K-Stability of Weighted Fano Hypersurfaces

• The paper establishes K-stability by employing test-configurations and precise invariants such as the Donaldson–Futaki, β, and δ-invariants.
• It reveals how wall-crossing phenomena and weight stratifications influence the birational models and moduli spaces of Fano varieties.
• The study combines GIT-theoretic methods and Abban–Zhuang flag techniques to derive concrete stability criteria for weighted hypersurfaces.

K-stability for weighted Fano hypersurfaces is a cornerstone in the modern theory of algebraic geometry, especially in the classification of Fano varieties, the existence of Kähler–Einstein metrics, and the construction of moduli spaces. Weighted projective hypersurfaces generalize classical projective hypersurfaces by allowing variable weights, introducing intricate singularity and moduli phenomena. For hypersurfaces in spaces such as $\mathbb{P}(1,2,n+2,n+3)$, the interplay between weights, degree, and moduli leads to deep wall-crossing behaviors and explicit birational models, as in the study of K-moduli spaces associated to families of log Fano pairs. The rigorous criteria for K-stability and K-polystability in this context rely on test-configurations, valuative and GIT-theoretic approaches, and explicit computations of invariants such as the Donaldson–Futaki, β, and δ-invariants.

1. Weighted Fano Hypersurfaces: Structure and Definitions

A weighted projective space $\mathbb{P}(a_0,\dots,a_n)$ is defined via homogeneous coordinates $[x_0:\dots:x_n]$ with $\deg x_i = a_i>0$, subject to projectivization by scaling. A hypersurface $X_d \subset \mathbb{P}(a_0,\dots,a_n)$ is the zero locus of a quasi-homogeneous polynomial $F(x_0,\dots,x_n)$ of degree $d$. When $X_d$ is well-formed (no codimension-one singular strata), quasi-smooth (affine cone smooth away from vertex), and Fano (the anticanonical divisor $-K_X$ is ample), it serves as a core object in K-stability studies. The Fano index is $I_X = \sum a_i - d$, and small index cases, notably index $1$ and $2$, have received concentrated research focus.

For the specific family $H \subset \mathbb{P}(1,2,n+2,n+3)$ of degree $2(n+3)$, hypersurfaces can be written in "standard form": $$H:\ t^2 + z^2y + a\,z\,x^{n+4} + a_0\,x^{2n+6} + a_1\,x^{2n+4}y + \cdots + a_{n+3}\,y^{n+3} = 0$$ The geometric and stability properties depend intricately on the weight stratification and the arrangement of coefficients in this normal form (Kim et al., 2024).

2. K-Stability Criteria: Test Configurations and Valuative Invariants

K-stability of a polarized pair $(X,\Delta;L)$ is defined via positivity of the Donaldson–Futaki invariant across all test-configurations. A test-configuration is a $\mathbb{C}^*$-equivariant flat family $(\mathcal{X},\mathcal{D};\mathcal{L})$ over $\mathbb{C}$ degenerating $(X,\Delta;L)$. The Donaldson–Futaki invariant is given in terms of asymptotic Hilbert weight or, equivalently, via valuative invariants for divisorial valuations $E$ over $X$: $\DF(\mathcal{X},\mathcal{D};\mathcal{L}) = \frac{d}{ds}\Big|_{s=0} \frac{\chi\big(\mathcal{X}_s,\,\lfloor k(-K_{\mathcal{X}_s}-\mathcal{D}_s)\rfloor\big)}{\chi\big(X,\lfloor k(-K_X-\Delta)\rfloor\big)}$ The Fujita–Li criterion relates K-stability to the β-invariant

$$\beta_{X,\Delta}(E) = A_{X,\Delta}(E) - S_{X,\Delta}(E)$$

where $A_{X,\Delta}(E)$ is the log discrepancy and $S_{X,\Delta}(E)$ encodes expected vanishing via integrals of volumes along $E$. K-semistability (resp. K-polystability) is equivalent to non-negativity (resp. strict positivity except for products) of $\beta_{X,\Delta}(E)$ for all divisorial $E$ (Kim et al., 2024).

For torus-invariant log Fano pairs, weighted criteria generalize via $g$-weighted β-invariants: $\beta^g(F) = A_{X,\Delta}(F) - S^g(F)$ with $S^g(F)$ computed via integrals weighted by a Duistermaat–Heckman density and moment polytope weighting $g$ satisfying vanishing conditions (Wang, 2024).

3. Wall-Crossing and K-Moduli Spaces of Weighted Hypersurfaces

Moduli of weighted Fano hypersurfaces exhibit wall-crossing behavior as the coefficients in associated log Fano pairs vary. For $H \subset \mathbb{P}(1,2,n+2,n+3)$, the projection $[x:y:z:t]\mapsto[x:y:z]$ realizes $H$ as a double cover over $W=\mathbb{P}(1,2,n+2)$ branched along $D$. K-polystability of $H$ is equivalent to K-polystability of $(W,\frac{1}{2}D)$, which generalizes to the family $(W,wD)$ for $w \in (0,\frac{n+5}{2n+6})$. Wall-crossings occur at finitely many rational values $w_i$ determined by GIT and a final exceptional K-wall $\xi_n$, manifested through explicit β-invariant formulas for toric–monomial valuations: $$\beta_{(W,wD_0)}(v_{(d,b)}) = \frac{1-n+2nw}{6} d + \frac{2n+1 - w(4n+6)}{3(n+2)} b$$ Critical walls align with GIT instability loci, while the divisorial contraction at $w=\xi_n$ is not detected by GIT. All K-polystable limits remain within the family of weighted hypersurfaces, with the wall-crossing diagram permitting explicit birational models of the moduli spaces (Kim et al., 2024).

4. Delta Invariants, Abban–Zhuang Flags, and Lower Bound Formulas

For the stability analysis of weighted Fano hypersurfaces, the δ-invariant functions as a sharp numerical threshold: $$\delta(X,\Delta;L) = \inf_{E} \frac{A_{X,\Delta}(E)}{S(L;E)}$$ K-stability is equivalent to $\delta > 1$ by the Li–Blum–Jonsson–Zhuang theorem. Abban–Zhuang's flag method enables reduction of δ-estimates to lower-dimensional strata by considering sequences of divisorial extractions (plt flags), even when non-admissible: $$\delta_{\eta}(X,\Delta;L) \geq \min\left\{\frac{A(Y_1)}{S(L;Y_1)}, \inf_{\eta' \in Y_1} \delta_{\eta'}(X,\Delta \supset Y_1;L)\right\}$$ For hypersurfaces $X_d \subset \mathbb{P}(a_0,\dots,a_{n+1})$ with $a_r \mid d$, the Abban–Zhuang method produces lower bounds

$\delta(X;\mathcal{O}_X(1)) \geq \frac{(n+1)a_r}{d}$

which guarantee K-stability for a wide class of index $1$ hypersurfaces, including large portions of the known Fano threefold and fourfold classifications (Sano et al., 2024, Campo et al., 2024, Campo et al., 6 Jan 2026).

5. Special Phenomena: Index, GIT, and K-Polystability in Weighted Contexts

The interplay between Fano index, weights, and stability yields remarkable phenomena unique to the weighted setting. For index-$2$ del Pezzo surfaces $X \subset \mathbb{P}(1,1,a,a)$ of degree $2a$, K-polystability is characterized via classical GIT stability of degree $2a$ binary forms: $X$ is K-polystable if and only if the associated binary form is GIT polystable under $\mathrm{SL}_2$-action; however, no such $X$ is K-stable. Quasi-smooth members are always K-polystable but never K-stable, due to nontrivial automorphism degenerations detected by the Futaki invariant (Liu et al., 2020).

In index $2$ weighted del Pezzo hypersurfaces, counterexamples to expected K-stability arise when the Futaki-type invariant $B_X(E)$ computed via Zariski decomposition and integration of nef parts is strictly negative, establishing K-instability for five explicit families (Kim et al., 2020).

6. Consequences, Moduli, and Birational Models

K-stability enables the construction of K-moduli spaces, which classify Fano hypersurfaces up to isomorphism and degeneration. The wall-crossing analysis for $\mathbb{P}(1,2,n+2,n+3)$ hypersurfaces establishes birational models for loci in the moduli of marked hyperelliptic curves, with divisorial contractions corresponding to collisions of marked points in the limit. The concrete structure of K-moduli and their walls forge links between singular Fano geometry, GIT, and classical algebraic curve theory (Kim et al., 2024). In higher-dimensional families, the stabilized moduli spaces remain weighted hypersurfaces, and the wall-crossing architecture offers a template for future classification in weighted complete intersections.

7. Outstanding Cases, Phenomenological Boundaries, and Open Directions

While broad classes of weighted Fano hypersurfaces are covered by the δ-bound and birational techniques, residual cases (sporadic families or those with high indices and multiple large weights) can exhibit borderline or non-classical behavior. Explicit K-unstable families have been constructed, and conjectural boundaries for semistability are posited (e.g., for $(1,1,n+1,n+1,2n+2)$ index-$2$ surfaces). The development of non-admissible plt flag techniques and convex geometric barycenter bounds has considerably strengthened the deductive apparatus for handling weighted singularities (Campo et al., 6 Jan 2026). Ongoing inquiries include refining δ-estimates for the remaining families, extending the moduli wall-crossing framework to higher Picard rank Fano orbifolds, and systematically describing the value distribution of δ-invariants in weighted settings (Campo et al., 2024, Sano et al., 2024, Kim et al., 2020).