Deformation of pairs of $\mathbb{P}^3$ and hypersurfaces

Abstract: Motivated by DeVleming's work on moduli of surfaces in $\mathbb{P}<sup>3$ and Chen-Hu-Jiang's work on moduli of threefolds with volume $2$ and geometric genus $4$, we study the deformation of pairs of $\mathbb{P}<sup>3$ and hypersurfaces using the classification of $\mathbb{Q}$-Gorenstein degenerations of $\mathbb{P}<sup>3$ with canonical singularities. We prove that if a degenerating threefold has canonical singularities, then the moduli space is smooth at the corresponding pair. Consequently, we find some boundary divisors of the moduli of smooth hypersurfaces. Finally, using the double cover method, we derive some information on the moduli space of threefolds $X$ with canonical singularities with the same volume and geometric genus as a double cover of $\mathbb{P}<sup>3$ branched over a hypersurface.

• The paper develops a rigorous framework for the deformation theory of (P3, B_d) pairs, identifying explicit Q-Gorenstein degenerations with canonical singularities.
• It classifies degenerations into four types (I-IV) and details the structure of boundary divisors in the compactified moduli via KSBA stability.
• The work employs toric geometry, vanishing theorems, and tangent sheaf computations, providing transferable tools for higher-dimensional moduli studies.

Deformation Theory of Pairs $(\mathbb{P}^3, \text{Hypersurface})$ and $Q$-Gorenstein Degenerations

Overview and Context

This paper develops a rigorous treatment of the deformation theory of pairs $(\mathbb{P}^3, B_d)$, where $B_d$ is a smooth hypersurface of degree $d$, with a focus on degenerations of $\mathbb{P}^3$ with canonical singularities. The work is motivated by questions in birational geometry concerning the compactification and boundary structure of moduli spaces of such pairs and their double covers, extending and formalizing results in the context of higher-dimensional algebraic varieties.

The authors build on the classification of $Q$-Gorenstein degenerations of $\mathbb{P}^3$ with canonical singularities, notably the results of H\"oring and Peternell, to systematically investigate the deformation and compactification theory of these pairs, and the associated moduli. Key cases (denoted as types I–IV) are analyzed, each exhibiting distinct features regarding their singularities, their behavior in moduli spaces, and the structure of their boundary divisors.

Classification of $Q$-Gorenstein Degenerations of $\mathbb{P}^3$

A central point of departure is the classification of normal projective threefolds $Q$0 which are $Q$1-Gorenstein degenerations of $Q$2 with canonical singularities. Four types arise:

• Type I: $Q$3.
• Type II: Projective cones over a smooth quadric surface, realized as images of certain projective bundles over $Q$4.
• Type III: Weighted projective spaces $Q$5, corresponding to cones over weighted projective planes, also described via degenerations involving Hirzebruch surfaces.
• Type IV: More intricate fibrations associated with $Q$6-bundles over $Q$7, whose anticanonical models have canonical singularities along a contracted section.

This precise classification sets the foundation for understanding which degenerations can arise as limits of the pairs $Q$8 in the moduli space.

Moduli Compactification for Pairs and Boundary Analysis

Given the moduli space $Q$9 of pairs $(\mathbb{P}^3, B_d)$0, the paper discusses the compactification of $(\mathbb{P}^3, B_d)$1 into a larger moduli space $(\mathbb{P}^3, B_d)$2 of log pairs $(\mathbb{P}^3, B_d)$3 under the framework of KSBA stability, imposing conditions such as semi-log-canonicity, ampleness of the log canonical class, and $(\mathbb{P}^3, B_d)$4-Cartier hypotheses. This enables the authors to precisely locate boundary divisors corresponding to limits where $(\mathbb{P}^3, B_d)$5 degenerates to a canonical singularity threefold $(\mathbb{P}^3, B_d)$6 of types II, III, or IV.

One notable assertion is the smoothness of the moduli space at boundary points corresponding to pairs with canonical singularities (Theorem 1), provided the specific type (particularly type IV) is addressed via refined deformation-theoretic arguments.

Double Covers and the Geometry of Even Degree Hypersurfaces

A substantial portion examines the relationship between the pair moduli and the moduli space of threefolds $(\mathbb{P}^3, B_d)$7 realized as double covers of $(\mathbb{P}^3, B_d)$8 branched along even degree hypersurfaces $(\mathbb{P}^3, B_d)$9. Explicitly, such threefolds $B_d$0 satisfy:

• $B_d$1,
• $B_d$2,

and their moduli are denoted $B_d$3 and $B_d$4 for canonical and smooth cases, respectively. Through the double cover correspondence, $B_d$5 is realized as a subset of $B_d$6 for $B_d$7 even.

The work details how type II and III degenerations induce well-behaved, explicit boundary divisors in $B_d$8, computing the relevant deformation spaces using tangent sheaf techniques, toric structures, and vanishing theorems (notably Kawamata–Viehweg).

Boundary Divisors: Strong Results and New Claims

The authors prove several strong and technically intricate results:

• For type II degenerations and even degrees $B_d$9, the boundary divisor in the compactified moduli corresponds to smooth complete intersections in $d$0 of multi-degrees $d$1.
• For odd $d$2, the pair $d$3 has a unique $d$4 singularity at the vertex—a fact significant for the finer structure of moduli.
• For type IV, the pair $d$5 can only appear as a boundary divisor if $d$6 (i.e., $d$7 is divisible by 4). Such divisors then parametrize smooth surfaces with a pencil structure, with intricate behavior relative to singularities and the conic fibration structure of the ambient threefold.

A sequence of vanishing and smoothness results is provided, ensuring that the moduli spaces are generically smooth at these boundary loci for canonical singularity cases. Detailed spectral-sequence-based computations of tangent and normal sheaf cohomologies underpin these results.

Theoretical and Practical Implications

The main theoretical import is a granular understanding of the deformation functors associated with degenerate pairs $d$8, showing the absence of obstructed deformations in the presence of canonical singularities (especially for types II and IV in certain cases). This allows for explicit identifications of divisorial boundaries within compactified moduli spaces. Importantly, the explicit realization of moduli points as complete intersections or pencil structures on projective bundles constrains the geometry of degenerations and informs the possible compactification scenarios in moduli theory.

On the practical level, the methods developed—combinatorial analysis of toric bundles, exact tangent sequence computations for double covers, and vanishing theorems—are transferable tools for studying higher-dimensional moduli problems and for constructing compactifications in related settings (such as K3 surfaces, Fano varieties, and higher genus analogues).

Directions for Future Development

Several avenues for further research arise:

• Extending these deformation-theoretic results to broader classes of Fano varieties or higher-dimensional analogues, potentially using similar $d$9-Gorenstein degeneration frameworks.
• Exploring the birational geometry of moduli spaces near these explicit boundary divisors, especially for more complex singularity types or in cases where the obstruction theory is nontrivial.
• Investigating arithmetic aspects, such as rationality of moduli spaces in higher degrees and the structure of their effective cones, using the explicit descriptions of boundary divisors.
• Leveraging the methods here to study wall-crossing phenomena in KSBA-type moduli, where the analysis of boundary behavior is crucial.

Conclusion

This paper offers a comprehensive and technically detailed account of the deformation theory of pairs $\mathbb{P}^3$0 through the lens of $\mathbb{P}^3$1-Gorenstein degenerations and canonical singularities, with a systematic classification of possible degenerations and explicit description of the associated boundary divisors in moduli. The analysis sets a benchmark for the application of rigorous deformation-theoretic and toric geometric methods in higher-dimensional moduli theory, and raises further questions about the general structure and compactification of moduli in algebraic geometry.