On K-moduli of quartic threefolds

This paper investigates the K-moduli space of quartic threefolds, a class of Fano 3-folds with Picard rank 1 and anticanonical volume 4. The significance of this work lies in its exploration of one of the simplest cases where the K-moduli compactification of a family of Fano varieties is expected to differ substantially from its classical GIT (Geometric Invariant Theory) moduli compactification. Smooth members of this family are either quartic hypersurfaces in P^4 or double covers of a smooth quadric threefold branched along an octic surface. All such smooth varieties are known to be K-stable and can be uniformly described as (2,4)-complete intersections in the weighted projective space P(1^5, 2) (where coordinates x_0,…,x_4 have weight 1, and y has weight 2).

The central question addressed is the nature of the K-polystable limits of these smooth quartic threefolds. While GIT moduli often provide a good compactification, K-stability offers a more intrinsic algebro-geometric framework, and the resulting K-moduli spaces can contain limits not found in GIT. This paper provides concrete examples and structural results illuminating this phenomenon for quartic threefolds.

Key Innovations and Main Results:

The paper presents several key findings, primarily encapsulated in its Theorems A and B:

1. New K-polystable limits (Theorem A): The authors demonstrate the existence of K-polystable Fano 3-folds that are smoothable to quartic 3-folds but are not themselves (2,4)-complete intersections in P(1^5, 2). This directly implies that the K-moduli compactification of quartic threefolds contains elements beyond those of the “expected” GIT type.

   • These new limits are identified as specific types of complete intersections, namely (2,2,4)-complete intersections in a different weighted projective space, P(1^5, 2^2) (where x_0,…,x_4 have weight 1, and y_0, y_1 have weight 2). These are termed “pure (2,2,4)-complete intersections” if they are not also (2,4)-complete intersections in P(1^5, 2).
   • Two types of examples are provided:
     • Toric examples: Three explicit toric Fano 3-folds are constructed. Their K-polystability is verified by checking that the barycenter of their anticanonical polytope is the origin. These were discovered using computer searches guided by predictions from Mirror Symmetry (specifically, by finding mutation-equivalent rigid Maximally Mutable Laurent Polynomials whose Newton polytopes correspond to K-polystable toric Fano varieties).
     • An infinite family: A non-toric infinite family of such (2,2,4)-complete intersections is also shown to be K-polystable under generic conditions. Their K-stability is established using lower bounds on the stability threshold (δ-invariant) derived via the Fujita-Li valuative criterion and the theory of refinements of anticanonical linear systems along flags.

2. Closedness of the new locus (Theorem B): The paper proves that the locus of these “pure (2,2,4)-complete intersections” in P(1^5, 2^2) is closed within the K-moduli space M^{Kps}_{3,4}. This is a significant structural result, indicating that these new types of limits do not degenerate further into other, more exotic varieties within this K-moduli space.

   • This is established by relating the K-moduli of these 3-folds to the K-moduli of pairs (S, cΔ), where S is a degree 4 del Pezzo surface and Δ ~ -4K_S is an anticanonical divisor (with c=1/16). The authors provide a detailed study of the wall-crossing behavior for such pairs.

Main Prior Ingredients Needed:

The results of this paper build upon a rich foundation of concepts and techniques in algebraic geometry:

1. K-stability and K-moduli Spaces: The fundamental framework is the theory of K-stability, an algebro-geometric notion of stability for Fano varieties conjectured to be equivalent to the existence of Kähler-Einstein metrics. Associated with K-stability are K-moduli stacks and their good moduli spaces, which provide well-behaved compactifications for families of Fano varieties.
2. GIT Moduli Spaces: Classical GIT constructions provide moduli spaces for many geometric objects, including hypersurfaces. The comparison between K-moduli and GIT moduli is a central theme.
3. Properties of Quartic Threefolds: Knowledge that smooth quartic threefolds and hyperelliptic threefolds (as defined in the paper) are K-stable and can be realized as (2,4)-complete intersections in P(1^5, 2) is crucial.
4. Tools for Proving K-stability:
   • Valuative Criterion (Fujita-Li): This criterion translates K-stability into conditions on valuations over the variety.
   • Stability Threshold (δ-invariant): A Fano variety X is K-stable if δ(X) > 1. The paper uses techniques to find lower bounds for δ(X).
   • Barycenter Criterion: For toric Fano varieties, K-polystability is equivalent to the barycenter of the anticanonical polytope (or polar polytope) being the origin.
   • Cyclic Cover Results: K-stability of a variety X can sometimes be deduced from the K-stability of a pair (Y, D) where X is a cyclic cover of Y branched along D.
   • Cone Construction and Interpolation: These are techniques to relate the K-stability of different varieties or pairs.
5. Mirror Symmetry for Fano Varieties: The conjectural correspondence between deformation families of Fano varieties and mutation-equivalence classes of special Laurent polynomials (rigid MMLPs) provided crucial guidance for finding the toric examples. The Newton polytope of such a Laurent polynomial defines a toric Fano variety.
6. Log Fano Pairs and their Moduli: The theory of K-stability extends to log Fano pairs (X, D). The paper heavily relies on the K-moduli of pairs (S, cΔ), where S is a del Pezzo surface, and their wall-crossing phenomena.
7. Weighted Projective Spaces and Complete Intersections: The explicit description of the varieties and their limits involves complete intersections in various weighted projective spaces.
8. Singularity Theory: Concepts like Q-factorial singularities, Kawamata log terminal (klt) singularities, and Gorenstein index are used to characterize the varieties involved.

In summary, this paper makes a significant contribution by explicitly identifying new types of K-polystable limits for quartic threefolds, demonstrating that the K-moduli space indeed captures a richer set of degenerations than previously understood via GIT for this class. It showcases the power of combining theoretical insights from K-stability, computational methods inspired by Mirror Symmetry, and detailed geometric analysis of specific constructions.