The boundary of K-moduli of prime Fano threefolds of genus twelve

Abstract: We study the K-moduli stack of prime Fano threefolds of genus twelve, known as $V_{22}$. We prove that its boundary, which parametrizes singular members, is purely divisorial and consists of four irreducible components corresponding to the four families of Prokhorov's one-nodal $V_{22}$. A key ingredient is a modular relation between Fano threefolds $X$ and their anticanonical K3 surfaces $S$. We prove that the forgetful morphism from the moduli of Fano--K3 pairs $(X,S)$ where $X$ is a K-semistable degeneration of $V_{22}$ to the moduli space of genus $12$ polarized K3 surfaces $(S,{-K_X}|S)$ is an open immersion. In particular, the K-moduli of $V{22}$ is governed by the moduli of their anticanonical K3 surfaces, providing a modular realization of Mukai's philosophy. Along the way, we develop a general deformation framework for Fano threefolds of large volume, which may be useful beyond the study of K-moduli.

• The paper establishes a precise divisorial description of the K-moduli boundary for V22, identifying four irreducible one-nodal degeneration families.
• It demonstrates an open immersion from the moduli of Fano–K3 pairs to the moduli of degree-22 polarized K3 surfaces, confirming Mukai’s conjectural framework.
• The study develops a deformation criterion generalizing Beauville’s theorem, effectively controlling singularities and smoothability in Fano threefold degenerations.

Boundary Structure of K-Moduli for Genus 12 Prime Fano Threefolds

Introduction and Motivation

The paper "The boundary of K-moduli of prime Fano threefolds of genus twelve" (2603.29827) addresses the explicit description of the boundary of the K-moduli stack for prime Fano threefolds of genus twelve, denoted $V_{22}$. Prime Fano threefolds are those with Picard rank and Fano index one; within this class, $V_{22}$ stands out as a $6$-dimensional family whose general member is known to be K-stable, but for which the K-polystable locus and degeneration behavior remain subtler than for more classically studied Fano threefolds.

The paper tackles two open problems: (1) giving a precise divisorial description of the boundary of the K-moduli stack for $V_{22}$, and (2) explicating the relationship between a Fano threefold and its anticanonical K3 divisors via their moduli. Central to their approach is the adoption of Mukai's philosophy, which posits that the geometry and deformation theory of a prime Fano threefold is governed by its anticanonical K3 surface sections.

Summary of Main Results

The authors prove that the boundary of the K-moduli stack for $V_{22}$ is purely divisorial with four irreducible components, each corresponding to one of Prokhorov's four families of one-nodal $V_{22}$ degenerations. Singular K-semistable degenerations are shown to lie in the closure of loci of one-nodal $V_{22}$, and terminal singularities are restricted to nodal ($A_1$) types.

A key modular result is established: the forgetful morphism from the moduli of Fano--K3 pairs $(X, S)$ (where $X$ is a K-semistable degeneration of $V_{22}$0 and $V_{22}$1 is an anticanonical K3 section) to the moduli of genus 12 polarized K3 surfaces is shown to be an open immersion. This is a concrete realization of Mukai's perspective—that the K-moduli of $V_{22}$2 is controlled by the moduli of the anticanonical K3 surfaces.

Additionally, a deformation-theoretic criterion is constructed, generalizing Beauville's theorem in the weak, mildly singular setting, showing that deformations of weak Fano-K3 pairs can be reduced to the deformation theory of their anticanonical K3 surfaces and associated lattices. This framework is shown to have broader applicability for other Fano threefolds of large anticanonical volume.

Boundary Classification and Degenerations

The boundary of the K-moduli stack $V_{22}$3 is shown to be divisorial, composed of four irreducible divisors associated with Prokhorov's four Sarkisov-link constructions of one-nodal $V_{22}$4 degenerations. The authors leverage the moduli continuity method and volume bounds for singularities, combined with geometric and lattice-theoretic techniques, to restrict possible degenerations:

• Terminal K-semistable $V_{22}$5: These have only isolated nodal singularities ($V_{22}$6).
• Non-terminal degenerations: Deformation theory, paired with the ampleness of anticanonical divisors, ensures smoothability or forces the degeneration to emerge from higher Picard rank families, which are excluded by moduli-theoretic purity and dimension counts.

A key contradiction established: for singularities worse than nodal, the Picard number increases, but a deformation subsequently restricts to the nodal Noether–Lefschetz divisor in the K3 moduli, clashing with the expected log discrepancy.

Modular Realization and Open Immersion

The stack of Fano-K3 pairs $V_{22}$7, with $V_{22}$8 a $V_{22}$9 degeneration and $6$0 an anticanonical K3 surface, maps to the moduli stack $6$1 of degree $6$2 polarized K3 surfaces. The forgetful morphism is demonstrated to be an open immersion whose image consists of the Brill–Noether general locus, plus four Noether–Lefschetz divisors corresponding to the four Prokhorov types.

Crucially, this implies that a K-semistable $6$3 is uniquely determined by its anticanonical polarized K3—extending Mukai's conjectural philosophy into a rigorous modular statement.

This modular realization, together with smoothness results, leads to the identification of the K-moduli space and stack for $6$4 as smooth, rational, and essentially governed by the moduli of associated K3 surfaces.

Deformation Framework

A general deformation-theoretic input is established: for Gorenstein terminal weak Fano threefolds with ADE K3 anticanonical divisors, a Beauville-type theorem shows that deformations are parametrized by deformations of the K3 with prescribed lattice data, with the forgetful morphism being smooth of relative dimension $6$5. The same approach gives the local constancy of Hodge numbers and Picard lattices in families, and guarantees good algebraization properties.

As a consequence, this lattice-polarized K3 moduli dominates the K-moduli of $6$6 and provides a strong structural handle on singularities and smoothing behavior: terminal $6$7 can only acquire nodal singularities, and any degeneration can be realized as a closure of one-nodal degenerations.

Implications and Connections

The results have substantial implications for the construction and interpretation of K-moduli spaces for Fano varieties, particularly those with large anticanonical volume. The identification of boundary divisors as Noether–Lefschetz strata in the K3 moduli demonstrates that the anticanonical K3 surface controls both local and global moduli, singularities, and degenerations.

The modular continuity method is systematized and extended, providing new tools for analyzing degenerations—especially in settings where geometric invariant theory fails to furnish useful compactifications.

Practical implications include improved recipes for constructing moduli of Fano threefolds, explicit birational models, and rationality results for moduli spaces ($6$8 is shown to be rational)—fostering connections between algebraic geometry, K-stability, and lattice theory.

Conclusion

The paper rigorously establishes that the boundary of K-moduli for prime Fano threefolds of genus twelve is divisorial, with four explicit irreducible components, and that the moduli is explicitly controlled by the associated anticanonical genus twelve K3 surfaces. The modular realization via an open immersion sharpens Mukai’s philosophy and extends Beauville’s deformation theory. The deformation framework developed is robust, applicable to broader classes of Fano threefolds, and yields strong restrictions on singularities. The results have significant consequences for both the theoretical structure and explicit computations within higher-dimensional moduli theory.