2.1 Foundation

In this paper we study log pairs consisting of weighted projective spaces and ${\mathbb {Q}}$ -divisors. We define the weighted projective space. Let N be a positive integer. For positive integers $a_0,\ldots , a_N$ , let

be the graded ring where the variable $x_i$ has weight $\operatorname {wt}(x_i) = a_i$ . We define

$$ \begin{align*} {\mathbb {P}}(a_0,\ldots, a_N) = \operatorname{Proj} R[a_0, \ldots, a_N] \end{align*} $$

and call it the weighted projective space with homogeneous coordinates $x_0,\ldots , x_N$ . We sometimes denote

$$ \begin{align*} {\mathbb {P}}(a_0,\ldots, a_N)_{x_0,\ldots , x_N} \end{align*} $$

in order to make it clear the homogeneous coordinates $x_0,\ldots , x_N$ . For $i=0,\ldots , N$ we denoted by

$$ \begin{align*} \mathsf {p}_{x_i}=[0:\cdots :1:\cdots : 0]\in {\mathbb {P}}(a_0,\ldots , a_N) \end{align*} $$

the coordinate point at which only the coordinate $x_i$ does not vanish. For $i=0,\ldots , N$ we define

and

We call $U_{x_i}$ the standard affine open subset of ${\mathbb {P}}(a_0,\ldots , a_N)$ containing the point $\mathsf {p}_{x_i}$ .

Definition 2.1. Let $(X,\Delta )$ be a pair, D an effective ${\mathbb {Q}}$ -divisor on X, and let $\mathsf {p}\in X$ be a point. Assume that $(X,\Delta )$ has at most log canonical singularities. We define the log canonical threshold of D at $\mathsf {p}$ with respect to the log pair $(X, \Delta )$ and the log canonical threshold of D with respect to the log pair $(X, \Delta )$ to be the numbers

$$ \begin{align*} \operatorname{lct}_{\mathsf {p}}(X, \Delta;D) &= \sup\{c\in{\mathbb {Q}}~|~(X,\Delta+cD) \textrm{~is log canonical at~}\mathsf {p}\}, \\ \operatorname{lct}(X, \Delta;D) &= \sup\{c\in{\mathbb {Q}}~|~(X,\Delta+cD) \textrm{~is log canonical}\}, \end{align*} $$

respectively. We set $\operatorname {lct}_{\mathsf {p}} (X, D) = \operatorname {lct}_{\mathsf {p}} (X, 0; D)$ and $\operatorname {lct}_{\mathsf {p}} (X, D) = \operatorname {lct}(X, \Delta; D)$ when $\Delta = 0$ .

Let S be a surface with at most cyclic quotient singularities and $\mathsf {p}$ a cyclic quotient singular point of type $\frac {1}{r}(a,b)$ on S, where $a,b,r$ are pairwise coprime integers. Then there is an orbifold chart $\phi \colon \widetilde {U}\to U$ for some open set $\mathsf {p}\in U$ on S such that $\widetilde {U}$ is smooth and $\phi $ is a cyclic cover of degree r branched over $\mathsf {p}$ . For any ${\mathbb {Q}}$ -divisor B on S we write $\operatorname {mult}_{\mathsf {p}}(B)$ for $\operatorname {mult}_{\mathsf {q}}(\phi ^*(B|_U))$ where $\mathsf {q}\in \widetilde {U}$ is a point such that $\phi (\mathsf {q}) = \mathsf {p}$ .

2.2 K-stability of log Fano pairs

Definition 2.4. Let E be a prime divisor over a normal variety X, that is, there is a proper birational morphism $\sigma :Y \rightarrow X$ from a normal variety Y and E is a divisor on Y. For a log pair $(X,\Delta )$ the log discrepancy $A_{X, \Delta }(E)$ of $(X,\Delta )$ along E is defined to be

$$ \begin{align*} A_{X, \Delta}(E) = \operatorname{\mathrm{ord}}_E(K_Y - \sigma^*(K_X + \Delta)) + 1. \end{align*} $$

We define the S-invariant $S_{X, \Delta }(E)$ of the log pair $(X, \Delta )$ along E as

(2.1) $$ \begin{align} S_{X,\Delta}(E)=\frac{1}{\operatorname{\mathrm{vol}}(-(K_X+\Delta))}\int^{\tau}_0\operatorname{\mathrm{vol}}(\sigma^*(-(K_X+\Delta))-tE) \mathop{}\!\mathrm{d} t \end{align} $$

where $\tau =\tau (E)=\mathrm {sup}\{t\in \mathbb {Q}_{\geq 0} \ | \ \sigma ^*(-(K_X+\Delta ))-tE \text { is big }\}$ is the pseudo-effective threshold of E with respect to $-(K_X+\Delta )$ . We set

$$ \begin{align*} \beta_{X,\Delta}(E)=A_{X,\Delta}(E)-S_{X,\Delta}(E). \end{align*} $$

We call $\beta _{X,\Delta }(E)$ the $\beta $ -invariant of the log pair $(X, \Delta )$ along E.

Proof. By (2.1), we have

$$\begin{align*}S_{X,\mathcal{O}(1)}(E)=\frac{1}{(\mathcal{O}(1))^2}\int^{\frac{1}{k}}_{0}\mathrm{vol}(\mathcal{O}(1)-tE)dt=\int^{\frac{1}{k}}_{0}(1-tk)^2dt=\frac{1}{3k}.\end{align*}$$

Since $\Delta \in |\mathcal {O}(r)|$ , we get $S_{X,\Delta }(E)=rS_{X,\mathcal {O}(1)}(E)$ . Hence the result is true.

K-stability of a log Fano pair was originally defined using test configurations (see, e.g., [Reference Xu62, Definition 2.6.]). The following theorem gives the Fujita–Li’s valuation criteria of K-stability which we also take as a definition.

We refer reader for to [Reference Ascher, DeVleming and Liu7, Definition 2.3] for the definition of test configuration and specially degeneration. The following theorem is a direct consequence of the openness of K-semistability [Reference Blum, Liu and Xu16, Reference Xu61].

We consider a klt log Fano pair $(X,\Delta )$ with an action by the maximal torus $\mathbb {T}$ in $\mathrm {Aut}(X,\Delta )$ .

The complexity of the $\mathbb {T}$ -action on $(X,\Delta )$ is the number $\mathrm {dim}(X)-\mathrm {dim}(\mathbb {T})$ . When the complexity of the $\mathbb {T}$ -action on $(X,\Delta )$ is one, we call $(X,\Delta )$ a complexity one $\mathbb {T}$ -pair. For a complexity one $\mathbb {T}$ -pair, the following theorem gives a criterion for K-polystability (cf. [Reference Ilten and Süss31], [Reference Araujo, Castravet, Cheltsov, Fujita, Kaloghiros, Martinez-Garcia, Shramov, Süß and Viswanathan2, Theorem 1.3.9], [Reference Liu40, Theorem 3.2]). In our paper, we use the criterion for log Fano surface pairs $(X,\Delta )$ .

We introduce another invariant of log Fano pairs which we call $\delta $ -invariant. It serves a criterion for K-stability.

Definition 2.11 [Reference Fujita and Odaka26, Reference Blum and Jonsson14].

Let $(X,\Delta )$ be a klt log Fano pair, we define the stability threshold to be

where the infimum is taken over all prime divisors E over X.

In this paper, we use both local and global methods to check K-stability. We now state local analogues of the stability threshold.

Definition 2.13 [Reference Abban and Zhuang9, Definition 2.5].

Let $(X,\Delta )$ be a log Fano pair and $\mathsf {p}$ be a point in X, we define the $\delta $ -invariant at a point $\mathsf {p}$ with respect to $(X,\Delta )$ to be

where infimum is taken all prime divisors E over X whose centers on X contain $\mathsf {p}$ .

2.3 Abban-Zhuang method

Let $(X,\Delta )$ be a klt log del Pezzo surface pair. We now explain how to apply Abban-Zhuang method to estimate $\delta (X,\Delta )$ . Let E be a prime divisor over X. Then there is a birational morphism $\phi \colon Y\to X$ where E is a prime divisor on Y. [Reference Abban and Zhuang9] gives the definition of the number $S(W^E_{\bullet ,\bullet };q)$ for $q\in E$ which can be calculated by the following theorem.

Theorem 2.14 [Reference Araujo, Castravet, Cheltsov, Fujita, Kaloghiros, Martinez-Garcia, Shramov, Süß and Viswanathan2, Equation (1.7.13)], [Reference Fujita29, Lemma 3.16].

Let $P(u)$ and $N(u)$ be the positive and negative parts respectively of the Zariski Decomposition of the divisor $\pi ^*(-K_X-\Delta )-uE$ . For each point q in E, we denote

(2.2) $$ \begin{align} h(u):=(P(u)\cdot E)\cdot \mathrm{ord}_q(N(u)|_E)+\int_0^{\infty}\operatorname{\mathrm{vol}}(P(u)|_E-vq)\mathop{}\!\mathrm{d} v. \end{align} $$

Then we have

(2.3) $$ \begin{align} S(W^E_{\bullet,\bullet};q)=\frac{2}{\operatorname{\mathrm{vol}}(-(K_X+\Delta))}\int_0^{\tau(E)}h(u) \mathop{}\!\mathrm{d} u. \end{align} $$

We denote the difference of on E by $\Phi $ such that , where is the strict transform of $\Delta $ . From [Reference Abban and Zhuang9, Theorems 3.2 and 3.4] we have the following result.

Theorem 2.15 [Reference Abban and Zhuang9].

We have

$$ \begin{align*} \delta_{\mathsf {p}}(W, \Delta)\geq \min\left\{\frac{A_{W, \Delta}(E)}{S_{W, \Delta}(E)}, \frac{A_{E, \Phi}(\mathsf {q})}{S(W^E_{\bullet,\bullet};\mathsf {q})}\right\}. \end{align*} $$

The following lemma allows us to compute the S-invariant of toric valuations.

Lemma 2.16. Let ${\mathbb {P}}(m_0,m_1,m_2)$ be a well-formed weighted projective plane with coordinates $x_0, x_1, x_2$ , and $v_{(a,b)}$ be the toric monomial valuation of weight $(a,b)$ in the affine coordinate $(x_j, x_k)$ and $x_i\neq 0$ , where a and b are two positive rational numbers and $\{i,j,k\}=\{0,1,2\}$ . Then we have

$$\begin{align*}S_{\mathcal{O}(1)}(v_{(a,b)}) = \frac{1}{3}\left(\frac{a}{m_j}+\frac{b}{m_k}\right). \end{align*}$$

Proof. Let $s_1, \cdots , s_{N_m}$ be a basis of $H^0(\mathcal {O}(m))$ . Denote by $v=v_{(a,b)}$ . We have $S_{m,\mathcal {O}(1)}(v)=\mathrm {sup}\frac {1}{mN_m}\sum _{\alpha =1}^{N_m}v(s_\alpha )$ , where the supremum is taken over all the bases of $H^0(\mathcal {O}(m))$ . The supremum is achieved by taking a basis consisting of monomials as v is a toric valuation. Let $s_\alpha =x_0^{p_\alpha }x_1^{q_\alpha }x_2^{r_\alpha }$ such that $m_0p_\alpha +m_1q_\alpha +m_2q_\alpha =m$ for $\alpha =1,\cdots ,N_m$ be a monomial basis in $H^0({\mathbb {P}}(m_0, m_1, m_2),\mathcal {O}(m))$ . We have $S_{m,\mathcal {O}(1)}(v)=\frac {1}{mN_m}\sum _{\alpha =1}^{N_m}v(x_0^{p_\alpha }x_1^{q_\alpha }x_2^{r_\alpha })$ . Without loss of generality, we take $(i,j,k)=(2,0,1)$ and consider in the affine chart $x_2=1$ . Let $\Delta _{m}\subset {\mathbb {R}}^2$ be the triangle with vertices $(0,0),(\frac {m}{m_0} ,0),(\frac {m}{m_1} ,0)$ and $\Delta ^{\mathbb {Z}}_{m}=\{p,q\in \mathbb {Z}_{\geq 0}\colon m_0p+m_1q\leq m \text { and } m_2| (m_0p+m_1q-m)\}$ be a set of a part of lattice points in $\Delta _{m}$ . We get

$$ \begin{align*}S_{m,\mathcal{O}(1)}(v)=(\frac{1}{mN_m}\sum_{p,q\in \Delta^{\mathbb{Z}}_{m}}(p,q) )\cdot (a,b).\end{align*} $$

Since $\frac {1}{mN_m}\sum _{p,q\in \Delta ^{\mathbb {Z}}_{m}}(p,q)$ approaches to the centroid of $\frac {1}{m}\Delta _m = \Delta _1$ when $m\to \infty $ (see, e.g., [Reference Blum and Jonsson14, Theorem 2.1]), we get

$$ \begin{align*}S_{\mathcal{O}(1)}(v)=\lim\limits_{m\to\infty}S_{m,\mathcal{O}(1)}(v)=( \frac{1}{3m_0},\frac{1}{3m_1} ) \cdot ( a,b)=\frac{a}{3m_0}+\frac{b}{3m_1},\end{align*} $$

where $( \frac {1}{3m_0},\frac {1}{3m_1})$ is the centroid of $\Delta _1$ .

2.4 K-moduli of log Fano pairs and wall crossing

Now we go over some background about the K-moduli space of log Fano pairs where the boundary divisors are proportional to the anticanonical divisor. The wall crossing for K-moduli spaces of such pairs with a single boundary divisor was first introduced in [Reference Ascher, DeVleming and Liu7] for ${\mathbb {Q}}$ -Gorenstein smoothable case. Later, using the advances from the algebraic K-stability theory, [Reference Zhou67] generalized [Reference Ascher, DeVleming and Liu7] to allow arbitrary klt singularities. Recently, [Reference Zhou66] generalizes the description of wall crossing in [Reference Zhou67, Theorem 1.2.] to log pairs with multiple boundaries. The definition of log pairs with multiple boundaries is stated in [Reference Loginov and Zhou47]. In our paper, we consider log pairs with two boundary divisors. Now we state the definition of log pairs used in the paper.

Definition 2.17 [Reference Loginov and Zhou47, Reference Zhou66].

Fix $n\in {\mathbb {N}}$ , $v\in {\mathbb {Q}}_{>0}$ , $h,d\in {\mathbb {Q}}_{>0}$ , let $\theta \in [0,\max \{1, h\})\cap {\mathbb {Q}}$ be a fixed number, we denote $\mathcal {E}:=\mathcal {E}(n, v, h, d,\theta , w)$ to be the set consisting of log pairs $(X, H+D)$ which satisfies the following conditions:

1. (1) X is a n-dimensional ${\mathbb {Q}}$ -Fano variety of volume $(-K_X)^n = v$ ;

2. (2) H and D are two ${\mathbb {Q}}$ -Cartier Weil divisors on X such that $-K_X\sim _{{\mathbb {Q}}} h^{-1}H \sim _{{\mathbb {Q}}} d^{-1} D$ ;

3. (3) there exists a number $w\in \Delta :=\{w\ |\ {w\in [0,1)\cap {\mathbb {Q}} \textit {and} h\theta + dw < 1}\}$ such that $(X, \theta H+wD)$ is K-semistable.

The K-semistable domain of $(X, H+D)\in \mathcal {E}$ is defined to be:

$$ \begin{align*}Kss(X, \theta H+wD):=\overline{\{w\in \Delta | (X, \theta H+wD)\ \textit{is}\ K\text{-}\textit{semistable}\}},\end{align*} $$

where the overline means taking the closure.

From the works of [Reference Jiang32, Reference Xu61, Reference Blum, Liu and Xu16], for $w\in \Delta $ , there is an Artin stack $\mathcal {M}^K_{n, v, h, d,\theta , w}$ of finite type, parametrizing the log pairs $(X, H+D)$ such that they satisfy the conditions (1), (2) in Definition 2.17, and $(X,\theta H+wD)$ is a K-semistable log Fano pair.

Furthermore, the Artin stack $\mathcal {M}^K_{n, v, h, d,\theta , w}$ descends to a projective scheme $M^K_{n, v, h, d,\theta , w}$ as a good moduli space which parametrizes K-polystable log Fano pairs $(X, \theta H + wD)$ where $h\theta + dw < 1$ [Reference Alper, Blum, Halpern-Leistner and Xu1, Reference Blum, Halpern-Leistner, Liu and Xu13, Reference Codogni and Patakfalvi21, Reference Xu and Zhuang63, Reference Liu, Xu and Zhuang45]. We call $\mathcal {M}^K_{n, v, h, d,\theta , w}$ (resp. $M^K_{n, v, h, d,\theta , w}$ ) a K-moduli stack (resp. K-moduli space).

In this paper, the description of wall crossing is based on the principle in the following theorem [Reference Zhou67, Theorem 1.2.], [Reference Zhou66, Theorem 1.6.].

Theorem 2.18. We have a list of rational numbers

$$ \begin{align*}0=w_0<w_1<w_2<\cdots<w_k<w_{k+1}=1\end{align*} $$

such that

1. (1) For any $w\in (w_i,w_{i+1})$ and $0\leq i\leq k$ , the Artin stacks $\mathcal {M}^{K}_{n, v, h, d,\theta , w}$ keep the same. Also there are open immersions

   $$ \begin{align*}\mathcal{M}^{K}_{n, v, h, d,\theta, w_i-\epsilon}\hookrightarrow \mathcal{M}^{K}_{n, v, h, d,\theta, w_i}\hookleftarrow \mathcal{M}^{K}_{n, v, h, d,\theta, w_i+\epsilon}\end{align*} $$
   for very small positive rational number $\epsilon \ll 1$ and $1\leq i\leq k$ .

2. (2) For any $w\in (w_i,w_{i+1})$ and $0\leq i\leq k$ , the good moduli space $M^{K}_{n, v, h, d,\theta , w}$ keeps the same. Also there are the following projective morphisms among good moduli spaces

   $$ \begin{align*}M^{K}_{n, v, h, d,\theta, w_i-\epsilon}\stackrel{\phi^-_i}{\rightarrow} M^{K}_{n, v, h, d,\theta, w_i}\stackrel{\phi^+_i}{\leftarrow}M^{K}_{n, v, h, d,\theta, w_i+\epsilon}\end{align*} $$
   for very small positive rational number $\epsilon \ll 1$ and $1\leq i\leq k$ . which are induced by the following diagram:
   

The following lemma displays interpolation of K-stability which is a helpful tool for checking K-stability. See [Reference Ascher, DeVleming and Liu7, Proposition 2.13] and [Reference Dervan23, Lemma 2.6] for related results on interpolation.

Here we use the following terminology for log Calabi-Yau pairs according to [Reference Odaka52]: it is K-stable if it is klt, and K-semistable if it is log canonical.

Proof. By interpolation [Reference Ascher, DeVleming and Liu7, Proposition 2.13], we know that $(X, \theta H + wD)$ is K-semistable for any $w\in [w_1, w_2]$ . Moreover, (1) also follows from [Reference Ascher, DeVleming and Liu7, Proposition 2.13]. We shall focus on (2).

By Theorem 2.18, there exists $\epsilon>0$ such that the K-polystable degeneration $(X_2, \theta H_2 + w D_2)$ of $(X, \theta H + w D)$ is independent of the choice of $w\in (w_1, w_1+\epsilon )$ . Denote the test configuration associated to this degeneration by $(\mathcal {X}, \theta \mathcal {H} + w\mathcal {D})/{\mathbb {A}}^1$ . Since $(X_2, \theta H_2 + w D_2)$ is K-polystable for every $w\in (w_1, w_1+\epsilon )$ , we know that $\mathrm {Fut}(\mathcal {X}, \theta \mathcal {H} + w\mathcal {D}) = 0$ . By linearity of Futaki invariants, we have $\mathrm {Fut}(\mathcal {X}, \theta \mathcal {H} + w\mathcal {D}) = 0$ for every $w\in [w_1, w_2]$ . Thus $(X_2, \theta H_2 + w D_2)$ is K-semistable for any $w\in [w_1, w_2]$ by [Reference Li, Wang and Xu43, Lemma 3.1].

Since K-semistable is a closed condition for coefficients, we know that $(X_2, \theta H_2 + w_1 D_2)$ is a K-semistable degeneration of $(X, \theta H + w_1 D)$ . By [Reference Li, Wang and Xu43, Theorem 1.3] we know that the K-polystable degeneration of $(X_2, \theta H_2 + w_1 D_2)$ is also $(X_1, \theta H_1 + w_1 D_1)$ . Thus

(2.4) $$ \begin{align} 1 = \mathrm{rank}\,\mathrm{Aut}(X_1, \theta H_1 + w_1 D_1)\geq \mathrm{rank}\,\mathrm{Aut}(X_2, \theta H_2 + w_1 D_2). \end{align} $$

If $(X_1, \theta H_1 + D_1)\cong (X_2, \theta H_2 + D_2)$ , then $(X_1, \theta H_1 + w_2 D_1)$ is K-semistable, a contradiction to our assumption. Thus $(X_1, \theta H_1 + D_1)$ is not isomorphic to $(X_2, \theta H_2 + D_2)$ , which implies that the inequality in (2.4) is strict by [Reference Li, Wang and Xu43, Theorem 1.4]. Thus $\mathrm {Aut}(X_2, \theta H_2 + w D_2)$ is finite for $w\in (w_1, w_2)$ by its rank being zero and reductivity [Reference Alper, Blum, Halpern-Leistner and Xu1]. As a result, $(X_2, \theta H_2 + w D_2)$ is K-stable for any $w\in (w_1, w_2)$ and so is $(X, \theta H + wD)$ by [Reference Blum and Xu18].

3.1 Set-up of w-GIT when n is odd

Let D be a curve in $W={\mathbb {P}}(1,2,n+2)_{x,y,z}$ defined by

(3.1) $$ \begin{align} z^2y + azx^{n+4} + a_0x^{2n+6} + a_1x^{2n+4}y + \cdots + a_{n+3}y^{n+3} = 0. \end{align} $$

We consider an affine space $\mathbb {A}^{n+5}$ with coordinates $a,a_0,\cdots ,a_{n+3}$ which parametrizes all curves D of the form (3.1). Consider the action of torus $(\mathbb {G}_m)_{s_1,s_2}^2$ on W given by

$$\begin{align*}(s_1, s_2) \cdot [x:y:z]:= [x:s_1^{-1} y: s_2^{-1} z]. \end{align*}$$

Then this induces an action of $(\mathbb {G}_m)_{s_1,s_2}^2$ on $\mathbb {A}^{n+5}$ by pushforward of D along the torus action. In particular, we have

$$\begin{align*}(s_1, s_2) \cdot (a, a_j) = (s_1^{-1}s_2^{-1} a, s_1^{j-1} s_2^{-2} a_j). \end{align*}$$

We have a universal family $\phi :(\mathcal {W},w\mathcal {D})\rightarrow \mathbb {A}^{n+5}$ and the fiber over each point $(a,a_0,\cdots ,a_{n+3})$ is $(W,wD)$ , where D is given by Equation (3.1). The CM ${\mathbb {Q}}$ -line bundle on $(\mathcal {W},w\mathcal {D})$ induces the trivial ${\mathbb {Q}}$ -line bundle $L_w$ on $\mathbb {A}^{n+5}$ with a $\mathbb {G}_m^2$ -linearization.

For each $(d,b)\in {\mathbb {Z}}^2$ we have a one-parameter subgroup $\lambda _{(d,b)}: \mathbb {G}_m \rightarrow \mathbb {G}_m^2$ given by $\lambda (t) = (t^d, t^b)$ . Moreover, $\lambda $ acts on ${\mathbb {A}}^{n+5}$ as

(3.2) $$ \begin{align} \lambda(t)\cdot a=t^{-d-b}a \text{ and } \lambda(t)\cdot a_j=t^{(j-1)d-2b}a_j \text{ for } j=0,\cdots,n+3. \end{align} $$

Let

$$\begin{align*}\beta^{w}_{(d,b)}:=\frac{1-n+2nw}{6}d+\frac{2n+1-w(4n+6)}{3(n+2)}b. \end{align*}$$

Proof. By Hilbert-Mumford criterion in the affine case (see, e.g., [Reference King33, Proposition 2.5]), it suffices to show that $\beta _{(d,b)}^w$ coincides with the GIT weight $\mu ^{L_w}(q, \lambda )$ of $\lambda =\lambda _{(d,b)}$ acting on the fiber of $L_w$ at the limit point $q:=\lim _{t\to 0}\lambda (t)\cdot p$ provided the existence of q. Since the $\mathbb {G}_m^2$ -action on ${\mathbb {A}}^{n+5}$ is linear, we know that the fixed locus of every $\lambda $ is an affine subspace of ${\mathbb {A}}^{n+5}$ and hence is connected. Therefore, we have $\mu ^{L_w}(q, \lambda ) = \mu ^{L_w}(0, \lambda )$ . Since the weight of the CM ${\mathbb {Q}}$ -line bundle coincides with the Futaki invariant, which also coincides with the $\beta $ -invariant for product test configurations, we have that

$$\begin{align*}\mu^{L_w}(q, \lambda_{(d,b)}) = \mu^{L_w}(0, \lambda_{(d,b)}) = \beta_{(W,wD_0)}(v_{(d,b)}), \end{align*}$$

where $v_{(d,b)}$ is the monomial valuation on W where in the affine chart $x=1$ we have $v_{(d,b)}(y) = d$ and $v_{(d,b)}(z) = b$ .

Let us assume $d,b\geq 0$ for a moment. Then we can compute $\beta _{(W,wD_0)}(v_{(d,b)})$ as follows. By Lemma 2.16, we get the S-invariant for $v=v_{(d,b)}$ :

$$ \begin{align*} S_{\mathcal{O}(1)}(v)=\frac{1}{3}\big(\frac{d}{2}+\frac{b}{n+2}\big). \end{align*} $$

We have $-K_{W}-wD_0=\mathcal {O}(n+5-w(2n+6))=r\mathcal {O}(1)$ and $S_{(W,wD_0)}(v)=\mathrm {deg}(-K_{W}-wD_0)\cdot S_{\mathcal {O}(1)}(v)=rS_{\mathcal {O}(1)}(v)$ , where $r=n+5-w(2n+6)$ . The log discrepancy $A_{W}(v)=d+b$ and $A_{(W,wD_0)}(v)=A_{W}(v)-wv(D_0)=d+b-w(d+2b)$ . Then we have $\beta $ -invariant

(3.3) $$ \begin{align} \beta_{(W,wD_0)}(v_{(d,b)})=A_{(W,wD_0)}(v)-S_{(W,wD_0)}(v)=\frac{1-n+2nw}{6}d+\frac{2n+1-w(4n+6)}{3(n+2)}b. \end{align} $$

Thus this implies that $\mu ^{L_w}(0, \lambda _{(d,b)}) = \beta ^w_{(d,b)}$ whenever $d,b\geq 0$ . By the standard representation theory of torus we know that $\mu ^{L_w}(0, \lambda _{(d,b)}) $ is linear in $(d,b)$ . Since $\beta ^w_{(d,b)}$ is linear in $(d,b)$ by definition, we conclude that

$$\begin{align*}\mu^{L_w}(q, \lambda_{(d,b)}) = \mu^{L_w}(0, \lambda_{(d,b)}) = \beta^w_{(d,b)} \end{align*}$$

for every $(d,b)\in {\mathbb {Z}}^2$ . Thus the proof is finished.

3.2 Set-up of w-GIT when n is even

Let $n=2l$ , where l is a positive integer. Let D be a curve in $W'={\mathbb {P}}(1,1,l+1)_{u,y,z}$ defined by

(3.4) $$ \begin{align} z^2y + azu^{l+2} + a_0u^{2l+3} + a_1u^{2l+2}y + \cdots + a_{2l+3}y^{2l+3} = 0. \end{align} $$

We consider an affine space $\mathbb {A}^{2l+5}$ with coordinates $a,a_0,\cdots ,a_{2l+3}$ which parametrizes all curves D of the form (3.4). We consider the action of torus $(\mathbb {G}_m)_{s_1,s_2}^2$ on $W'$ given by

$$\begin{align*}(s_1, s_2) \cdot [u:y:z]:= [u:s_1^{-1} y: s_2^{-1} z]. \end{align*}$$

Then this induces an action of $(\mathbb {G}_m)_{s_1,s_2}^2$ on $\mathbb {A}^{2l+5}$ by pushforward of D along the torus action. In particular, we have

$$\begin{align*}(s_1, s_2) \cdot (a, a_j) = (s_1^{-1}s_2^{-1} a, s_1^{j-1} s_2^{-2} a_j) \end{align*}$$

for $j=0,1,\cdots ,2l+3.$

We have a universal family $\phi :(\mathcal {W}',\frac {1}{2}\mathcal {H}+w\mathcal {D})\rightarrow \mathbb {A}^{2l+5}$ and the fiber over each point $(a,a_0,\cdots ,a_{2l+3})$ is $(W',\frac {1}{2}H+wD)$ , where D is given by Equation (3.4). The CM ${\mathbb {Q}}$ -line bundle on $(\mathcal {W}',\frac {1}{2}\mathcal {H}+w\mathcal {D})$ induces the trivial ${\mathbb {Q}}$ -line bundle $L_w$ on $\mathbb {A}^{2l+5}$ with a $\mathbb {G}_m^2$ -linearization.

For each $(d,b)\in {\mathbb {Z}}^2$ we have a one-parameter subgroup $\lambda _{(d,b)}: \mathbb {G}_m \rightarrow \mathbb {G}_m^2$ given by $\lambda (t) = (t^d, t^b)$ . Moreover, $\lambda $ acts on ${\mathbb {A}}^{2l+5}$ as

(3.5) $$ \begin{align} \lambda(t)\cdot a=t^{-d-b}a \text{ and } \lambda(t)\cdot a_j=t^{(j-1)d-2b}a_j \text{ for } j=0,\cdots,2l+3. \end{align} $$

Let

$$\begin{align*}\beta_{(d,b)}^w:=\frac{1-2l+4lw}{6}d+\frac{4l+1-w(8l+6)}{6(l+1)}b.\end{align*}$$

Proof. By Hilbert-Mumford criterion in the affine case (see, e.g., [Reference King33, Proposition 2.5]), it suffices to show that $\beta _{(d,b)}^w$ coincides with the GIT weight $\mu ^{L_w}(q, \lambda )$ of $\lambda =\lambda _{(d,b)}$ acting on the fiber of $L_w$ at the limit point $q:=\lim _{t\to 0}\lambda (t)\cdot p$ provided the existence of q. Since the $\mathbb {G}_m^2$ -action on ${\mathbb {A}}^{2l+5}$ is linear, we know that the fixed locus of every $\lambda $ is an affine subspace of ${\mathbb {A}}^{2l+5}$ and hence is connected. Therefore, we have $\mu ^{L_w}(q, \lambda ) = \mu ^{L_w}(0, \lambda )$ . Since the weight of the CM ${\mathbb {Q}}$ -line bundle coincides with the Futaki invariant, which also coincides with the $\beta $ -invariant for product test configurations, we have that

$$\begin{align*}\mu^{L_w}(q, \lambda_{(d,b)}) = \mu^{L_w}(0, \lambda_{(d,b)}) = \beta_{(W',\frac{1}{2}H+wD_0)}(v_{(d,b)}), \end{align*}$$

where $D_0$ is given by $z^2y=0$ , $v_{(d,b)}$ is the monomial valuation on $W'$ where in the affine chart $u=1$ we have $v_{(d,b)}(y) = d$ and $v_{(d,b)}(z) = b$ .

Let us assume $d,b\geq 0$ for a moment. Then we can compute $\beta _{(W',\frac {1}{2}H+wD_0)}(v_{(d,b)})$ as follows. By Lemma 2.16, we get the S-invariant for $v=v_{(d,b)}$ :

$$ \begin{align*} S_{\mathcal{O}(1)}(v)=\frac{1}{3}\big(d+\frac{b}{l+1}\big). \end{align*} $$

We have $-K_{W'}-\frac {1}{2}H-wD_0=r\mathcal {O}(1)$ and $S_{(W',\frac {1}{2}H+wD_0)}(v)=rS_{\mathcal {O}(1)}(v)$ , where $r=l+3-\frac {1}{2}-w(2l+3)$ . The log discrepancy $A_{W'}(v)=d+b$ and $A_{(W',\frac {1}{2}H+wD_0)}(v)=A_{W'}(v)-v(\frac {1}{2}H+wD_0)=d+b-w(d+2b)$ since $v(H)=0$ . Then we have $\beta $ -invariant $\beta _{(W',\frac {1}{2}H+wD_0)}(v_{(d,b)})$ equals

(3.6) $$ \begin{align} A_{(W',\frac{1}{2}H+wD_0)}(v)-S_{(W',\frac{1}{2}H+wD_0)}(v)=\frac{1-2l+4lw}{6}d+\frac{4l+1-w(8l+6)}{6(l+1)}b. \end{align} $$

Thus this implies that $\mu ^{L_w}(0, \lambda _{(d,b)}) = \beta ^w_{(d,b)}$ whenever $d,b\geq 0$ . By the standard representation theory of torus we know that $\mu ^{L_w}(0, \lambda _{(d,b)}) $ is linear in $(d,b)$ . Since $\beta ^w_{(d,b)}$ is linear in $(d,b)$ by definition, we conclude that

$$\begin{align*}\mu^{L_w}(q, \lambda_{(d,b)}) = \mu^{L_w}(0, \lambda_{(d,b)}) = \beta^w_{(d,b)} \end{align*}$$

for every $(d,b)\in {\mathbb {Z}}^2$ . Thus the proof is finished.

3.3 w-GIT moduli spaces

In Sections 3.1 and 3.2, we set up w-GIT in the affine case. The goal of this section is to describe the w-GIT moduli spaces $M_w^{GIT}$ . We first give an explicit criterion for w-GIT stability.

Let $(d,b)$ , $u=( \frac {1-n+2nw}{6},\frac {2n+1-w(4n+6)}{3(n+2)})$ , $v=(-1,-1)$ , $v_j=(j-1,-2)$ for $j=0,\cdots ,n+3$ be vectors in $\mathbb {R}^2$ . For a point $p=(a,a_0,\cdots ,a_{n+3})$ , we denote $T_p$ as a subset of $\{v,v_0,\cdots ,v_{n+3}\}$ such that $v\in T_p$ if and only if $a\neq 0$ , and $v_j\in T_p$ if and only if $a_j\neq 0$ for $j=0,\cdots ,n+3$ . That is to say,

$$\begin{align*}T_p:=\begin{cases} \{v_j\mid a_j\neq 0 \text{ for } j=0,\cdots,n+3\} & \textrm{ if } a=0;\\ \{v\}\cup\{v_j\mid a_j\neq 0 \text{ for } j=0,\cdots,n+3\} & \textrm{ if }a\neq 0. \end{cases} \end{align*}$$

Let $\Gamma _p$ be the cone generated by all the vectors in $T_p$ . We denote the dual cone of $\Gamma _p$ to be

$$ \begin{align*}\Gamma_p^{\vee}=\{r\in\mathbb{R}^2\mid r\cdot r'\geq 0 \text{ for any } r'\in \Gamma_p\}.\end{align*} $$

Now we can reinterpret the conditions for w-GIT stability as follows.

Proof. Let $J=\{j\in \{0,\cdots ,n+3\} \mid a_j\neq 0\}$ . Let $\lambda =\lambda _{(d,b)}$ be a one-parameter subgroup. Note (3.2) and (3.5). When $a=0$ , we have $\lim \limits _{t \to 0}\lambda (t)p$ exists if and only if

$$ \begin{align*} v_j\cdot (d,b)=(j-1)d-2b\geq 0\text{ for } j\in J. \end{align*} $$

When $a\neq 0$ , we have $\lim \limits _{t \to 0}\lambda (t)p$ exists if and only if

$$ \begin{align*} v\cdot (d,b) =-d-b\geq 0\text{ and } v_j\cdot (d,b)=(j-1)d-2b\geq 0\text{ for } j\in J. \end{align*} $$

Then the existence of $\lim \limits _{t \to 0}\lambda (t)p$ is equivalent to that the vector $(d,b) \in \Gamma _p^{\vee }$ . By Propositions 3.2 and 3.4, if $(d,b)\in \Gamma _p^{\vee }$ , then $u\cdot (d,b)=\beta ^w_{(d,b)} \geq 0$ , which is equivalent to $u\in \Gamma _p$ .

Let p be a w-GIT semistable point. If $\Gamma _p$ has dimension one, then $T_p$ contains only one vector which is denoted by r, that is, p has only one nonzero coordinate corresponding to r. Since p is w-GIT semistable, we have $u\in \Gamma _p$ is proportional to r. By $(d,b) \cdot u=\beta ^w_{(d,b)}=0$ , we have $ (d,b) \cdot r=0$ . Thus $\lim \limits _{t \to 0}\lambda (t)p$ is in the same orbit as p. Hence p is w-GIT polystable.

Assume $\Gamma _p$ has dimension two, that is to say, p has at least two nonzero coordinates. By Propositions 3.2 and 3.4, the point p is w-GIT polystable if u is in the interior of the cone $\Gamma _p$ . Otherwise, we suppose u is on a bound of $\Gamma _p$ . Then u is proportional to a vector $r\in T_p$ generating the bound. By $( d,b) \cdot u=\beta ^w_{(d,b)}=0$ , we have $(d,b) \cdot r=0$ and $ (d,b) \cdot r'>0$ for all $r'\in T_p$ such that $r'\neq r$ . Under the action of the one-parameter group $\lambda _{(d,b)}$ , as t goes to zero, the coordinate corresponding to r goes to a nonzero number, and other nonzero coordinates go to zero. Thus $\lim \limits _{t \to 0}\lambda (t)p$ is not in the same orbit as p, which leads to a contradiction.

Now we describe the w-GIT moduli spaces $M^{GIT}_w:= {\mathbb {A}}^{n+5}//{}_{L_w} \mathbb {G}_m^2$ parameterizing w-GIT polystable points. Recall that we always assume $w\in (0, \frac {n+5}{2n+6})\cap {\mathbb {Q}}$ .

Theorem 3.6. Let n be an odd positive integer. Then the walls for $M^{GIT}_w$ are given by

$$\begin{align*}w_{i}:=\frac{(n+2)^2 - (2n+1)i}{(n+2)(2n+6)-(4n+6)i}, \quad \textrm{ where }0\leq i\leq \frac{n+1}{2}. \end{align*}$$

Moreover, we have the following explicit description of $M^{GIT}_w$ .

1. (1) When $w\in (0, w_{0}=\frac {n+2}{2n+6})$ , the w-GIT moduli space $M^{GIT}_w$ is empty.

2. (2) The w-GIT moduli space $M^{GIT}_{w_0}$ consists of a single point parameterizing the w-GIT polystable points $(a,a_0,\cdots ,a_{n+3})$ such that $a=a_0=\cdots =a_{n+2}=0$ and $a_{n+3}\neq 0$ .

3. (3) The w-GIT moduli space $M^{GIT}_{w_i}$ parameterizes all points $(a,a_0,\cdots ,a_{n+3})$ from one of the following two cases:

   • ○ $\{a,a_0,\cdots ,a_{n+2-i}\}$ are not all zero, $a_{n+3-i}$ is arbitrary and $\{a_{n+4-i},\cdots ,a_{n+3}\}$ are not all zero;

   • ○ $a=a_0=\cdots =a_{n+2-i}=0$ , $a_{n+3-i}\neq 0$ , and $a_{n+4-i}=\cdots =a_{n+3}=0$ .

4. (4) If $w\in (w_{i},w_{i+1})$ where $0\leq i\leq \frac {n+1}{2}$ , then $M^{GIT}_w$ parameterizes all points $(a,a_0,\cdots ,a_{n+3})$ such that $\{a,a_0,\cdots ,a_{n+2-i}\}$ are not all zero, and $\{a_{n+3-i},\cdots ,a_{n+3}\}$ are not all zero. Note that $w_{\frac {n+3}{2}}=\frac {n+5}{2n+6}$ .

Proof. If $w\in (0,\frac {2n+1}{4n+6}]$ , then $u=(\frac {1-n+2nw}{6},\text { }\frac {2n+1-w(4n+6)}{3(n+2)})$ is not in the lower half plane of $\mathbb {R}^2$ . For any nonzero point $p=(a,a_0,\cdots ,a_{n+3})$ , cone $\Gamma _p$ is in the lower half plane of $\mathbb {R}^2$ . So u is not in $\Gamma _p$ , which implies p is not w-GIT semistable by Lemma 3.5. Now we consider $w>\frac {2n+1}{4n+6}$ . Multiplying by $-2\frac {3(n+2)}{2n+1-w(4n+6)}>0$ , the vector u becomes

$$ \begin{align*}(\frac{(n+2)(n-1)-2n(n+2)w}{2n+1-(4n+6)w}, \text{ } -2)\end{align*} $$

which we still denote by u. For $0\leq i\leq \frac {n+1}{2}$ , we have

$$\begin{align*}\frac{(n+2)(n-1)-2n(n+2)w}{2n+1-(4n+6)w}=n+2-i \text{ if and only if } w=w_i.\end{align*}$$

When $w\in (\frac {2n+1}{4n+6},w_0)$ , we have $\frac {(n+2)(n-1)-2n(n+2)w}{2n+1-(4n+6)w}>n+2$ . Thus u is not in $\Gamma _p$ for any nonzero point $p=(a,a_0,\cdots ,a_{n+3})$ . Hence (1) holds.

If $w=w_0$ , then $u=(n+2, -2)=v_{n+3}$ . By (2) of Lemma 3.5, a point $p=(a,a_0,\cdots ,a_{n+3})$ is w-GIT semistable if $T_p$ contains $v_{n+3}$ . If $T_p=\{v_{n+3}\}$ , then $\Gamma _p$ has dimension one and contains u. So p is w-GIT polystable. If $T_p$ contains a vector more than $v_{n+3}$ , then u is not in the interior of $T_p$ . Thus p is not w-GIT polystable. Hence p is w-GIT polystable iff $T_p=\{v_{n+3}\}$ , that is, $a=a_0=\cdots =a_{n+2}=0$ and $a_{n+3}\neq 0$ . Therefore (2) holds.

If $w=w_i$ for $1\leq i\leq \frac {n+1}{2}$ , then $u=(n+2-i, -2)=v_{n+3-i}$ . By (2) of Lemma 3.5, a point $p=(a,a_0,\cdots ,a_{n+3})$ is w-GIT semistable if $T_p$ contains $v_{n+3}$ , that is, $\{a,a_0,\cdots ,a_{n+3-i}\}$ are not all zero and $\{a_{n+3-i},\cdots ,a_{n+3}\}$ are not all zero. If p is polystable, then $v_{n+3}$ cannot lie on a bound of $\Gamma _p$ , that is, $T_p$ must contain a vector in $\{v,v_0,\cdots ,v_{n+2-i}\}$ and a vector in $\{v_{n+4-i},\cdots ,v_{n+3}\}$ . Thus (3) holds.

If $w\in (w_{i},w_{i+1})$ for $0\leq i\leq \frac {n+1}{2}$ , then u is between vector $v_{n+3-i}$ and $v_{n+2-i}$ . By Lemma 3.5, a point $p=(a,a_0,\cdots ,a_{n+3})$ is w-GIT semistable if $T_p$ contains a vector in $\{v,v_0,\cdots ,v_{n+2-i}\}$ and a vector in $\{v_{n+3-i},\cdots ,v_{n+3}\}$ . We also have that u is in the interior of $\Gamma _p$ . So p is w-GIT polystable. Hence (4) holds.

With similar arguments, we can describe the w-GIT moduli spaces $M^{GIT}_w$ when n is an even positive integer as follows.

Theorem 3.7. Let $n=2l$ be an even positive integer for $l\leq 1$ . Then the walls for $M^{GIT}_w$ are given by

$$\begin{align*}w_{i}:=\frac{(n+2)^2 - (2n+1)i}{(n+2)(2n+6)-(4n+6)i}, \quad \textrm{ where }0\leq i\leq l+1. \end{align*}$$

Moreover, we have the following explicit description of $M^{GIT}_w$ .

1. (1) When $w\in (0, w_{0}=\frac {l+1}{2l+3})$ , the w-GIT moduli space $M^{GIT}_w$ is empty.

2. (2) The w-GIT moduli space $M^{GIT}_{w_0}$ consists of a single point parameterizing the w-GIT polystable points $(a,a_0,\cdots ,a_{2l+3})$ such that $a=a_0=\cdots =a_{2l+2}=0$ and $a_{2l+3}\neq 0$ .

3. (3) The w-GIT moduli space $M^{GIT}_{w_i}$ parameterizes all points $(a,a_0,\cdots ,a_{2l+3})$ from one of the following two cases:

   • ○ $\{a,a_0,\cdots ,a_{2l+2-i}\}$ are not all zero, $a_{2l+3-i}$ is arbitrary, and $\{a_{2l+4-i},\cdots ,a_{2l+3}\}$ are not all zero;

   • ○ $a=a_0=\cdots =a_{2l+2-i}=0$ , $a_{2l+3-i}\neq 0$ and $a_{{2l+4-i}}=\cdots =a_{2l+3}=0$ .

4. (4) If $w\in (w_{i},w_{i+1})$ , where $0\leq i\leq l$ , then the w-GIT moduli space $M^{GIT}_{w}$ parameterizes all points $(a,a_0,\cdots ,a_{2l+3})$ such that $\{a,a_0,\cdots ,a_{2l+2-i}\}$ are not all zero, and $\{a_{2l+3-i},\cdots ,a_{2l+3}\}$ are not all zero. Note that $w_{l+1}=\frac {1}{2}$ .

5. (5) If $w\in (w_{l+1},\frac {2l+5}{4l+6})$ , then the w-GIT moduli space $M^{GIT}_{w}$ parameterizes all points $(a,a_0,\cdots ,a_{2l+3})$ such that $\{a,a_0,\cdots ,a_{l+1}\}$ are not all zero, and $\{a_{l+2},\cdots ,a_{2l+3}\}$ are not all zero.

Proof. The proofs for (1)-(4) are the same as the proofs for (1)-(4) in Theorem 3.6.

Now we consider (5). When $w\in (w_{l+1},\frac {2l+5}{4l+6})$ , we have $w>\frac {2n+1}{4n+6}$ . Similar as the proof for Theorem 3.6, multiplying by $-2\frac {3(n+2)}{2n+1-w(4n+6)}>0$ , the vector u becomes

$$ \begin{align*}(\frac{(n+2)(n-1)-2n(n+2)w}{2n+1-(4n+6)w}, \text{ } -2)\end{align*} $$

which we still denote by u. Let $h(w)= \frac {(n+2)(n-1)-2n(n+2)w}{2n+1-(4n+6)w}$ . Since $h(w_{l+1})=l+1$ , $h(\frac {2l+5}{4l+6})=l+\frac {1}{2}$ and $h(w)$ decreases when $w\in (w_{l+1},\frac {2l+5}{4l+6})$ , we have $h(w)\in (l+\frac {1}{2},l+1)$ for $w\in (w_{l+1},\frac {2l+5}{4l+6})$ . Thus u is between vectors $v_{l+1}=(l,-2)$ and $v_{l+2}=(l+1,-2)$ . By Lemma 3.5, a point $p=(a,a_0,\cdots ,a_{n+3})$ is w-GIT semistable if $T_p$ contains a vector in $\{v,v_0,\cdots ,v_{l+1}\}$ and a vector in $\{v_{l+2},\cdots ,v_{n+3}\}$ . We also have u is in the interior of $\Gamma _p$ , so p is w-GIT polystable. Hence (5) holds.

To conclude this section, we introduce the relevant projective GIT following similar approach as [Reference Ascher, DeVleming and Liu7, Definition 5.7 and Theorem 5.8]. Let n be a positive integer. Let $\lambda _{(0,-1)}$ be the one-parameter subgroup of $\mathbb {G}_m^2$ acting on coordinates $a,a_1,\cdots ,a_{n+3}$ with positive weights $1,2,\cdots ,2$ . Taking quotient of $\mathbb {A}^{n+5}\setminus \{0\}$ by the action of $\lambda _{(0,-1)}$ , we get the weight projective stack $\mathcal {P}(1,2^{n+4}):= [(\mathbb {A}^{n+5}\setminus \{0\})/\mathbb {G}_m]$ whose coarse moduli space ${\mathbb {P}}(1,2^{n+4})$ is isomorphic to ${\mathbb {P}}^{n+4}$ via $[a:a_0:\cdots :a_{n+3}]\mapsto [a^2:a_0:\cdots :a_{n+3}]$ . Recall that $\phi : (\mathcal {W}, w\mathcal {D})\to {\mathbb {A}}^{n+5}$ (resp. $\phi : (\mathcal {W}', \frac {1}{2}\mathcal {H} + w\mathcal {D})\to {\mathbb {A}}^{n+5}$ ) is the universal family of pairs when n is odd (resp. even). After quotienting out the diagonal action of $\lambda _{(0,-1)}$ we can descend $\phi $ to a universal family over $\mathcal {P}(1, 2^{n+4})$ , where the CM ${\mathbb {Q}}$ -line bundle $L_w$ over ${\mathbb {A}}^{n+5}$ also descends to a CM ${\mathbb {Q}}$ -line bundle $\Lambda _w$ over ${\mathbb {P}}(1, 2^{n+4})$ . By standard GIT we know that the degree of $\Lambda _w$ is equal to the GIT weight $\mu ^{L_w}(0, \lambda _{(0,-1)})$ which coincides with $\beta _{(0,-1)}^w$ by the proof of Propositions 3.2 and 3.4. Thus we know that $\Lambda _w$ is ample if and only if $\beta _{(0,-1)}^w> 0$ which is equivalent to $w> \frac {2n+1}{4n+6}$ . Thus we can define the projective GIT for the $\mathbb {G}_m^2/\langle \lambda _{(0,-1)}\rangle $ -action on ${\mathbb {P}}(1,2^{n+4})$ with respect to $\Lambda _w$ .

Proposition 3.8. Let $w\in (\frac {2n+1}{4n+6}, \frac {n+5}{2n+6})\cap {\mathbb {Q}}$ . Then a point $p = (a,a_0, \cdots , a_{n+5})\in {\mathbb {A}}^{n+5}$ is w-GIT (semi/poly)stable if and only if $p\neq 0$ and $[p]$ is GIT (semi/poly)stable for the $\mathbb {G}_m^2/\langle \lambda _{(0,-1)}\rangle $ -action on ${\mathbb {P}}(1,2^{n+4})$ with respect to the ample linearization $\Lambda _w$ . In particular, we have

$$\begin{align*}M_w^{GIT}\cong {\mathbb {P}}(1,2^{n+4})//{}_{\Lambda_w} (\mathbb{G}_m^2/\langle\lambda_{(0,-1)}\rangle) \end{align*}$$

is a normal projective variety.

4.1 Odd cases

Let $W= {\mathbb {P}}(1,2,n+2)_{x,y,z}$ be the weighted projective plane where n is a positive odd integer. And let D be an effective divisor on W defined by quasi-homogeneous polynomial

(4.1) $$ \begin{align} z^2y + azx^{n+4} + a_0x^{2n+6} + a_1x^{2n+4}y + \cdots + a_{n+3}y^{n+3} = 0 \end{align} $$

of degree $2n+6$ , where a is a constant and $a_0,\ldots , a_{n+3}$ are not all zero constants. We set $t = \max \{i\in {\mathbb {Z}}\mid a_i \neq 0\}$ . Then the quasi-homogeneous polynomial of the curve D is represented by

$$ \begin{align*} z^2y + azx^{n+4} + a_0x^{2n+6} + a_1x^{2n+4}y + \cdots + a_tx^{2n+6-2t}y^t = 0. \end{align*} $$

To calculate the log canonical threshold of the log pair $(W, D)$ we need to divide it into two cases whether the constant a is zero or not. We first consider the case that a is nonzero. We have $H_y\cap D = \{\mathsf {p}_z, [1:0:-\frac {a_0}{a}]\}$ . On the affine open set $U_z$ of W given by $z=1$ , the divisor D is locally defined by the equation

$$ \begin{align*} y + ax^{n+4} + a_0x^{2n+6} + a_1x^{2n+4}y + \cdots + a_tx^{2n+6-2t}y^t = 0. \end{align*} $$

The curve in $\mathbb {C}^2 = \mathrm {Spec}(\mathbb {C}[x,y])$ defined by this equation is smooth at the origin. Since there is an orbifold chart $\mathbb {C}^2\to U_z$ on W, it follows that the log pair $(W, D)$ is log canonical at the point $\mathsf {p}_z$ . Similarly, on the affine open set $U_x$ of W defined by $x=1$ , the local equation of D is given by

$$ \begin{align*} z^2y + az + a_0 + a_1y + \cdots + a_ty^t = 0. \end{align*} $$

Since the curve in $\mathbb {C}^2=\mathrm {Spec}(\mathbb {C}[y, z])$ defined by this equation is smooth at the point $(y,z) = (0,-\frac {a_0}{a})$ , it follows that the log pair $(W, D)$ is log canonical at the corresponding point $[1:0:-\frac {a_0}{a}]$ . Therefore, the log pair $(W, D)$ is log canonical along $H_y$ .

On the affine open set $U_y$ of W defined by $y=1$ the curve D is defined by

$$ \begin{align*} & \left(z+\frac{a}{2}x^{n+4}\right)^2 - \frac{a^2}{4}x^{2n+8} + a_0x^{2n+6} + a_1x^{2n+4} + \cdots + a_{t}x^{2n+6-2t}\\ =& \left(z+\frac{a}{2}x^{n+4}\right)^2 - \frac{a^2}{4} x^{2n+6-2t}(x^2 - b_1)^{m_1}\cdots (x^2 - b_k)^{m_k} = 0\\ \end{align*} $$

where $b_1,\ldots , b_k$ are nonzero constants. We set

(4.2) $$ \begin{align} m = \max\{m_1,\ldots , m_k\}. \end{align} $$

By the definition of the log canonical threshold and [Reference Kuwata35, Theorem 1.1], we have

(4.3) $$ \begin{align} \operatorname{lct}(W, D) = \begin{cases} \min\left\{\frac{m+2}{2m}, 1\right\} &\text{ for } t\geq n+2\\ \min\left\{\frac{n+4-t}{2n+6-2t}, \frac{m+2}{2m}, 1\right\} &\text{ for } t<n+2 \end{cases} \end{align} $$

Definition 4.1. Let be the curve in W defined by

(4.4)

Then in the affine chart $U_y = \{y=1\}$ the curve can be expressed as

$$ \begin{align*}\left(z-x^{n+4}\right)^2 - (x^2 -1)^{n+4} =0.\end{align*} $$

We choose the notation based on the observation that the pair will be replaced (at the last wall) by pairs coming from Weierstraß double marked hyperelliptic curves (see Section 7.3).

From the above discussions we obtain the following lemma.

Proof. From the above discussion we see that $m\leq \sum _{j=1}^k m_j = t+1 \leq n+4$ . The first statement follows from the lct computations (4.3). For the second statement, if $m\geq n+4$ then combining with the previous inequality we get $m = n+4$ , $k = 1$ , and $t = n+3$ . In particular, in the affine open chart $U_y$ one has

$$\begin{align*}D: \left(z+\frac{a}{2}x^{n+4}\right)^2 - \frac{a^2}{4} (x^2 - b_1)^{n+4} = 0, \end{align*}$$

where a and $b_1$ are both nonzero. Thus by a rescaling of the coordinates we may assume $ a= -2$ and $b_1=1$ which shows that D and are projectively equivalent. Clearly has an $A_{n+3}$ -singularity at the nontoric point $[1, 1 , 1]$ . The last statement on lct again follows from (4.3).

We next consider the case that a is zero. Then the curve D is defined by

$$ \begin{align*} z^2y + a_0x^{2n+6} + a_1x^{2n+4}y + \cdots + a_{t}x^{2n+6-2t}y^{t} = 0. \end{align*} $$

Under this condition we first consider the case that $a_0$ is not zero. Then D is irreducible. We have $H_y\cap D = \{\mathsf {p}_z\}$ . Since the equation of the curve D has the monomial $z^2y$ , the log pair $(W, D)$ is log canonical at the point $\mathsf {p}_z$ . On the affine open set $U_y$ the curve D is defined by

$$ \begin{align*} z^2 + a_0x^{2n+6-2t}(x^2 - b_1)^{m_1}\cdots (x^2 - b_k)^{m_k} = 0 \end{align*} $$

where $b_1,\ldots , b_k$ are distinct nonzero constants. We set $m' = \max \{m_1,\ldots , m_k\}$ . Then we have

$$ \begin{align*} \operatorname{lct}(W, D) = \begin{cases} \min\left\{\frac{m' + 2}{2m'}, 1\right\} &\text{ for } t\geq n+2\\ \min\left\{\frac{n+4-t}{2n+6-2t}, \frac{m' + 2}{2m'}, 1\right\} &\text{ for }t<n+2 \end{cases} \end{align*} $$

Finally we consider the case $a = a_0 = 0$ . Then there is a nonzero nonnegative integer $m_0$ such that $a_{m_0+1}\neq 0$ and $a_0=\ldots =a_{m_0}=0$ . The curve D is defined by

$$ \begin{align*} &y(z^2 + x^{2n+6-2t}y^{m_0}(a_{m_0+1}x^{2t-2m_0-2}+a_{m_0+2}x^{2t-2m_0-4}y+\cdots + a_ty^{t-m_0})) \\ =&y(z^2 + a_t x^{2n+6-2t}y^{m_0}(y - b_1x^2)^{m_1}\cdots (y - b_kx^2)^{m_k} = 0. \end{align*} $$

Since the equation of D is divided by y the curve D is reducible. We set $m" = \max \{m_1,\ldots , m_k\}$ . Then we have

$$ \begin{align*} \operatorname{lct}(W, D) = \begin{cases} \min\left\{\frac{m_0 + 2}{2m_0+2}, \frac{m" + 2}{2m"}, 1\right\} &\text{ for } t\geq n+2\\ \min\left\{\frac{m_0 + 2}{2m_0+2}, \frac{n+4-t}{2n+6-2t}, \frac{m" + 2}{2m"}, 1\right\} &\text{ for } t<n+2 \end{cases} \end{align*} $$

Then we have the following lemma:

4.2 Even cases

Let $W' = {\mathbb {P}}(1,1,l+1)_{u,y,z}$ be the weighted projective plane where l is a positive integer. And Let H and D be the effective divisors on $W'$ defined by $u=0$ and $z^2y + azu^{l+2} + a_0u^{2l+3} + a_1u^{2l+2}y + \cdots + a_{2l+3}y^{2l+3} = 0$ , respectively, where a is a constant and $a_0,\ldots , a_{2l+3}$ are not all zero constants. We set $t = \max \{i\in {\mathbb {Z}}\mid a_i \neq 0\}$ . Then the quasi-homogeneous polynomial of the curve D is represented by

(4.5) $$ \begin{align} z^2y + azu^{l+2} + a_0u^{2l+3} + a_1u^{2l+2}y + \cdots + a_tu^{2l+3-t}y^t = 0. \end{align} $$

In this subsection we calculate the log canonical threshold of the divisor D with respect to the log pair $(W', \frac {1}{2}H)$ . We first consider the locus $H\cap D$ which is defined by $u = z^2y + a_{2l+3}y^{2l+3} = 0$ on $W'$ . The point $\mathsf {p}_z$ is contained in $H\cap D$ . Since the equation of the effective divisor $H+D$ has the monomial $uz^2y$ , the log pair $(W', H+D)$ is log canonical at $\mathsf {p}_z$ . It implies that the log pair $(W', \frac {1}{2}H+D)$ is log canonical at $\mathsf {p}_z$ . We next consider two cases whether $a_{2l+3}$ is zero or not. If $a_{2l+3}$ is not zero then we can assume that $a_{2l+3} = 1$ . It implies that $H\cap D = \{\mathsf {p}_z, \mathsf {p}_1, \mathsf {p}_2\}$ where $\mathsf {p}_1 = [0:1:i]$ and $\mathsf {p}_2 = [0:1:-i]$ . Since the intersection number $H\cdot D$ is $\frac {2l+3}{l+1}$ , we have the local intersection numbers $(H\cdot D)_{\mathsf {p}_1} = (H\cdot D)_{\mathsf {p}_2} = 1$ and $(H\cdot D)_{\mathsf {p}_z} = \frac {1}{l+1}$ . These imply that H and D intersect transversely. Thus the log pair $(W', \frac {1}{2}H+D)$ is log canonical along $H\cap D$ . If $a_{2l+3}$ is zero then $H\cap D = \{\mathsf {p}_y, \mathsf {p}_z\}$ . On the affine open set $U_y$ defined by $y=1$ the equation (4.5) is represented by

$$ \begin{align*} \left(z + \frac{a}{2}u^{l+2}\right)^2 + u^{2l+3-t}\left(-\frac{a^2}{4}u^{1+t} + a_0u^t + a_1u^{t-1} + \cdots + a_t\right) = 0. \end{align*} $$

We use the local coordinates $(u',z')=(u, z+ \frac {a}{2}u^{l+2})$ at $\mathsf {p}_y$ . Let $\phi \colon Y\to W'$ be the weighted blow-up at $\mathsf {p}_y$ with weights $\operatorname {wt}(u') = 2$ and $\operatorname {wt}(z') = 2l+3-t$ . Then we have the equation

$$ \begin{align*} K_Y + \frac{1}{2}\widetilde{H} + c\widetilde{D} \equiv \phi^*\left(K_W + \frac{1}{2}H + cD\right) + (2l+4-t - 1 - c(4l+6-2t))E, \end{align*} $$

where $\widetilde {H}$ and $\widetilde {D}$ are the strict transform of H and D, respectively, E is the $\phi $ -exceptional and c is a positive rational number. From this equation we have the following lemma:

Finding the log canonical threshold $\operatorname {lct}_{\mathsf {p}}(W', \frac {1}{2}H; D)$ where $\mathsf {p}\in W'\setminus H$ is similar to the odd cases. Since $\operatorname {lct}_{\mathsf {p}}(W', D) = \operatorname {lct}_{\mathsf {p}}(W', \frac {1}{2}H; D)$ we calculate the log canonical threshold $\operatorname {lct}_{\mathsf {p}}(W', D)$ instead of $\operatorname {lct}_{\mathsf {p}}(W', \frac {1}{2}H; D)$ . We first consider the case that a is not zero. We have $H_y\cap D = \{\mathsf {p}_z, [1:0:-\frac {a_0}{a}]\}$ . Since the curve in $\mathbb {C}^2$ defined by the local equation of D on the affine open set $U_z$ of $W'$ defined by $z=1$ is smooth at the origin and there is an orbifold chart $\mathbb {C}^2\to U_z$ on $W'$ , the log pair $(W', D)$ is log canonical at the point $\mathsf {p}_z$ . Similarly, the curve in $\mathbb {C}^2$ defined by the local equation of D on the affine open set $U_u$ of $W'$ defined by $u=1$ is smooth at the point $(y,z) = (0, -a_0/a)$ , the log pair $(W', D)$ is log canonical at the corresponding point $[1:0:-\frac {a_0}{a}]$ . Therefore, $\operatorname {lct}_{\mathsf {p}}(W', \frac {1}{2}H; D) = 1$ where $\mathsf {p}\in H_y\setminus H$ . On the affine open set $U_y$ defined by $y=1$ the equation (4.5) of D is represented by

$$ \begin{align*} \left(z + \frac{a}{2}u^{l+2}\right)^2 - \frac{a^2}{4}u^{2l+3-t}(u - b_1)^{m_1}\cdots (u - b_k)^{m_k} = 0 \end{align*} $$

where $b_1\ldots , b_k$ are nonzero constants. We set

(4.6) $$ \begin{align} m = \max\{m_1,\ldots , m_k\}\leq 2l+4. \end{align} $$

We can obtain the following:

$$ \begin{align*} \inf\left\{\operatorname{lct}_{\mathsf {p}}(W', D) ~\middle|~ \mathsf {p}\in W'\setminus H\right\} = \min\left\{\frac{m+2}{2m}, 1\right\} \end{align*} $$

Definition 4.6. Let be the curve in $W'$ defined by

(4.7)

Then in the affine chart $U_y = \{y=1\}$ the curve can be expressed as

$$ \begin{align*}\left(z-u^{l+2}\right)^2 - (u -1)^{2l+4} =0.\end{align*} $$

If $a=0$ and $a_0\neq 0$ then the equation (4.5) is presented by

$$ \begin{align*} z^2 + a_tu^{2l+3-t}(u - b_1)^{m_1}\cdots (u - b_k)^{m_k} = 0 \end{align*} $$

on the affine open set $U_y$ , where $b_1,\ldots , b_k$ are nonzero constants. We have the following:

$$ \begin{align*} \inf\left\{\operatorname{lct}_{\mathsf {p}}(W', D) ~\middle|~ \mathsf {p}\in W'\setminus H\right\} = \min\left\{\frac{m'+2}{2m'}, 1\right\}\geq \frac{2l+5}{4l+6} \end{align*} $$

where $m' = \max \{m_1,\ldots , m_k\}\leq 2l+3$ .

Finally we consider the case $a=a_0=0$ . Then there is a nonzero nonnegative integer $m_0$ such that $a_{m_0+1}\neq 0$ and $a_0=\ldots =a_{m_0}=0$ . The curve D is defined by

$$ \begin{align*} &y(z^2 + u^{2l+3-t}y^{m_0}(a_{m_0+1}u^{t-m_0-1}+a_{m_0+2}x^{t-m_0-2}y+\cdots + a_ty^{t-m_0})) \\ =&y(z^2 + a_t u^{2l+3-t}y^{m_0}(y + b_1u)^{m_1}\cdots (y + b_ku)^{m_k}) = 0, \end{align*} $$

where $b_1,\ldots , b_k$ are nonzero constants. We have the following:

$$ \begin{align*} \inf\left\{\operatorname{lct}_{\mathsf {p}}(W', D) ~\middle|~ \mathsf {p}\in W'\setminus H\right\} = \min\left\{\frac{m_0+2}{2m_0 + 2}, \frac{m"+2}{2m"}, 1\right\}, \end{align*} $$

where $m" = \max \{m_1,\ldots , m_k\}\leq 2l+3$ . Then we have the following lemma: