_j=H^0({\mathbb{P}}^3,\mathcal O(j))$, it claims the fixed-ambient normal quotient \[ V_5/(\ell V_4+q_4V_1)\simeq H^0(C,\mathcal O_C(5))\simeq H^0(C,T^1_{X_0}),\quad\dim=18. \] Second, the octic double plane $z^2=q_4^2$ is the union of two plane components along a smooth quartic $C$. For $V_j=H^0({\mathbb{P}}^2,\mathcal O(j))$ in this second formula, it claims \[ V_8/(q_4V_4)\simeq H^0(C,\mathcal O_C(8))\simeq H^0(C,T^1_{X_0}),\quad\dim=30. \] Here $T^1$ is the sheaf of first-order local deformation directions, $\mathcal E xt^1(\Omega_{X_0},\mathcal O_{X_0})$, supported on the double curve.

The aid reports that general equation smoothings give Gorenstein KSBA families with exactly $20$ and $32$ ordinary threefold double points in the respective total spaces. It attributes the unavoidable singular total space to the positive degree of $T^1$: neither unchanged reduced SNC central fiber admits a regular-total-space smoothing, including after higher-order or ramified base change. In the projectivized normal spaces, it claims that directions with multiple zeros on $C$ form irreducible discriminant hypersurfaces of degrees $44$ and $68$, respectively. These are fixed-ambient deformation calculations; they are not a complete quotient-moduli description or a classification of stable limits of all bidouble and abelian covers. The genericity and family hypotheses and full proof are not supplied.

Short English statement

Construct explicit compactified surface-moduli examples and describe their stable limits. The aid treats two SNC limits, computing normal deformation directions, singularities of general smoothing total spaces and discriminants, while leaving the broader cover families open.

Sources

• Problem 4.7, "Explicit examples," AIM Compact moduli spaces and birational geometry: https://aimath.org/WWN/birational/birational.pdf. Canonical location: aim-algebraic-geometry-notes.json, notes[340]. The source names Catanese, Manetti and Pardini for covers, without complete bibliographic entries. No PDF was consulted.
• The attached machine-generated, non-peer-reviewed aid proposes the paired normal-direction calculations with low novelty confidence. It supplies no full primary references or precise equations and genericity hypotheses for the smoothing families; original remarks and literature are empty.
• Local registry: data/unsolvedmath/problems.json, permanent catalogue entry, exact source code catalogue entry.