Intrinsic visibility in K-moduli

Math research based on GenAI.

Intrinsic visibility of coefficient walls in K-moduli

Shengrong Wu
National University of Singapore

Automatic Pipeline with ChatGPT

Abstract

We formulate and prove an intrinsic visibility theorem for fixed-ambient wall crossing in \(K\)-moduli. The theorem has two parts. First, for every liftable coefficient face, the GIT wall equation is exactly the corresponding special-value equation \[ \mu_F^{\mathrm{GIT}}=-\beta. \] Second, an interior strictly \(K\)-semistable parameter is forced away from every locally product open set on which a local \(\delta\)-gap holds, hence any such hidden wall is detected over the non-log-smooth locus of a special fiber. This upgrades the coefficient-rigidity hypothesis of the abstract visibility package to an intrinsic condition.

We then apply the theorem in two directions. In the Kim–Liu–Wang weighted branch family, we recover the full dichotomy: every pre-last wall is coefficient-visible, while the last wall is pair-singularity-visible and is detected by a local weighted blow-up over the unique non-toric \(A\)-singularity of the special divisor \(D_\beta\). As a second application, we study the family \[ \bigl(\mathbb P^2,\alpha \Lambda + c C_p\bigr), \qquad \Lambda=(y=0), \qquad C_p:\ z^d+a z x^{d-1}+\sum_{j=0}^d a_j x^{d-j}y^j=0. \] On a natural dense open coefficient locus \(B^\circ\), we compute the complete coefficient wall arrangement, prove the exact identities \[ \mu_e^{\mathrm{GIT}}(\alpha,c)=-\beta_{(\mathbb P^2,\alpha\Lambda+cC_e)}(v_e), \] and show that there is no hidden wall on \(B^\circ\). Thus the second family exhibits the opposite behavior from the weighted branch family: on the generic locus, all walls are coefficient-visible.

1. Introduction

Two complementary analyses isolate a precise phenomenon in non-proportional wall crossing for \(K\)-stability. A wall can arise in two very different ways.

The first mechanism is fixed-ambient and coefficient-theoretic. One starts with a common affine coefficient space \(B\) carrying a torus action, a family of log Fano pairs over \(B\), and an affine CM character map \[ \chi:P\to M_\mathbb R \] from the coefficient domain. A wall appears when \(\chi(\mathbf c)\) crosses a supporting hyperplane of a coefficient face. If the face is liftable, then the corresponding wall model carries a product special test configuration and the wall equation is exactly a \(\beta\)-equation.

The second mechanism is invisible to the coefficient arrangement. A parameter may lie in the interior of a coefficient chamber and still be strictly \(K\)-semistable. In that case the destabilizing object comes from a non-product special test configuration. The problem is then to show that the associated special center cannot stay inside a locally rigid part of the family, and therefore must be detected over the non-log-smooth locus of a special fiber.

A family-specific analysis of the weighted branch example proves this dichotomy in the Kim–Liu–Wang family. A parallel abstract analysis proves the coefficient-visible part, the hidden-wall criterion, and an extended visibility theorem under a coefficient-rigidity hypothesis. The present article unifies those two strands and adds two genuinely new ingredients.

First, we replace coefficient-rigidity by an intrinsic criterion. The key input is a local product-neighborhood transport theorem: if the center of a hidden special divisor lies in a neighborhood isomorphic to \(V\times \mathbb A^1\), then the same divisor restricts to a divisorial valuation on \(V\). Combined with a local \(\delta\)-gap \[ \delta_x>1, \] this excludes zero-\(\beta\) valuations from every such product neighborhood.

Second, we add a second substantial application. The naive plane-branch family \[ (\mathbb P^2,cC_p) \] does not produce a nontrivial wall arrangement: the corresponding \(\beta\)-functions are all proportional to \(c\). The correct family is the non-proportional variant \[ \bigl(\mathbb P^2,\alpha \Lambda + c C_p\bigr), \qquad \Lambda=(y=0). \] For this family we compute the coefficient walls explicitly and prove that on a natural dense open coefficient locus there is no hidden wall at all.

The center theorem of the paper is the following.

Theorem A (Intrinsic visibility theorem). In the fixed-ambient setup of Sections 2 and 3, assume that every exposed coefficient face met by \(\chi(P)\) is liftable. Assume moreover that for every hidden wall point, every point of the corresponding special fiber outside its non-log-smooth locus admits a neighborhood isomorphic over \(\mathbb A^1\) to \(V\times \mathbb A^1\), where \(V\) is an open subset of the original fiber satisfying \[ \delta_x(X,\Delta)>1 \qquad\text{for all }x\in V. \] Then every wall point represented by a semistable coefficient point is of exactly one of the following two types:

1. coefficient-visible, detected by a liftable coefficient face and satisfying the exact identity \[ \mu_F^{\mathrm{GIT}}=-\beta \] on the corresponding wall model;

2. pair-singularity-visible, detected by a special divisorial valuation with \(\beta=0\) whose special center lies over the non-log-smooth locus of a special fiber.

The weighted-branch family realizes both alternatives. The plane-branch-with-line family on the regular open locus \(B^\circ\) realizes only the first one.

The proof road map is short. Section 3 establishes the cone criterion and toric wall centers. Section 4 proves liftability criteria, the facewise identity \(\mu^{\mathrm{GIT}}=-\beta\), the hidden-wall criterion, and the local product transport theorem. Section 5 assembles these ingredients into Theorem A. Sections 6 and 7 treat the two applications.

2. Standard inputs

We use the following facts from the literature.

1. If a \(1\)-PS of \(\operatorname{Aut}(X,\Delta;L)\) induces a product test configuration, then the weight on the CM line is the generalized Futaki invariant of that product test configuration.

2. If a log Fano pair is \(K\)-semistable but not \(K\)-polystable, then there exists a non-product special test configuration with vanishing generalized Futaki invariant whose central fiber is the unique \(K\)-polystable degeneration.

3. For a special test configuration, the generalized Futaki invariant equals the \(\beta\)-invariant of the associated special divisorial valuation.

4. For a log Fano pair, \(K\)-semistability is equivalent to the nonnegativity of \(\beta(v)\) for all prime divisors over the pair.

5. Since \(\mathbb P^2\) is \(K\)-polystable, one has \[ S_{-K_{\mathbb P^2}}(v)\le A_{\mathbb P^2}(v) \] for every divisorial valuation \(v\) over \(\mathbb P^2\). Equivalently, \[ S_H(v)\le \frac{A_{\mathbb P^2}(v)}{3}, \qquad H=\mathcal O_{\mathbb P^2}(1). \]

6. For a semi-quasihomogeneous plane curve singularity whose weighted initial form is \(Z^d+X^s\) with \(d,s\ge 2\), the local log canonical threshold equals \[ \operatorname{lct}(Z^d+X^s)=\frac1d+\frac1s. \]

Items (1)–(3) are standard in Li–Xu and in the real-coefficient \(K\)-moduli theory of Liu–Zhou. Item (4) is the valuative criterion for \(K\)-semistability. Item (6) is the standard semi-quasihomogeneous computation of the local log canonical threshold.

3. Fixed-ambient setup and coefficient faces

Let \(T\) be an algebraic torus with character lattice \(M=X^*(T)\) and cocharacter lattice \(N=X_*(T)\). Let \[ B=\mathbb A^s_{z_1,\dots,z_s} \] carry a diagonal \(T\)-action with nonzero weights \[ r_1,\dots,r_s\in M. \] Let \[ \pi:(\mathcal X,\mathcal D_1,\dots,\mathcal D_m)\to B \] be a \(T\)-equivariant family, and let \(P\subset \mathbb R^m\) be a convex coefficient domain such that for every \(\mathbf c=(c_1,\dots,c_m)\in P\) and every relevant fiber \(X_p\), the pair \[ (X_p,\Delta_{p,\mathbf c}), \qquad \Delta_{p,\mathbf c}:=\sum_{j=1}^m c_j D_{j,p}, \] is log Fano. Assume the CM line is \(T\)-linearized by an affine character map \[ \chi:P\to M_\mathbb R. \]

For \(p\in B\), set \[ \operatorname{Supp}(p):=\{i\mid z_i(p)\neq 0\}, \qquad \Gamma(p):=\operatorname{Cone}\{r_i\mid i\in\operatorname{Supp}(p)\}\subset M_\mathbb R. \]

3.1. Coefficient faces

Fix an exposed face \(F\) of some support cone. Choose a primitive supporting covector \(\nu_F\in N\) with \[ \langle m,\nu_F\rangle=0 \quad (m\in F), \qquad \langle m,\nu_F\rangle>0 \quad (m\in \Gamma\setminus F) \] for a support cone \(\Gamma\) having \(F\) as an exposed face. Define the supporting hyperplane \[ H_F:=\{\mathbf c\in P\mid \langle \chi(\mathbf c),\nu_F\rangle=0\}. \] The connected components of \[ P\setminus \bigcup_F H_F \] are the coefficient chambers.

For an exposed face \(F\), define \[ I_F:=\{i\mid r_i\in F\}, \qquad B_F:=\{z_j=0\text{ for }j\notin I_F\}\cong \mathbb A^{I_F}, \] and for \(\mathbf c\in H_F\) define \[ W_F:=\{\mathbf c\in P\mid \chi(\mathbf c)\in \operatorname{relint}(F)\}, \] \[ B_F^{ss}(\mathbf c):=\{q\in B_F\mid \chi(\mathbf c)\in \Gamma(q)\}, \] \[ Z_F(\mathbf c):=B_F^{ss}(\mathbf c)//T. \]

A point \(\mathbf c_*\in P\) is a wall point if some fiber \((X_p,\Delta_{p,\mathbf c_*})\) is strictly \(K\)-semistable.

A wall point is called coefficient-visible if \(\mathbf c_*\in W_F\) for some liftable face \(F\). It is called hidden if \(\mathbf c_*\) lies in the interior of a coefficient chamber. It is called pair-singularity-visible if it is hidden and the associated special divisorial valuation with \(\beta=0\) is centered over the non-log-smooth locus of a special fiber.

3.2. The cone criterion

Proposition 3.1. For any \(p\in B\) and \(\mathbf c\in P\), \[ p\text{ is }\chi(\mathbf c)\text{-semistable} \iff \chi(\mathbf c)\in \Gamma(p). \]

Proof. Let \(\nu\in N_\mathbb R\), and let \(\lambda_\nu:\mathbb G_m\to T\) be the corresponding \(1\)-PS. The limit \[ \lim_{t\to 0}\lambda_\nu(t)\cdot p \] exists if and only if \[ \langle r_i,\nu\rangle\ge 0 \qquad\text{for all }i\in \operatorname{Supp}(p), \] that is, if and only if \(\nu\in \Gamma(p)^\vee\). By Hilbert–Mumford, \[ p\text{ is semistable } \iff \langle \chi(\mathbf c),\nu\rangle\ge 0 \quad\text{for all }\nu\in \Gamma(p)^\vee. \] By the bipolar theorem, this is equivalent to \(\chi(\mathbf c)\in \Gamma(p)\). \(\square\)

Proposition 3.2. Let \(p\in B\), let \(F\) be an exposed face of \(\Gamma(p)\) with supporting covector \(\nu_F\), and suppose \[ \chi(\mathbf c)\in \operatorname{relint}(F). \] Then the limit \[ q:=\lim_{t\to 0}\lambda_F(t)\cdot p \] exists, lies in \(B_F\), satisfies \[ \Gamma(q)=F, \] and is semistable at \(\mathbf c\).

Proof. Because \(F\) is an exposed face of \(\Gamma(p)\), \[ \langle r_i,\nu_F\rangle=0 \quad (r_i\in F), \qquad \langle r_i,\nu_F\rangle>0 \quad (r_i\in \Gamma(p)\setminus F). \] Hence \(\lambda_F\) kills exactly the coordinates whose weights are not in \(F\). Therefore \(q\) exists and belongs to \(B_F\). The surviving support weights are exactly the support weights of \(p\) lying on \(F\), so they generate \(F\), and thus \[ \Gamma(q)=F. \] Since \(\chi(\mathbf c)\in \operatorname{relint}(F)\subset F=\Gamma(q)\), Proposition 3.1 implies that \(q\) is semistable at \(\mathbf c\). \(\square\)

Corollary 3.3. For every exposed face \(F\) and every \(\mathbf c\in W_F\), the quotient \[ Z_F(\mathbf c)=B_F^{ss}(\mathbf c)//T \] is a semiprojective toric variety. If \(F\) is a ray, then \(Z_F(\mathbf c)\) is a weighted projective space.

Proof. The coordinate subspace \(B_F\) is an affine space with diagonal torus action. The semistable locus is cut out by Proposition 3.1, so the quotient is the standard toric GIT quotient. If \(F\) is a ray generated by a primitive character \(r_F\) and \[ r_i=m_i r_F \qquad (i\in I_F),\qquad m_i\in \mathbb Z_{>0}, \] then the action factors through a one-dimensional torus acting with weights \(m_i\), and the quotient is \[ \mathbb P(m_i\mid i\in I_F). \] \(\square\)

4. Liftability, the identity \(\mu^{\mathrm{GIT}}=-\beta\), and hidden walls

4.1. Liftable faces

An exposed face \(F\) is called liftable if there exists a primitive cocharacter \[ \lambda_F:\mathbb G_m\to T \] with differential \(\nu_F\) such that for every \(\mathbf c_0\in W_F\) and every polystable point \(q\in B_F^{ss}(\mathbf c_0)\), there is an open set \(U_{F,q}\subset P\) containing \(W_F\) with the following property: for every \(\mathbf c\in U_{F,q}\), the induced \(\mathbb G_m\)-action on \((X_q,\Delta_{q,\mathbf c})\) is a product special test configuration whose associated special valuation is divisorial. We denote that valuation by \[ v_{F,q,\mathbf c}. \]

In fixed-ambient linear-system problems, liftability is often automatic once the supporting \(1\)-PS fixes the wall model and has divisorial weight valuation.

Proposition 4.1 (Fixed-ambient liftability criterion). Let \(Y\) be a normal projective \(T\)-variety, let \(V_1,\dots,V_m\) be finite-dimensional \(T\)-stable linear systems on \(Y\), and let \[ B:=V_1\times\cdots\times V_m. \] Assume that every relevant fiber pair \((Y,\Delta_{p,\mathbf c})\) is log Fano. Let \(F\) be an exposed coefficient face met by \(\chi(P)\), let \(\lambda_F\) be the supporting \(1\)-PS, and let \(\mathbf c_0\in W_F\), \(q\in B_F^{ss}(\mathbf c_0)\) be polystable. Assume the weight valuation of \(\lambda_F\) on \(K(Y)\) is divisorial. Then \(F\) is liftable.

Proof. Since \(q\in B_F\), every nonzero coefficient of \(q\) has weight in \(F\), and \(\nu_F\) vanishes on \(F\). Hence \(\lambda_F\) fixes \(q\) and therefore fixes every divisor appearing in the boundary of the wall model. For every \(\mathbf c\), the induced action on \((Y,\Delta_{q,\mathbf c})\) is therefore a product test configuration. Because the pair is log Fano, it is a product special test configuration. By assumption the associated weight valuation is divisorial, so the definition of liftability is satisfied. \(\square\)

A useful local criterion for divisoriality is weighted blow-up divisoriality.

Corollary 4.2 (Weighted blow-up criterion). In the situation of Proposition 4.1, suppose that near the generic point of an irreducible \(\lambda_F\)-fixed subvariety \(Z\subset Y\), the variety \(Y\) is smooth with local coordinates \[ (u_1,\dots,u_r,v_1,\dots,v_s) \] such that \[ Z=(v_1=\cdots=v_s=0), \] \(\lambda_F\) acts trivially on the \(u_i\), and \[ \lambda_F(t)\cdot v_j=t^{w_j}v_j, \qquad w_j\in \mathbb Z_{>0}, \qquad \gcd(w_1,\dots,w_s)=1. \] Then the weight valuation of \(\lambda_F\) is the divisorial valuation of the exceptional divisor of the weighted blow-up of \(Z\) with weights \((w_1,\dots,w_s)\).

Proof. The valuation ideals are generated by monomials in the \(v_j\) of weighted degree at least \(m\), so their Rees algebra is the Rees algebra of the weighted blow-up of \(Z\) with weights \((w_1,\dots,w_s)\). The exceptional divisor is prime, and by construction its valuation equals the weight valuation. \(\square\)

4.2. The facewise identity

For a liftable exposed face \(F\) with supporting covector \(\nu_F\), define \[ \mu_F^{\mathrm{GIT}}(\mathbf c):=-\langle \chi(\mathbf c),\nu_F\rangle. \]

Theorem 4.3 (Facewise identity). Let \(F\) be a liftable face, let \(\mathbf c_0\in W_F\), let \(q\in B_F^{ss}(\mathbf c_0)\) be polystable, and let \(U_{F,q}\) be as above. Then for every \(\mathbf c\in U_{F,q}\), \[ \mu_F^{\mathrm{GIT}}(\mathbf c) = -\beta_{(X_q,\Delta_{q,\mathbf c})}(v_{F,q,\mathbf c}). \]

Proof. Because \(q\in B_F\), the \(1\)-PS \(\lambda_F\) fixes \(q\). By liftability, for every \(\mathbf c\in U_{F,q}\) the induced action on \((X_q,\Delta_{q,\mathbf c})\) gives a product special test configuration whose associated special divisorial valuation is \(v_{F,q,\mathbf c}\).

By definition of the CM linearization, the weight of \(\lambda_F\) on the CM line over \(q\) equals \[ \langle \chi(\mathbf c),\nu_F\rangle. \] By the CM/Futaki dictionary, this equals the generalized Futaki invariant of the induced product special test configuration. By the special test configuration formula, the same Futaki invariant equals \[ \beta_{(X_q,\Delta_{q,\mathbf c})}(v_{F,q,\mathbf c}). \] Therefore \[ \beta_{(X_q,\Delta_{q,\mathbf c})}(v_{F,q,\mathbf c}) = \langle \chi(\mathbf c),\nu_F\rangle = -\mu_F^{\mathrm{GIT}}(\mathbf c). \] \(\square\)

4.3. Hidden walls

Theorem 4.4 (Hidden-wall criterion). Let \(\mathbf c_*\) lie in the interior of a coefficient chamber. Suppose that for some fiber \(X_p\) the pair \((X_p,\Delta_{p,\mathbf c_*})\) is strictly \(K\)-semistable. Then:

1. there exists a special divisorial valuation \(v_*\) with \[ \beta_{(X_p,\Delta_{p,\mathbf c_*})}(v_*)=0; \]

2. the wall point \(\mathbf c_*\) is not explained by any liftable coefficient face.

In particular, every strictly \(K\)-semistable interior point is a hidden wall point.

Proof. Since \((X_p,\Delta_{p,\mathbf c_*})\) is \(K\)-semistable but not \(K\)-polystable, there exists a non-product special test configuration with vanishing generalized Futaki invariant. Let \(v_*\) be the associated special divisorial valuation. Then \[ \beta(v_*)=0. \] This proves (1).

Suppose that \(\mathbf c_*\) were explained by a liftable face \(F\). Then there would exist a wall model \((X_q,\Delta_{q,\mathbf c_*})\) and a divisorial valuation \(v_{F,q,\mathbf c_*}\) with \[ 0=\beta(v_{F,q,\mathbf c_*}). \] By Theorem 4.3, \[ 0=\beta(v_{F,q,\mathbf c_*})=-\mu_F^{\mathrm{GIT}}(\mathbf c_*), \] hence \(\mathbf c_*\in H_F\). This contradicts the assumption that \(\mathbf c_*\) lies in the interior of a coefficient chamber. Therefore no liftable face explains \(\mathbf c_*\). \(\square\)

4.4. Intrinsic exclusion of hidden centers

We now isolate the intrinsic replacement for coefficient-rigidity.

Proposition 4.5 (Local \(\delta\)-gap). Let \((X,\Delta)\) be a log Fano pair and let \[ L:=-(K_X+\Delta). \] Suppose \(V\subset X\) is an open subset such that \[ \delta_x(X,\Delta)>1 \qquad\text{for all }x\in V. \] Then every divisorial valuation \(v\) centered in \(V\) satisfies \[ \beta_{(X,\Delta)}(v)>0. \]

Proof. By definition of \(\delta_x\), \[ \frac{A_{X,\Delta}(v)}{S_L(v)}\ge \delta_x(X,\Delta)>1 \] for every divisorial valuation \(v\) centered at \(x\in V\). Hence \[ A_{X,\Delta}(v)>S_L(v), \] that is, \[ \beta_{(X,\Delta)}(v)>0. \] \(\square\)

The next proposition transports a hidden special divisor through a local product neighborhood.

Proposition 4.6 (Local product transport). Let \((X,\Delta)\) be an \(n\)-dimensional log Fano pair, and let \[ (\mathfrak X,\mathfrak \Delta)\to \mathbb A^1_t \] be a non-product special test configuration of \((X,\Delta)\). Let \(E\) be the prime divisor over \(\mathfrak X\) defining the associated special valuation \(w\) on \(K(X)(t)\), normalized by \[ w(t)=1. \] Let \(z\in \mathfrak X_0\) be the center of \(E\) on the special fiber. Assume there exists an open neighborhood \(U\ni z\) and an isomorphism of pairs over \(\mathbb A^1\) \[ (U,\mathfrak \Delta|_U)\cong (V,\Delta_V)\times \mathbb A^1 \] for some open subset \(V\subset X\). Then the restriction \[ v:=w|_{K(V)} \] is a nontrivial divisorial valuation on \(K(V)\) centered at the point of \(V\) corresponding to \(z\).

Proof. Set \(K:=K(V)\). The valuation \(w\) is a rank-one divisorial valuation on the \((n+1)\)-dimensional function field \(K(t)\). Its residue field has transcendence degree \[ \operatorname{trdeg}_k \kappa(w)=n. \] The restriction \(v=w|_K\) is nontrivial because \(w\) is the valuation associated with a non-product special test configuration.

Since \(v(K^\times)\) is a nonzero subgroup of \(\mathbb Z\), there exists \(d\in \mathbb Z_{>0}\) and \(f\in K^\times\) with \[ v(f)=d. \] Set \[ u:=\frac{t^d}{f}. \] Then \[ w(u)=d-d=0. \] Moreover \(t\) is algebraic over \(K(u)\) because it satisfies the equation \[ T^d-fu=0. \] Hence \(K(t)\) is algebraic over \(K(u)\). Passing to residue fields, \(\kappa(w)\) is algebraic over \(\kappa(v)(\bar u)\), and therefore \[ \operatorname{trdeg}_k \kappa(w)\le \operatorname{trdeg}_k \kappa(v)+1. \] Thus \[ n\le \operatorname{trdeg}_k \kappa(v)+1. \]

On the other hand, \(v\) is a rank-one valuation on the \(n\)-dimensional field \(K\), so Abhyankar’s inequality gives \[ \operatorname{trdeg}_k \kappa(v)+1\le n. \] Hence \[ \operatorname{trdeg}_k \kappa(v)=n-1, \] so \(v\) is divisorial.

Finally, because \(w\) dominates the local ring at \(z\) on \(V\times \mathbb A^1\), its restriction \(v\) dominates the local ring of the corresponding point on \(V\). Therefore \(v\) is centered there. \(\square\)

5. The intrinsic visibility theorem

We can now prove the center theorem.

Theorem 5.1 (Intrinsic visibility theorem). Assume the setup of Section 3. Assume that every exposed coefficient face met by \(\chi(P)\) is liftable. Assume moreover that for every hidden wall point, every point of the corresponding special fiber outside its non-log-smooth locus admits a neighborhood isomorphic over \(\mathbb A^1\) to \(V\times \mathbb A^1\), where \(V\) is an open subset of the original fiber satisfying \[ \delta_x(X,\Delta)>1 \qquad\text{for all }x\in V. \] Then every wall point represented by a semistable coefficient point is of exactly one of the following two types:

1. coefficient-visible, detected by a liftable coefficient face;

2. pair-singularity-visible, detected by a special divisorial valuation with \(\beta=0\) centered over the non-log-smooth locus of a special fiber.

Moreover, on every liftable coefficient face one has the exact identity \[ \mu_F^{\mathrm{GIT}}=-\beta. \]

Proof. Let \(\mathbf c_*\) be a wall point represented by a semistable coefficient point.

If \(\mathbf c_*\) lies on the coefficient hyperplane arrangement, then since the representing coefficient point is semistable, Proposition 3.1 implies that \(\chi(\mathbf c_*)\) lies in an exposed face \(F\) of its support cone. By assumption \(F\) is liftable, so \(\mathbf c_*\) is coefficient-visible. The identity \[ \mu_F^{\mathrm{GIT}}=-\beta \] on the corresponding wall model is exactly Theorem 4.3.

Now suppose that \(\mathbf c_*\) lies in the interior of a coefficient chamber. Then by Theorem 4.4 there exists a special divisorial valuation \(v_*\) with \[ \beta(v_*)=0, \] and \(\mathbf c_*\) is hidden. Let \[ (\mathfrak X,\mathfrak \Delta)\to \mathbb A^1 \] be the corresponding special test configuration, and let \(z\) be the center of the associated prime divisor on the special fiber. If \(z\) lay outside the non-log-smooth locus, then by hypothesis \(z\) would admit a product neighborhood \[ U\cong V\times \mathbb A^1 \] with \[ \delta_x(X,\Delta)>1 \qquad (x\in V). \] By Proposition 4.6, the special divisor would restrict to a divisorial valuation on \(V\). By Proposition 4.5, every divisorial valuation centered in \(V\) has strictly positive \(\beta\). This contradicts \[ \beta(v_*)=0. \] Therefore the special center of \(v_*\) must lie over the non-log-smooth locus of the special fiber. By definition, \(\mathbf c_*\) is pair-singularity-visible.

The two alternatives are disjoint because pair-singularity-visibility is defined only for hidden interior points. \(\square\)

6. Application I: the weighted branch family

We now recover the full structural theorem for the weighted branch family as an application of Theorem 5.1.

6.1. The family and its coefficient data

The family is the Kim–Liu–Wang weighted branch family. In odd parity one works on \[ W=\mathbb P(1,2,n+2)_{x,y,z} \] with divisors \[ D_p:\ z^2y+a z x^{n+4}+\sum_{j=0}^{n+3} a_j x^{2n+6-2j} y^j=0, \] and pairs \((W,wD_p)\) for \[ 0<w<\frac{n+5}{2n+6}. \] In even parity one writes \(n=2\ell\) and works on \[ W'=\mathbb P(1,1,\ell+1)_{u,y,z}, \qquad H=(u=0), \] with divisors \[ D_p:\ z^2y+a z u^{\ell+2}+\sum_{j=0}^{2\ell+3} a_j u^{2\ell+3-j} y^j=0, \] and pairs \[ \left(W',\frac12H+wD_p\right), \qquad 0<w<\frac{2\ell+5}{4\ell+6}. \]

The common coefficient space is \[ B_n=\mathbb A^{n+5}_{a,a_0,\dots,a_{n+3}}, \] with residual torus \[ T=(\mathbb G_m)^2 \] acting with weights \[ v=(-1,-1), \qquad v_j=(j-1,-2) \qquad (0\le j\le n+3), \] and CM character \[ u_w= \left( \frac{1-n+2nw}{6}, \frac{2n+1-(4n+6)w}{3(n+2)} \right). \]

These are exactly the coefficient data computed in the weighted branch analysis.

6.2. The slice and the VGIT walls

For \[ w>\frac{2n+1}{4n+6}, \] one rescales \(u_w\) to the affine slice \(y=-2\) and obtains \[ \widetilde u_w=(\rho(w),-2), \] where \[ \rho(w)=\frac{(n+2)(n-1)-2n(n+2)w}{2n+1-(4n+6)w}. \] All coefficient weights \(v_j\) lie on the same affine line \(y=-2\), and the ray through \(v=(-1,-1)\) meets that slice at \((-2,-2)\). Therefore semistability is interval containment on the slice, and the walls are exactly the values \[ w_i=\frac{(n+2)^2-(2n+1)i}{(n+2)(2n+6)-(4n+6)i} \] for which \[ \rho(w_i)=n+2-i. \] Before the last \(K\)-moduli wall, each new wall orbit is a coordinate orbit.

6.3. Liftable wall rays and the identity \(\mu^{\mathrm{GIT}}=-\beta\)

Fix a pre-last wall and let \[ e=n+3-i \] in the odd case and \[ e=2\ell+3-i \] in the even case. Let \(p_e\) be the coordinate point with only \(a_e\neq 0\).

The associated wall model is \[ D_e:\ z^2y+x^{2n+6-2e}y^e=0 \] in odd parity and \[ D_e:\ z^2y+u^{2\ell+3-e}y^e=0 \] in even parity. It is obtained as the limit under the supporting \(1\)-PS \[ \mu_e(\tau)[x:y:z]=[x:\tau^{-2}y:\tau^{1-e}z] \] in odd parity and the analogous \(u\)-version in even parity.

Each such \(1\)-PS fixes the wall model and acts with divisorial weight valuation. The corresponding intrinsic complexity-one valuation is \[ v_e=v(1,n+3-e) \] in odd parity and \[ v_e=v(2,2\ell+3-e) \] in even parity.

Therefore every pre-last coefficient face is liftable by Proposition 4.1, and Theorem 4.3 applies. In this family the resulting equality is the explicit identity \[ \mu_e^{\mathrm{GIT}}(w)=-\beta_{D_e}(v_e). \] Equivalently, \[ \mu_e^{\mathrm{GIT}}(w)=0 \iff \beta_{D_e}(v_e)=0 \iff w=w_i. \]

Thus every pre-last wall is coefficient-visible.

6.4. The last wall is not a coefficient wall

Let \[ m=\frac{n+1}{2} \] when \(n\) is odd, and let \(m=\ell+1\) when \(n=2\ell\) is even. The weighted branch computation shows that the last \(K\)-moduli wall occurs at \[ \xi_n=\frac{n^3+11n^2+31n+23}{2n^3+18n^2+50n+42} \] in odd parity and at \[ \xi_{2\ell}=\frac{2\ell^2+8\ell+3}{4\ell^2+12\ell+6} \] in even parity.

The same computation gives the strict inequalities \[ w_m<\xi_n<\frac{n+5}{2n+6} \] in odd parity and \[ \frac12<\xi_{2\ell}<\frac{2\ell+5}{4\ell+6} \] in even parity. Equivalently, \[ \rho(\xi_n)\in \left(\frac{n+1}{2},\frac{n+3}{2}\right), \qquad \rho(\xi_{2\ell})\in (\ell,\ell+1). \] Therefore the CM character lies strictly inside the last VGIT chamber at the last \(K\)-wall. In particular, the last wall is hidden from the fixed-ambient coefficient arrangement.

6.5. The last wall is pair-singularity-visible

The weighted branch computation also identifies a special fixed-ambient model \(D_\beta\) carrying the unique non-toric \(A\)-singularity relevant to the last wall. Let \(E\) be the exceptional divisor of the corresponding weighted blow-up.

In odd parity, \(E\) arises from the local weighted blow-up of weights \((2,n+4)\) and has \[ \beta_E(w)= \frac{n^3+11n^2+31n+23-(2n^3+18n^2+50n+42)w}{3(n+2)(n+3)}. \] In even parity, \(E\) arises from the local weighted blow-up of weights \((1,\ell+2)\) and has \[ \beta_E(w)= \frac{2\ell^2+8\ell+3-(4\ell^2+12\ell+6)w}{6(\ell+1)}. \] Hence \[ \beta_E(w)=0 \iff w=\xi_n \quad\text{or}\quad w=\xi_{2\ell}. \]

Because the last wall is not coefficient-visible, Theorem 4.4 identifies it as hidden. The special divisor \(E\) has \(\beta_E=0\) and is centered over the non-log-smooth point of the special fiber \(D_\beta\). Therefore the last wall is pair-singularity-visible.

We have proved the following.

Theorem 6.1. In the Kim–Liu–Wang weighted branch family, every wall before the last one is coefficient-visible, and on each pre-last wall model one has \[ \mu_e^{\mathrm{GIT}}(w)=-\beta_{D_e}(v_e). \] The last wall lies strictly inside the final VGIT chamber and is detected instead by the local weighted-blow-up divisor over the unique non-toric \(A\)-singularity of the special divisor \(D_\beta\). In particular, the weighted branch family realizes both alternatives of Theorem 5.1.

7. Application II: plane branches with a fixed line

We now give a second substantial application, in which the generic picture is purely coefficient-visible.

7.1. The family and its coefficient geometry

Fix an integer \(d\ge 3\) and a real number \(0<\alpha<1\). Let \[ \Lambda=(y=0)\subset \mathbb P^2_{x,y,z}, \] and let \[ B_d=\mathbb A^{d+2}_{a,a_0,\dots,a_d}. \] For \[ p=(a,a_0,\dots,a_d)\in B_d \] define \[ C_p:\ z^d+a z x^{d-1}+\sum_{j=0}^d a_j x^{d-j} y^j=0. \] We study the log Fano pairs \[ (\mathbb P^2,\alpha\Lambda+cC_p), \qquad 0<c<\frac{3-\alpha}{d}. \]

Let \[ T=(\mathbb G_m)^2 \] act on coordinates by \[ (s,t)\cdot [x:y:z]=[sx:ty:z]. \] The coefficient weights are \[ r_a=(-(d-1),0), \qquad r_j=(-(d-j),-j) \qquad (0\le j\le d). \] All \(r_j\) lie on the affine line \[ u+v=-d. \] The ray through \(r_a\) meets the same affine line at \(r_0\), so after taking the common slice the support points are indexed by \[ 0,1,\dots,d. \]

For \(e>0\), let \(F_e=\mathbb R_{\ge 0}r_e\). For \(e=0\), the ray \(F_0\) contains both \(r_a\) and \(r_0\).

The supporting quotient for \(F_0\) is \[ Z_{F_0}\cong \mathbb P(d-1,d), \] while for \(e>0\) the ray quotient is a point.

7.2. Liftable wall rays

For \(0\le e\le d\), define \[ \lambda_e(\tau)[x:y:z]=[\tau^e x:\tau^{e-d}y:z]. \] If \(a_e\neq 0\) and \(a_j=0\) for \(j>e\), then \[ \lim_{\tau\to 0}\lambda_e(\tau)\cdot C_p = C_e, \] where \[ C_e:\ z^d+x^{d-e}y^e=0 \qquad (e>0), \] and for \(e=0\) one gets the weighted-projective wall family \[ C_{[u:v]}:\ z^d+u z x^{d-1}+v x^d=0, \qquad [u:v]\in \mathbb P(d-1,d). \]

On the affine chart \(y=1\) with coordinates \[ X=\frac{x}{y}, \qquad Z=\frac{z}{y}, \] the same \(1\)-PS acts by \[ X\mapsto \tau^d X, \qquad Z\mapsto \tau^{d-e} Z. \] Thus the associated valuation is the weighted-blow-up valuation \[ v_e=v\!\left(\frac d{g_e},\frac{d-e}{g_e}\right), \qquad g_e:=\gcd(d,e). \] Each ray is therefore liftable by Corollary 4.2.

7.3. The \(\beta\)-functions on the wall models

For \(e>0\), the center of \(v_e\) is the point \[ P=[0:1:0], \] which does not lie on \(\Lambda\). Hence \[ v_e(\Lambda)=0. \] Moreover \[ A_{\mathbb P^2}(v_e)=\frac{2d-e}{g_e}, \qquad S_H(v_e)=\frac{2d-e}{3g_e}, \qquad v_e(C_e)=\frac{d(d-e)}{g_e}. \] Since \[ -(K_{\mathbb P^2}+\alpha\Lambda+cC_e)=(3-\alpha-dc)H, \] one obtains \[ \beta_e(\alpha,c) := \beta_{(\mathbb P^2,\alpha\Lambda+cC_e)}(v_e) = \frac{\alpha(2d-e)+dc(2e-d)}{3g_e}. \]

For \(e=0\), the wall family is \(C_{[u:v]}\). The relevant valuation is the ordinary blow-up at \(P\), namely \(v_0=v(1,1)\), and \[ \beta_0(\alpha,c)=\frac{2\alpha-dc}{3}. \]

Therefore Theorem 4.3 gives the exact wall identity \[ \mu_e^{\mathrm{GIT}}(\alpha,c) = -\beta_e(\alpha,c) \qquad (0\le e\le d). \]

7.4. The common CM character and the coefficient walls

The equations for \(e=0\) and \(e=1\) determine the common CM character uniquely. One finds \[ \chi_{\alpha,c} = \left( -\frac{\alpha+dc}{3}, \frac{2\alpha-dc}{3} \right). \] For \[ c>\frac{\alpha}{2d}, \] the ray through \(\chi_{\alpha,c}\) meets the affine line \(u+v=-d\). After positive rescaling one obtains \[ \widetilde\chi_{\alpha,c} = \bigl(-(d-\rho(c)),-\rho(c)\bigr), \] with \[ \rho(c)=\frac{d(dc-2\alpha)}{2dc-\alpha}. \] A direct derivative computation gives \[ \rho'(c)=\frac{3\alpha d^2}{(2dc-\alpha)^2}>0, \] so \(\rho\) is strictly increasing.

Because the slice points are exactly \(0,1,\dots,d\), semistability is interval containment: \[ p\text{ is semistable at }(\alpha,c) \iff \rho(c)\in [j_{\min}(p),j_{\max}(p)]. \] In particular, semistable coefficient points exist if and only if \[ \rho(c)\in [0,d], \] that is, if and only if \[ c\ge \frac{2\alpha}{d}. \] The walls occur exactly when \(\rho(c)\) equals an integer. Solving \[ \rho(c_e)=e \] gives \[ c_e=\frac{\alpha(2d-e)}{d(d-2e)}. \] These are positive exactly for \[ 0\le e<\frac d2, \] and strictly increasing in \(e\) because \[ \frac{d}{de}\left(\frac{2d-e}{d-2e}\right)=\frac{3d}{(d-2e)^2}>0. \]

Let \[ c_{\max}:=\frac{3-\alpha}{d}. \] Then \[ c_e<c_{\max} \iff e<\frac{d(1-\alpha)}{2-\alpha}. \] Define \[ m:=\max\left\{e\in \mathbb Z_{\ge 0}\ \middle|\ e<\frac{d(1-\alpha)}{2-\alpha}\right\}. \] Then the coefficient walls in the log Fano interval are exactly \[ c_0<c_1<\cdots<c_m, \] with \[ c_0=\frac{2\alpha}{d}, \qquad c_m<c_{\max}\le c_{m+1} \] whenever \(m+1<d/2\).

7.5. A natural regular open locus

Let \[ P=[0:1:0]\in \mathbb P^2. \] Define \(B^\circ\subset B_d\) to be the open set of coefficient points \(p\) such that:

1. \(C_p\) is smooth on \(\mathbb P^2\setminus \{P\}\);

2. \(C_p\) meets \(\Lambda\) transversely.

This is a nonempty \(T\)-invariant Zariski open subset.

For \(p\in B^\circ\), the pair \((\mathbb P^2,\alpha\Lambda+cC_p)\) is log smooth away from \(P\). Thus any hidden wall on \(B^\circ\) would have to come from a divisorial valuation centered at \(P\).

We now show that this never happens at any non-wall semistable parameter.

7.6. Two local lemmas on \(\mathbb P^2\)

Lemma 7.2. For every divisorial valuation \(v\) over \(\mathbb P^2\), \[ S_H(v)\le \frac{A_{\mathbb P^2}(v)}{3}. \]

Proof. Since \(\mathbb P^2\) is \(K\)-polystable, \[ A_{\mathbb P^2}(v)\ge S_{-K_{\mathbb P^2}}(v). \] Now \(-K_{\mathbb P^2}=3H\), so \[ S_{-K_{\mathbb P^2}}(v)=3S_H(v). \] Hence \[ S_H(v)\le \frac{A_{\mathbb P^2}(v)}{3}. \] \(\square\)

Lemma 7.3. Let \(x\) be a smooth point of a smooth surface \(X\), let \(u,v\) be regular parameters at \(x\), and let \(w\) be a divisorial valuation centered at \(x\). Then \[ A_X(w)\ge w(u)+w(v). \]

Proof. Choose a model \(\pi:Y\to X\) on which the corresponding prime divisor \(E\) appears, and let \(w=\operatorname{ord}_E\). At the generic point of \(E\) choose a local equation \(s=0\) for \(E\). Write \[ u=s^a \phi, \qquad v=s^b \psi \] with \(\phi,\psi\) not divisible by \(s\), so \[ w(u)=a, \qquad w(v)=b. \] Then \[ \pi^*(du\wedge dv) = d(s^a\phi)\wedge d(s^b\psi) \] has vanishing order at least \(a+b-1\) along \(E\). Since \(du\wedge dv\) trivializes \(K_X\) locally, the coefficient of \(E\) in \(K_Y-\pi^*K_X\) is at least \(a+b-1\). Therefore \[ A_X(w)=1+\operatorname{ord}_E(K_Y-\pi^*K_X)\ge a+b=w(u)+w(v). \] \(\square\)

7.7. No hidden wall on \(B^\circ\)

We now prove the second application.

Theorem 7.4. Let \(p\in B^\circ\), and let \[ c\in (0,c_{\max}) \] be a coefficient-semistable value which is not one of the wall values \(c_e\). Then the pair \[ (\mathbb P^2,\alpha\Lambda+cC_p) \] is \(K\)-stable. In particular, there is no hidden wall on \(B^\circ\).

Proof. Let \(E\) be any prime divisor over \(\mathbb P^2\). We show \[ \beta_{(\mathbb P^2,\alpha\Lambda+cC_p)}(E)>0. \] Write \[ A(E):=A_{\mathbb P^2}(E), \qquad S(E):=S_H(E). \] Then \[ \beta(E)=A(E)-\alpha E(\Lambda)-c E(C_p)-(3-\alpha-dc)S(E). \] By Lemma 7.2, \[ \beta(E)\ge \frac{\alpha+dc}{3}A(E)-\alpha E(\Lambda)-c E(C_p). \]

Because \(c\) is coefficient-semistable and not a wall value, the interval criterion implies \[ c>c_0=\frac{2\alpha}{d}. \]

We distinguish cases according to the center of \(E\) on \(\mathbb P^2\).

First, if the center is outside \(\Lambda\cup C_p\), then \[ E(\Lambda)=E(C_p)=0, \] so \[ \beta(E)\ge \frac{\alpha+dc}{3}A(E)>0. \]

Second, suppose the center lies on a smooth point of \(\Lambda\setminus C_p\). Choose regular parameters \(u_1,u_2\) with \(\Lambda=(u_1=0)\). Then \(E(C_p)=0\), and Lemma 7.3 gives \[ A(E)\ge E(u_1)+E(u_2). \] Hence \[ \beta(E)\ge \frac{\alpha+dc}{3}\bigl(E(u_1)+E(u_2)\bigr)-\alpha E(u_1) = \frac{dc-2\alpha}{3}E(u_1)+\frac{\alpha+dc}{3}E(u_2)>0. \]

Third, suppose the center lies on a smooth point of \(C_p\setminus \Lambda\). Choose regular parameters \(u_1,u_2\) with \(C_p=(u_2=0)\). Then Lemma 7.3 gives \[ A(E)\ge E(u_1)+E(u_2), \] and therefore \[ \beta(E)\ge \frac{\alpha+dc}{3}\bigl(E(u_1)+E(u_2)\bigr)-c E(u_2) = \frac{\alpha+dc}{3}E(u_1)+\frac{\alpha+c(d-3)}{3}E(u_2)>0. \]

Fourth, suppose the center lies on \(\Lambda\cap C_p\). Because \(p\in B^\circ\), the intersection is transverse. Choose regular parameters \(u_1,u_2\) with \[ \Lambda=(u_1=0), \qquad C_p=(u_2=0). \] Then Lemma 7.3 gives \[ A(E)\ge E(u_1)+E(u_2), \] and \[ \beta(E)\ge \frac{\alpha+dc}{3}\bigl(E(u_1)+E(u_2)\bigr)-\alpha E(u_1)-c E(u_2) = \frac{dc-2\alpha}{3}E(u_1)+\frac{\alpha+c(d-3)}{3}E(u_2)>0. \]

It remains to consider valuations centered at \[ P=[0:1:0]. \] Let \[ e_+(p):=\max\{j\mid a_j\neq 0\}. \] Set \[ k:=\lfloor \rho(c)\rfloor+1. \] Since \(c\) is not a wall value, \(\rho(c)\notin \mathbb Z\). By semistability, \[ \rho(c)\in [j_{\min}(p),j_{\max}(p)], \] hence \[ e_+(p)=j_{\max}(p)\ge k. \] Set \[ s:=d-e_+(p). \] On the chart \(y=1\) the local equation of \(C_p\) at \(P\) is \[ f_P(X,Z)=Z^d+a Z X^{d-1}+X^s u(X), \] where \(u(0)\neq 0\) if \(s\ge 1\).

If \(s=0\), then \(P\notin C_p\), already covered by the first case. If \(s=1\), then \(P\) is a smooth point of \(C_p\), already covered by the third case. So we may assume \(s\ge 2\).

Consider the weights \[ \operatorname{wt}(X)=d, \qquad \operatorname{wt}(Z)=s. \] The weighted degree of the principal part \(Z^d+X^s\) is \(ds\), while the term \(Z X^{d-1}\) has weighted degree \[ s+d(d-1)>ds \] because \(s<d\). Therefore \(f_P\) is semi-quasihomogeneous with weighted initial form \[ Z^d+X^s. \] By the standard semi-quasihomogeneous theorem, \[ \operatorname{lct}_P(C_p)=\frac1d+\frac1s. \] Hence for every prime divisor \(E\) centered at \(P\), \[ E(C_p)\le \frac{ds}{d+s}A(E). \] Therefore \[ \beta(E)\ge \left(\frac{\alpha+dc}{3}-c\frac{ds}{d+s}\right)A(E). \] Since \(s=d-e_+(p)\), this coefficient equals \[ \frac{\alpha(2d-e_+(p))+dc(2e_+(p)-d)}{3(2d-e_+(p))}. \]

We now check that this number is positive. If \(e_+(p)\ge d/2\), then \(2e_+(p)-d\ge 0\), so positivity is immediate. If \(e_+(p)<d/2\), then \(k<d/2\) as well, because \(e_+(p)\ge k\). Since \(\rho\) is strictly increasing and \(\rho(c)<k\), one has \[ c<c_k. \] Because the wall values are strictly increasing for \(e<d/2\) and \(e_+(p)\ge k\), we get \[ c<c_k\le c_{e_+(p)}. \] Thus \[ \alpha(2d-e_+(p))+dc(2e_+(p)-d)>0. \] Hence \(\beta(E)>0\) in every case.

We have proved that every prime divisor over \(\mathbb P^2\) has strictly positive \(\beta\)-invariant. By the valuative criterion, the pair is \(K\)-stable. In particular, there is no hidden wall on \(B^\circ\). \(\square\)

We combine the previous calculations.

Theorem 7.5. On the regular open locus \(B^\circ\), the wall structure of the family \[ (\mathbb P^2,\alpha\Lambda+cC_p) \] is completely coefficient-visible. More precisely:

1. the coefficient walls in the log Fano interval are exactly \[ c_e=\frac{\alpha(2d-e)}{d(d-2e)} \] for \[ 0\le e<\frac{d(1-\alpha)}{2-\alpha}; \]

2. the wall center for \(e=0\) is \[ Z_{F_0}\cong \mathbb P(d-1,d), \] while for \(e>0\) the wall center is a point;

3. on every wall model one has the exact identity \[ \mu_e^{\mathrm{GIT}}(\alpha,c)=-\beta_e(\alpha,c); \]

4. there is no hidden wall on \(B^\circ\).

Proof. Parts (1)–(3) are Sections 7.2–7.4. Part (4) is Theorem 7.4. \(\square\)

8. Final remarks

The weighted branch family and the plane-branch-with-line family illustrate two opposite outcomes of the same abstract theorem.

In the weighted branch family, the fixed-ambient coefficient problem controls every wall before the last one, but the final wall is forced out of the coefficient arrangement and is detected by a singularity valuation over a special fiber. This is the genuinely two-mechanism picture.

In the plane-branch-with-line family, the coefficient arrangement is again nontrivial and the exact identities \[ \mu^{\mathrm{GIT}}=-\beta \] hold on all liftable wall models. However, on the natural dense regular locus \(B^\circ\), there is no hidden wall at all. Thus the second application shows that hidden walls are not forced by non-proportionality alone; they arise only when the special singularity geometry is strong enough to overcome the local \(\delta\)-gap mechanism.

This is exactly the conceptual content of Theorem 5.1. The theorem is not merely a reformulation of a family-specific calculation. It separates three independent tasks:

1. identify the coefficient faces and prove liftability;

2. compare product CM weights with \(\beta\)-invariants on wall models;

3. show that a hidden special divisor cannot remain in the locally product, \(\delta\)-rigid part of the family.

Once these are separated, the weighted branch family and the plane-branch-with-line family become two clean test cases for the same theorem. The first family exhibits both coefficient-visible and pair-singularity-visible walls. The second family exhibits only coefficient-visible walls on its regular coefficient locus. Together they show that the visibility theorem is both flexible and sharp.

References

[1] C. Li and C. Xu, Special test configuration and \(K\)-stability of Fano varieties, Ann. of Math. (2) 180 (2014), no. 1, 197–232.

[2] Y. Liu and C. Zhou, \(K\)-moduli with real coefficients, arXiv:2412.15723.

[3] Y. Liu and C. Zhou, Non-proportional wall crossing for \(K\)-stability, arXiv:2412.15725.

[4] I.-K. Kim, Y. Liu, and C. Wang, Wall-crossing for \(K\)-moduli spaces of certain families of weighted projective hypersurfaces, arXiv:2406.07907.

[5] J. Kollár, Singularities of the Minimal Model Program, Cambridge Tracts in Mathematics 200, Cambridge University Press, 2013.