Proof. By definition, $\mathsf {p} \in X$ is a hypersurface singularity, therefore without loss of generality we assume that $X$ is given by $g=0$ in $\mathbb {A}^4$ and $\mathsf {p} = (0,0,0,0)$. Since general hyperplane section of $X$ has the $A_1$-singularity the general form of $g$ is

\[ x^2 + y^2 + z^2 + t h(x,y,z,t) = 0. \]

By the splitting lemma [Reference Greuel, Lossen and ShustinGLS07, Theorem I.2.47] we may change $g$ into

\[ x^2 + y^2 + z^2 + t^k = 0. \]

Proof. Let $i \colon X \to W$ and $\tilde {i} \colon \tilde {X} \to \tilde {W}$ be the embeddings agreeing with the maps $\varphi$ and $\varphi _W$. Let $e \in A_1 (E_W)$ be the class of a line. Then in $A_1 (\tilde {W})$, we have

(3.1)\begin{equation} \varphi_W^* i_* \Gamma \equiv \varphi_*^{-1} i_* \Gamma + m e. \end{equation}

Since $D_i$ is a Cartier divisor on $X$, it is defined by a single element in the residue ring $\mathcal {O}_{X, \mathsf {p}} = \mathcal {O}_{W, \mathsf {p}}/(x y + z^2 + w^M)$, hence we can take a Cartier divisor $G_i$ on $W$ such that $G_i|_X = D_i$. It follows that

\begin{align*} \varphi_W^* i_* \Gamma &= \varphi_W^* i_* (D_1 \cdot D_2) \\ &\equiv \varphi_W^* (G_1 \cdot G_2 \cdot X) \\ &\equiv \varphi^*_W G_1 \cdot \varphi_W^*G_2 \cdot \varphi_W^* X \\ &\equiv \varphi^*_W G_1 \cdot \varphi_W^*G_2 \cdot \tilde{X} \\ &\equiv \tilde{i}_* \bigl( \varphi^*D_1 \cdot \varphi^*D_2 \bigr), \end{align*}

and thus

(3.2)\begin{align} \tilde{i}_* (\varphi_*^{-1} D_1 \cdot \varphi_*^{-1} D_2) &\equiv \tilde{i}_* \bigl( (\varphi^* D_1 - \nu(\mathcal{M}) E) \cdot (\varphi^* D_2 - \nu(\mathcal{M}) E) \bigr)\nonumber\\ &\equiv \tilde{i}_* \bigl( \varphi^*D_1 \cdot \varphi^* D_2 \bigr) + \tilde{i}_* (\nu(\mathcal{M})^2 E^2 )\nonumber\\ &\equiv \varphi_W^* i_* \Gamma - 2 \nu(\mathcal{M})^2 e, \end{align}

where the last equivalence follows from (3.1). By (3.2), the $1$-cycle $Z$ is rationally equivalent to $(m-2 \nu (\mathcal {M})^2) e$ and the claim follows.

Proof. We prove the claims by induction. The statements clearly hold for $i = 1$, now suppose the statements hold for all $k < i$.

First, we define values $a_i = a(E_i,X_{i-1})$ for all $i$, clearly

\[ a_i = \begin{cases} 1, & i\le K,\\ 2, & K < i \le L,\\ 1, & L \le N. \end{cases} \]

By assumption of induction we have

\[ K_{X^{(i-1)}} \sim \varphi_{i-1,0}^* (K_{X}\!) + \sum_{k=1}^{i-1} \biggl( \sum_{j=1}^k a_j P_{k,j} \biggr) E_k^{(i-1)}. \]

Next we compute $K_{X^{(i)}}$:

\begin{align*} K_{X^{(i)}} &\sim \varphi_i^* (K_{X^{(i-1)}}) + a_i E_i \sim \varphi_{i,0}^* (K_{X}\!) + \sum_{k=1}^{i-1} \biggl( \sum_{j=1}^k a_j P_{k,j} \biggr) \varphi_i^* E_k^{(i-1)} + a_i E_i\\ & \sim \varphi_{i,0}^* (K_{X}\!) + \sum_{k=1}^{i-1} \biggl( \sum_{j=1}^k a_j P_{k,j} \biggr) E_k^{(i)} + \Biggl( a_i P_{i,i} + \sum_{k \mid i\to k} \biggl( \sum_{j=1}^k a_j P_{k,j} \biggr) \Biggr)E_i. \end{align*}

It remains to compute the coefficient at $E_i$,

\begin{align*} a(E_i,X) &= a_i P_{i,i} + \sum_{k\mid i\to k} \sum_{j=1}^k a_j P_{k,j} = a_i P_{i,i} + \sum_{k\mid i\to k} \sum_{j=1}^{i-1} a_j P_{k,j}\\ &= a_i P_{i,i} + \sum_{j=1}^{i-1} a_j \sum_{k\mid i\to k} P_{k,j} = \sum_{j=1}^i a_j P_{i,j}, \end{align*}

where the second equality follows from $P_{k,j} = 0$ for $j > k$ and the last equality follows from

\[ P_{i,j} = \sum_{k\mid i\to k} P_{k,j}. \]

This proves the first assertion and its special case. The proof of the second assertion is analogous.

Proof. Set $d_i = \deg _{E_i} Z_i$. By Lemmas 3.2 and 3.6, we obtain a system of equalities corresponding to the sequence of cycles (3.4):

(3.6)\begin{equation} \begin{aligned} 2 \nu_1^2 + d_1 & = m_{0,1}, \\ 2 \nu_2^2 + d_2 & = m_{0, 2} + m_{1, 2}, \\ & \cdots, \\ 2 \nu_K^2 + d_K & = m_{0, K} + \cdots + m_{K-1. K}, \\ \nu_{K+1}^2 + d_{K+1} & = m_{0, K+1} + \cdots + m_{K, K+1}, \\ & \cdots, \\ \nu_L^2 + d_L & = m_{0, L} + \cdots + m_{L-1, L}. \end{aligned} \end{equation}

We multiply the $i$th equality in (3.6) by $p_i$ and sum them up:

(3.7)\begin{equation} 2 \sum_{i=1}^K p_i \nu_i^2 + \sum_{i=K+1}^L p_i \nu_i^2 + \sum_{i=1}^L p_i d_i = \sum_{j=1}^L p_j m_{0, j} + \sum_{i = 1}^{L-1} \sum_{j = i+1}^L p_j m_{i, j}. \end{equation}

For each $i$ with $0 \le i \le L-1$, we consider the expression

\[ \sum_{j = i+1}^L p_j m_{i, j} \]

which appears in the right-hand side of the summation (3.7). We have $m_{i, j} \le d_i$ for any $i, j$ with $L \ge j > i \ge 1$ by [Reference PukhlikovPuk13, Lemma 2.4].Footnote ¹ Note that, for $i \ge 1$, we have $p_i \ge \sum _{j \to i} p_j$, and if $m_{i, j} \neq 0$, then $j \to i$. It follows that, for $i = 1, \ldots, L-1$, we have

\[ \sum_{j = i + 1}^L p_j m_{i, j} = \sum_{\substack{i + 1 \le j \le L \\ m_{i,j} \ne 0}} p_j m_{i,j} \le d_i \sum_{\substack{i + 1 \le j \le L \\ j \to i}} p_j \le p_i d_i. \]

Thus, for $i = 1, \ldots, L-1$, we can remove all $m_{i, *}$ and $p_i d_i$ from the summation (3.7) if we replace $=$ with $\le$, and obtain

(3.8)\begin{equation} 2 \sum_{i=1}^K p_i \nu_i^2 + \sum_{i = K+1}^L p_i \nu_i^2 + p_L d_L \le \sum_{i=1}^L p_i m_{0,i}. \end{equation}

It remains to give a lower bound for $d_L$ in terms of $\nu _i$. First, we decompose

\[ Z_L = \Gamma_L + \alpha_L B_L, \]

where $\operatorname {Supp} \Gamma _L$ does not contain $B_L$ and $\alpha _L \ge 0$. Thus, we may decompose

\[ D_1^{(L+1)}\cdot D_2^{(L+1)} = Z_0^{(L+1)} + \ldots + Z_{L-1}^{(L+1)} + \Gamma_L^{(L+1)} + Z_{L+1}, \]

where $\operatorname {Supp} Z_{L+1} \subset E_{L+1}$. We apply Lemma 3.2(2) to $\varphi _{L+1}$:

\[ D_1^{(L)}\cdot D_2^{(L)} = (\varphi_{L+1})_* \bigl( Z_1^{(N)} + \cdots + Z_{L-1} + \Gamma_L^{(L+1)} \bigr) + \bigl( \nu_{L+1}^2 + \deg_{E_{L+1}} Z_{L+1} \bigr) B_L. \]

Thus, we have $\alpha _L = \nu _{L+1}^2 + \deg _{E_{L+1}} Z_{L+1}$ and

\[ d_L = \deg_{E_L} \bigl(\Gamma_L + (\nu_{L+1}^2 + \deg_{E_{L+1}} Z_{L+1}) B_L \bigr) \ge \bigl( \nu_{L+1}^2 + \deg_{E_{L+1}} Z_{L+1} \bigr) \deg_{E_L} B_L. \]

Iterating the application of Lemma 3.2(2) for $\varphi _i$ with $i = L + 2, \ldots, N$, we obtain

\[ d_L \ge \sum_{i = L + 1}^N \nu_i^2 \deg_{E_L} (\varphi_{i-1, L})_* B_{i-1} \ge \sum_{i= L + 1}^N \nu_i^2. \]

Combining this inequality with (3.8), we obtain the inequality (3.5).

Proof. The minimum of the left-hand side of (3.5), viewed as a quadratic form in $\nu _i$, under the constraint

\[ \sum_{i=1}^N p_i \nu_i = \nu (\mathcal{M}), \]

is attained at

\[ 2 \nu_1 = \cdots = 2 \nu_K = \nu_{K+1} = \cdots = \nu_N = \frac{2 \nu (\mathcal{M})}{\sum_{i=1}^K p_i + 2 \sum_{i = K+1}^L p_i}. \]

Calculating the minimum, we get the required inequality.

Proof. By the assumption, there exists a divisorial valuation $\nu$ of $\mathbb {C} (X)$ centered at $\mathsf {p}$ such that $(X, ({1}/{n}) \mathcal {M})$ is not canonical at $\nu$, that is, the inequality $\nu (\mathcal {M}) > n a(\nu,X)$ holds for $\nu$. Thus, by Lemma 3.7 we have

(3.10)\begin{equation} \nu (\mathcal{M}) > n (\Sigma_0 + 2 \Sigma_1 + \Sigma_2). \end{equation}

Note that

(3.11)\begin{equation} \operatorname{mult}_{\mathsf{p}} (D_1 \cdot D_2) = m_{0,1} \ge m_{0, i} \end{equation}

for any $1 \le i \le L$. By (3.10), (3.11) and Corollary 3.13, we obtain

(3.12)\begin{equation} \operatorname{mult}_{\mathsf{p}} (D_1 \cdot D_2) > 2 n^2 \frac{(\Sigma_0 + 2 \Sigma_1 + \Sigma_2)^2}{(\Sigma_0 + \Sigma_1)(\Sigma_0 + 2 \Sigma_1 + 2 \Sigma_2)}. \end{equation}

It is then easy to get the required inequality.

Proof. Consider a fan corresponding to $\mathbb {A}^4$ with the following description

\begin{align*} v_x &= (1,0,0,0),\\ v_y &= (0,1,0,0),\\ v_z &= (0,0,1,0),\\ v_w &= (0,0,0,1). \end{align*}

Then the toric morphism corresponding to adding the ray $\rho = (s, 2t-s, t, 1)$ to the fan is the $(s, 2t-s, t, 1)$-weighted blow-up of $\mathbb {A}^4$. We can also add the ray $\rho$ by performing a sequence of regular blow-ups at points, curves, and surfaces corresponding to adding the following rays to the fan:

\begin{align*} u_1 &= (1,1,1,1),\\ u_2 &= (2,2,2,1),\\ &\cdots\\ u_s &= (s,s,s,1),\\ u_{s+1} &= (s,s+1,s+1,1),\\ &\cdots \\ u_{t} &= (s,t,t,1),\\ u_{t+1} &= (s,t+1,t,1),\\ &\cdots\\ u_{2t-s} &= (s,2t-s,t,1).\\ \end{align*}

From this description we can see that the first $s$ morphisms are the blow-ups at a points, first at $(x=y=z=w=0)$, then at $(x=y=z=u_{i}=0)$, the next $t-s$ morphisms are blow-ups at curves $(y=z=u_i=0)$, and the last $t-s$ morphisms are blow-ups at surface $(y=u_i=0)$. The restriction of the morphisms to the proper transforms $X_i$ of $X$ gives us the $s$ blow-ups at singular points of $X$ followed by $t-s$ blow-ups at a curve and $t-s$ blow-ups at a divisor as can be seen from the local equations of $X_i$:

\begin{align*} X_0:& \quad xy + z^2 + w^M = 0,\\ X_1:& \quad xy + z^2 + u_1^{M-2} w^M = 0,\\ &\cdots \\ X_s:& \quad xy + z^2 + u_s^{M-2s} w^M = 0,\\ X_{s+1}:& \quad xy + u_{s+1} z^2 + u_{s+1}^{M-2s-1} w^M = 0,\\ & \cdots\\ X_{t}:& \quad xy + u_{t}^{t-s} z^2 + u_{t}^{M-s-t} w^M= 0,\\ X_{t+1}:& \quad xy + u_{t+1}^{t-s-1} z^2 + u_{t+1}^{M-s-t-1} w^M =2,\\ & \cdots \\ X_{2t-s}:& \quad xy + z^2 + u_{2t-s}^{M-2t} w^M = 0. \end{align*}

We can see from the toric description that for every blow-up we have $B_i \subset E_i\setminus E_{i-1}^{(i)}$, so we can drop $u_j$ in the local equation of $X_i$ for $j< i$. This fact also implies that the graph corresponding to the tower of blow-ups is a simple chain and we have $p_i = 1$ for all $i$. Thus, we have $N = t$, $\Sigma _0 = K = L = s$, $\Sigma _1 = 0$, and $\Sigma _2 = t - s$. The required inequality is obtained by substituting these values into the inequality (3.12).

Proof. We first note that by Lemma 3.7

\[ a (E_{\infty}, X) = \Sigma_0 + 2 \Sigma_1 + \Sigma_2, \]

where we recall that $E_{\infty } = E_N$ is the prime exceptional divisor realizing $\nu$. By the assumption that the pair (3.13) is not canonical at $\nu$, we have

(3.14)\begin{equation} \nu (\mathcal{M}) > n \biggl( \Sigma_0 + 2 \Sigma_1 + \Sigma_2 + \sum_{j = 1}^k \gamma_j \nu (F_j) \biggr). \end{equation}

For $j = 1, \ldots, l$, we set

\[ K_j = \begin{cases} 0, & \text{if }\mathsf{p} \in F_j, \\ \max \bigl\{i \le L \mid B_{i-1} \notin F_j^{(i-1)} \bigr\}, & \text{otherwise}, \end{cases} \]

and then set

\[ \Xi_j = \sum_{i=1}^{K_j} p_i. \]

Then by Lemma 3.7 we have

\[ \nu (F_j) = \sum_{i = 1}^{K_j} p_i \operatorname{mult}_{B_{i-1}} F_j^{(i-1)} \ge \sum_{i = 1}^{K_j} p_i = \Xi_j. \]

Observe that

\[ m_{i,0} = \operatorname{mult}_{B_{i-1}} Z_0^{(i-1)} = \operatorname{mult}_{B_{i-1}} Z_h^{(i-1)} + \sum_{j=1}^l \operatorname{mult}_{B_{i-1}} Z_j^{(i-1)}, \]

and that if $i > K_j$, then $\operatorname {mult}_{B_{i-1}} Z_j^{(i-1)} = 0$. This decomposition implies that

(3.15)\begin{align} \sum_{i = 1}^L p_i m_{0, i} &= \sum_{i = 1}^L p_i \operatorname{mult}_{B_{i-1}} (Z_h^{(i-1)}) + \sum_{j = 1}^l \sum_{i=1}^{K_j} p_i \operatorname{mult}_{B_{i-1}} (Z_j^{(i-1)})\nonumber\\ &\le (\Sigma_0 + \Sigma_1) \operatorname{mult}_{\mathsf{p}} Z_h + \sum_{j=1}^l \Xi_j \operatorname{mult}_{\mathsf{p}} Z_j, \end{align}

since $\operatorname {mult}_{B_{i-1}} (Z_h^{(i-1)}) \le \operatorname {mult}_{\mathsf {p}} Z_h$ and $\operatorname {mult}_{B_{i-1}} (Z_j^{(i-1)} )\le \operatorname {mult}_{\mathsf {p}} Z_j$ for every $1 \le i \le L$.

By the inequality in Corollary 3.13 combined with two inequalities (3.14) and (3.15), we get the inequality

(3.16)\begin{align} & (\Sigma_0 + \Sigma_1) \operatorname{mult}_{\mathsf{p}} Z_h + \sum_{j = 1}^l \Xi_j \operatorname{mult}_{\mathsf{p}} Z_j \nonumber\\ &\qquad \ge \sum_{i=1}^L p_i m_{0,i} \nonumber\\ &\qquad > 2 n^2 \Sigma_0 + 4 n^2 \Sigma_1 + 4 n^2 \sum_{j = 1}^l \gamma_j \nu (F_j) + 2 n^2 \frac{(\Sigma_2 - \sum_{j=1}^l \gamma_j \nu (F_j))^2}{\Sigma_0 + 2 \Sigma_1 + 2 \Sigma_2}\nonumber\\ &\qquad \ge 2 n^2 (\Sigma_0 + \Sigma_1) + 4 n^2 \sum_{j=1}^l \gamma_j \Xi_j, \end{align}

where the last inequality follows from the inequality $\nu (F_j) \ge \Xi _j$.

Dividing (3.16) by $\Sigma _0 + \Sigma _1$ and setting

\[ t_j = \frac{\Xi_j}{\Sigma_0 + \Sigma_1}, \]

we get the required inequality.

Proof. Let $\Gamma$ be a curve on $X$. Suppose that $\Gamma$ is a maximal center. Then $(- K_X \cdot \Gamma ) < (-K_X)^3 = 2$ by Lemma 2.7. It follows that $\Gamma$ is a curve of degree $1$ on $X$. By the assumption, there exists a movable linear system $\mathcal {M} \sim - n K_X$ on $X$ such that the pair $(X, ({1}/{n}) \mathcal {M})$ is not canonical along $\Gamma$. In particular, we have $\operatorname {mult}_{\Gamma } \mathcal {M} > n$.

In the following, we merely repeat the arguments in the proof of [Reference Cheltsov and ParkCP10, Lemma 3.8] for readers’ convenience. Suppose that $\pi (\Gamma )$ is contained in the sextic surface $B \subset \mathbb {P}^3$. We choose and fix a smooth point $\mathsf {p} \in \Gamma$ which is contained in the smooth locus of $X$ and which is not contained in any curve in $\operatorname {Bs} \mathcal {M}$ other than $\Gamma$. Let $L \subset \mathbb {P}^3$ be a general line tangent to $B$ at $\pi (\mathsf {p})$ and let $\Delta$ be the inverse image of $L$ by $\pi$. We see that $\Delta$ is a curve of degree $2$ which is singular at $\mathsf {p}$. Moreover, by our choice of $\mathsf {p}$, $\Delta$ is not contained in the base locus of $\mathcal {M}$. Then we obtain

\[ 2 n = (\mathcal{M} \cdot \Delta) \ge \operatorname{mult}_{\mathsf{p}} \mathcal{M} \operatorname{mult}_{\mathsf{p}} \Delta \ge \operatorname{mult}_{\Gamma} \mathcal{M} \operatorname{mult}_{\mathsf{p}} \Delta > 2 n. \]

This is a contradiction.

Proof. We choose homogeneous coordinates of $\mathbb {P} := \mathbb {P} (1, 1, 1, 1, 3)$ as above, that is, $\mathsf {p} = (1:0:0:0:0)$, $X$ is defined by the (4.3), and $\Gamma$ is as given in (4.4). In the following, we assume that a general member $S \in \mathcal {P}$ is cut out by the equation $t = \theta z$ for a general $\theta \in \mathbb {C}$. We set $U_x := \mathbb {P} (1, 1, 1, 1, 3) \setminus (x = 0)$ and, by a slight abuse of notation, we identify $U_x$ with the affine space $\mathbb {A}^4$ with affine coordinates $y, z, t, w$.

We first prove assertion (1). Suppose $\operatorname {mult}_{\mathsf {p}} \Gamma \cap \Delta = 1$, that is, $\lambda \ne 0$. In view of the (4.5), we can write

\[ f_2 = \lambda^2 y^2 + y \ell (z, t) + q (z, t), \]

where $\ell (z, t)$ and $q (z, t)$ are linear and quadratic forms, respectively. Note that at least one of $\ell (z, t)$ and $q (z, t)$ is non-zero since $\mathsf {p} \in X$ is of type $cA_1$. By eliminating the variable $t = \theta z$, $S$ is defined in $\mathbb {P} (1, 1, 1, 1, 3)$ by the equation

\[ - w^2 + x^4 (\lambda^2 y^2 + \zeta y z + \eta z^2) + x^3 f_3 (y, z, \theta z) + \cdots = 0, \]

where $\zeta, \eta \in \mathbb {C}$ satisfy $(\zeta, \eta ) \ne (0,0)$. By setting $x = 1$, $S \cap U_x$ is isomorphic to the hypersurface in $\mathbb {A}^3_{y, z, w}$ defined by

\[ - w^2 + \lambda^2 y^2 + \zeta y z + \eta z^2 + h_{\ge 3} (y, z) = 0, \]

where $h_{\ge 3} (y, z)$ is a polynomial of order at least $3$. The point $\mathsf {p} = (1:0:0:0:0)$ corresponds to the origin and the curve $\Gamma$ is defined in $\mathbb {A}^3$ by the equations

(4.6)\begin{equation} z = w - \lambda y - \mu y^2 - \nu y^3 = 0. \end{equation}

It is easy to see that $\mathsf {p} \in S$ is of type $A_1$ and the proper transform via the blow-up of $\mathsf {p} \in S$ of $\Gamma$ intersects the exceptional divisor transversally at one point. It follows that $G (S, \mathsf {p}, \Gamma )$ is of type $A_{1, 1}$.

We prove assertion (2). Suppose $\operatorname {mult}_{\mathsf {p}} \Gamma \cap \Delta = 2$, that is, $\lambda = 0$ and $\mu \ne 0$. We can write

\[ f_2 = y \ell (z, t) + q (z, t), \]

where $\ell (z, t)$ and $q (z, t)$ are linear and quadratic forms, respectively. Let $\zeta, \eta \in \mathbb {C}$ be such that $\ell (z, \theta z) = \zeta z$ and $q (z, \theta z) = \eta z^2$. Then, by eliminating $t$ and setting $x = 1$, $S \cap U_x$ is isomorphic to the hypersurface in $\mathbb {A}^3_{y, z, w}$ defined by the equation

(4.7)\begin{equation} {-}w^2 + \zeta y z + \eta z^2 + z (h_2 (y, z) + h_3 (y, z)) +\mu^2 y^4 + h_{\ge 5} (y, z) = 0, \end{equation}

where $h_2, h_3 \in \mathbb {C} [y, z]$ are homogeneous polynomials of degree $2, 3$, respectively, and $h_{\ge 5} \in \mathbb {C} [y, z]$ is a polynomial of order at least $5$. The curve $\Gamma$ is defined in $\mathbb {A}^3$ by the equations

(4.8)\begin{equation} z = w - \mu y^2 - \nu y^3 = 0. \end{equation}

Suppose that $\ell (z, t) \ne 0$. Then $\zeta \ne 0$, and it is easy to see that $\mathsf {p} \in S$ is of type $A_1$ and $G (S, \mathsf {p}, \Gamma )$ is of type $A_{1, 1}$.

Suppose that $\ell (z, t) = 0$. Then $\zeta = 0$ and $\eta \ne 0$. The exceptional divisor $E$ of the blow-up $\tilde {S} \to S$ of $S$ at $\mathsf {p}$ is isomorphic to the hypersurface

\[ (- w^2 + \eta z^2 = 0) \subset \tilde{\mathbb{P}}^2 := \mathbb{P}^2_{y, z, w}, \]

where the polynomial $-w^2 + \eta z^2$ is the least degree part of (4.7). It follows that $E = E_+ + E_-$, where $E_{\pm } = (w \pm \sqrt {\eta } z = 0)$ are smooth rational curves which meet only at $\mathsf {q} := (1:0:0) \in \tilde {\mathbb {P}}^2$. By (4.8), the proper transform $\tilde {\Gamma } \subset \tilde {S}$ of $\Gamma$ intersect $E$ only at $\mathsf {q}$. The point $\tilde {\mathsf {p}}$ is in the $y$-chart, denoted by $\tilde {S}_y$, of the blow-up $\tilde {S} \to S$, and $\tilde {S}_y$ is the hypersurface in $\tilde {\mathbb {A}}^3 := \mathbb {A}^3_{y, z, w}$ defined by

\[ - w^2 + \eta z^2 + z y (h_2 (1, z) + y h_3 (1, z)) + \mu^2 y^2 + h_{\ge 5} (y, y, z)/y^2 = 0. \]

Note that $\tilde {\mathsf {p}}$ corresponds to the origin, and it is an ordinary double point of $\tilde {S}$. Moreover, on $\tilde {S}_y$, we have

\begin{align*} \tilde{\Gamma} &= (z = w - \mu y - \nu y^2 = 0) \subset \tilde{\mathbb{A}}^3, \\ E_{\pm} &= (y = w - \sqrt{\eta} z = 0) \subset \tilde{\mathbb{A}}^3. \end{align*}

Let $\hat {S} \to \tilde {S}$ be the blow-up of $\tilde {S}$ at $\tilde {\mathsf {p}}$ with exceptional curve $F$. Then $\hat {S}$ is smooth along $F$,

\[ F \cong (-w^2 + \eta z^2 + \mu^2 y^2 = 0) \subset \hat{\mathbb{P}}^2 = \mathbb{P}^2_{y, z, w} \]

is a smooth rational curve, and $\hat {\Gamma } \cap F = \{(1:0:\mu )\}$, $F \cap \hat {E}_{\pm } = \{(0:1: \pm \sqrt {\eta })\}$, $\hat {\Gamma } \cap \hat {E}_{\pm } = \emptyset$. This shows that $G (S, \mathsf {p}, \Gamma )$ is of type $A_{3, 2}$.

We prove assertion (3). Suppose $\operatorname {mult}_{\mathsf {p}} \Gamma \cap \Delta = 3$, that is, $\lambda = \mu = 0$ and $\nu \ne 0$. We can write

\[ f_2 = y \ell (z, t) + q (z, t) \]

as before. If $\ell (z, t) \ne 0$, then $G (S, \mathsf {p}, \Gamma )$ is of type $A_{1, 1}$ as in the previous case. We assume $\ell (z, t) = 0$. Then $q (z, t) \ne 0$, and by eliminating $t$ and setting $x = 1$, $S \cap U_x$ is isomorphic to the hypersurface in $\mathbb {A}^3_{y, z, w}$ defined by the equation

\[ -w^2 + \eta z^2 + z h (y, z) + \nu^2 y^6 = 0, \]

where $\eta \in \mathbb {C}$ is non-zero and $h (y, z)$ is a polynomial consisting of monomials of degree $2, 3, 4, 5$. The curve $\Gamma$ is defined in $\mathbb {A}^3$ by the equations

(4.9)\begin{equation} z = w - \nu y^3 = 0. \end{equation}

We write

\[ h (y, 0) = \alpha_2 y^2 + \alpha_3 y^3 + \alpha_4 y^4 + \alpha_5 y^5, \]

where $\alpha _i \in \mathbb {C}$. Let $\tilde {S} \to S$ be the blow-up of $S$ at $\mathsf {p}$ with exceptional divisor $E$. We see that $E = E_+ + E_-$, where

\[ E_{\pm} := (w \pm \sqrt{\eta} z = 0) \subset \tilde{\mathbb{P}}^2 = \mathbb{P}^2_{y, z, w}, \]

are smooth rational curves which meet only at $\tilde {\mathsf {p}} := (1:0:0) \in \tilde {\mathbb {P}}^2$. Moreover the proper transform $\tilde {\Gamma } \subset \tilde {S}$ of $\Gamma$ meet $E_{\pm }$ only at $\tilde {\mathsf {p}}$. The $y$-chart $\tilde {S}_y$ of the blow-up $\tilde {S} \to S$ is the hypersurface in $\tilde {\mathbb {A}}^3 := \mathbb {A}^3_{y, z, w}$ defined by

\[ - w^2 + \eta z^2 + z y (\alpha_2 + \alpha_3 y + \alpha_4 y^2 + \alpha_5 y^3 + \cdots) + \nu^2 y^4 = 0, \]

where the omitted terms consist of monomials divisible by $z$. The point $\tilde {\mathsf {p}}$ corresponds to the origin of $\tilde {\mathbb {A}}^3$. On this chart $\tilde {S}_y$, we have

\begin{align*} \tilde{\Gamma} &= (z = w - \nu y^2 = 0) \subset \tilde{\mathbb{A}}^3, \\ E_{\pm} &= (y = w \pm \sqrt{\eta} z = 0) \subset \tilde{\mathbb{A}}^3. \end{align*}

Now let $\hat {S} \to \tilde {S}$ be the blow-up of $\tilde {S}$ at $\tilde {\mathsf {p}}$ with exceptional divisor

\[ F \cong (- w^2 + \eta z^2 + \alpha_2 z y = 0) \subset \hat{\mathbb{P}}^2 := \mathbb{P}^2_{y, z, w}. \]

The proper transform $\hat {\Gamma } \subset \hat {S}$ of $\Gamma$ intersects $F$ only at the point $\hat {\mathsf {p}} := (1:0:0) \in \hat {\mathbb {P}}^2$. We denote by $\hat {S}_y$ the $y$-chart of the blow-up $\hat {S} \to \tilde {S}$ at $\tilde {\mathsf {p}}$, which is the hypersurface in $\hat {\mathbb {A}}^3$ defined by

\[ - w^2 + \eta z^2 + z (\alpha_2 + \alpha_3 y + \alpha_4 y^2 + \alpha_5 y^3 + \cdots) + \nu^2 y^2 = 0. \]

The point $\hat {\mathsf {p}}$ corresponds to the origin of $\hat {\mathbb {A}}^3$, and $\hat {S}$ has an ordinary double point at $\hat {\mathsf {p}}$. Note that the proper transforms $\hat {E}_{\pm } \subset \hat {S}$ of $E_{\pm }$ are both disjoint from the $y$-chart $\hat {S}_y$.

Suppose that $\alpha _2 \ne 0$. In this case $\tilde {S}$ has an ordinary double point at $\tilde {\mathsf {p}}$. We see that $\hat {\Gamma }$ intersects $F$ transversally at $\hat {\mathsf {p}}$, $F \cap \hat {E}_{\pm } = \{(0:1:\pm \sqrt {\eta })\}$, and $\hat {E}_{\pm } \cap \hat {\Gamma } = \emptyset$. This shows that $G (S, \mathsf {p}, \Gamma )$ is of type $A_{3, 2}$.

Suppose that $\alpha _2 = 0$. In this case we have $F = F_+ + F_-$, where

\[ F_{\pm} \cong (w \pm \sqrt{\eta} z = 0) \subset \hat{\mathbb{P}}^2, \]

and we have $F_+ \cap F_- \cap \hat {\Gamma } = \{\hat {\mathsf {p}}\}$. On the $y$-chart $\hat {S}_y$, we have

\begin{align*} \hat{\Gamma} &= (z = w - \nu y = 0) \subset \hat{\mathbb{A}}^3, \\ F_{\pm} &= (y = w \pm \sqrt{\eta} z = 0) \subset \hat{\mathbb{A}}^3. \end{align*}

Let $\breve {S} \to \hat {S}$ be the blow-up of $\hat {S}$ at $\hat {\mathsf {p}}$ with exceptional divisor $G$. We have an isomorphism

\[ G \cong (- w^2 + \eta z^2 + \alpha_3 z y + \nu^2 y^2 = 0) \subset \breve{\mathbb{P}}^2_{y, z, w}, \]

and, hence, $G$ is a smooth rational curve. Let $\breve {\Gamma }$, $\breve {E}_{\pm }$, and $\breve {F}_{\pm }$ be proper transforms of $\Gamma$, $E_{\pm }$, and $F_{\pm }$ on $\breve {S}$, respectively. We have $G \cap \breve {\Gamma } = \{(1:0:\nu )\}$ and $G \cap \breve {F}_{\pm } = \{(0:1:\pm \sqrt {\eta })\}$. This shows that $G (S, \mathsf {p}, \Gamma )$ is of type $A_{5, 3}$. This completes the proof.

Proof. We use the same notation as in the previous proof.

We prove assertion (1). Suppose $\operatorname {mult}_{\mathsf {p}} \Gamma \cap \Delta = 1$, that is, $\lambda \ne 0$. Since $\mathsf {p} \in X$ is of type $cA_2$, we have $f_2 = \lambda ^2 y^2$ and $f_3 (0, z, t) \ne 0$ as a polynomial. Then, by eliminating $t$ and setting $x = 1$, $S \cap U_x$ is isomorphic to the hypersurface in $\mathbb {A}^3_{y, z, w}$ defined by the equation

\[ - w^2 + \lambda^2 y^2 + \zeta z^3 + \alpha y z^2 + \beta y^2 z + h_{\ge 4} (y, z) = 0, \]

where $\alpha, \beta, \zeta \in \mathbb {C}$ with $\zeta \ne 0$ and $h_{\ge 4} (y, z)$ is a polynomial of order at least $4$. The point $\mathsf {p} = (1:0:0:0:0)$ corresponds to the origin and the curve $\Gamma$ is defined in $\mathbb {A}^3$ by the (4.6). Let $\tilde {S} \to S$ be the blow-up of $S$ at $\mathsf {p}$ with exceptional divisor $E$. Then $E = E_+ + E_-$, where

\[ E_{\pm} \cong (w \pm \lambda y = 0) \subset \tilde{\mathbb{P}}^2 := \mathbb{P}^2_{y, z, w}. \]

The surface $\tilde {S}$ is smooth outside the singular locus of $E$. The singular locus of $E$ is the intersection point $\mathsf {q}$ of $E_+$ and $E_-$, where $\mathsf {q} = (0:1:0) \in \tilde {\mathbb {P}}^2$. The point $\mathsf {q}$ corresponds to the origin of the $z$-chart $\tilde {S}_z$ which is the hypersurface in $\tilde {\mathbb {A}}^3 := \mathbb {A}^3_{y, z, w}$ defined by

\[ - w^2 + \lambda^2 y^2 + \zeta z + \alpha y + \beta y^2 z + h_{\ge 4} (y z, z)/z^2 = 0. \]

It follows that $\tilde {S}$ is smooth at $\tilde {\mathsf {p}}$. Then, since $\Gamma$ is defined by the (4.6), we have $\tilde {\Gamma } \cap E = \{\mathsf {q}'\}$, where $\mathsf {q}' := (1:0:\lambda ) \in \tilde {\mathbb {P}}^2$. We have $\mathsf {q}' \in E_-$ and $\mathsf {q}' \notin E_+$. This shows that $G (S, \mathsf {p}, \Gamma )$ is of type $A_{2,1}$.

We prove assertion (2). Suppose $\operatorname {mult}_{\mathsf {p}} \Gamma \cap \Delta = 2$, that is, $\lambda = 0$ and $\nu \ne 0$. We see that $f_2$ is a square of a linear form in $z$ and $t$. We may assume $f_2 = z^2$ by replacing $z$ and $t$. By eliminating $t$ and setting $x = 1$, $S \cap U_x$ is isomorphic to the hypersurface in $\mathbb {A}^3_{y, z, w}$ defined by the equation

\[ -w^2 + z^2 + z h_2 (y, z) + \mu^2 y^4 + h_{\ge 5} (y, z) = 0, \]

where $h_2 (y, z)$ is a homogeneous polynomial of degree $2$ and $h_{\ge 5} (y, z)$ is a polynomial of order at least $5$. The curve $\Gamma$ is defined in $\mathbb {A}^3$ by the (4.8). By repeating similar computations as in assertion (2) of Lemma 4.6, we see that $\mathsf {p} \in S$ is of type $A_3$ and $G (S, \mathsf {p}, \Gamma )$ is of type $A_{3, 2}$.

Table 1.

Types of $G (S, \mathsf {p}, \Gamma )$.

We prove assertion (3). Suppose $\operatorname {mult}_{\mathsf {p}} \Gamma \cap \Delta = 3$, that is, $\lambda = \mu = 0$ and $\nu \ne 0$. As in previous arguments, $S \cap U_x$ is isomorphic to the hypersurface in $\mathbb {A}^3_{y, z, w}$ defined by the equation

\[ -w^2 + z^2 + z h (y, z) + \nu^2 y^6 = 0, \]

where $h (y, z)$ is a polynomial consisting of monomials of degree $2, 3, 4, 5$. The curve $\Gamma$ is defined in $\mathbb {A}^3$ by the (4.9). By the same argument as in the proof of Lemma 4.6(3), we conclude that $G (S, \mathsf {p}, \Gamma )$ is of type $A_{3, 2}$ or $A_{5, 3}$.

Proof. By Lemma 4.4, it remains to consider a curve $\Gamma$ of degree $1$ such that $\Gamma \cap \operatorname {Sing} X \ne \emptyset$ and $\pi (\Gamma ) \not \subset B$.

Let $\mathcal {P} \subset | - K_X |$ be the pencil as above, and let $S, T \in \mathcal {P}$ be general members. By [Reference OkadaOka20b, Lemma 10.4], $S$ and $T$ are smooth outside $\Gamma \cap \operatorname {Sing} X$. We have $T|_S = \Gamma + \Delta$, where $\Delta \ne \Gamma$ is also a curve of degree $1$. By (4.2), we see that $\Gamma \cap \Delta$ consists of $3$ points counting with multiplicity.

We compute the self-intersection number $(\Gamma ^2)_S$. Let $\mathsf {p}_1, \ldots, \mathsf {p}_l$ be the points in $\Gamma \cap \operatorname {Sing} (X)$. By Remark 4.5, these points are contained in $\Gamma \cap \Delta$. We set $m_i = \operatorname {mult}_{\mathsf {p}_i} \Gamma \cap \Delta$. Then $l \le \sum m_i \le 3$. Let $A_{n_i, k_i}$ be the type of $G (S, \mathsf {p}_i, \Gamma )$ which is described in Lemmas 4.6 and 4.7 (see also Table 1). Since $K_S = 0$ and $\Gamma \cong \mathbb {P}^1$, we have

\[ (\Gamma^2)_S = - 2 + \sum_{i = 1}^l \frac{k_i (n_i - k_i + 1)}{n_i + 1} \]

by Lemma 2.11. By considering all the possible combinations of singular points $\mathsf {p}_1, \ldots, \mathsf {p}_l$ satisfying ${\sum } m_i \le 3$, we have the upper bound

\[ (\Gamma^2)_S \le - 2 + \bigl(\tfrac{2}{3} + \tfrac{2}{3} + \tfrac{2}{3} \bigr) = 0, \]

where this maximum (of the correction term corresponding to singular points) is attained when $\Gamma \cap \Delta$ consists of $3$ multiplicity $1$ points of $\Gamma \cap \Delta$ which are $cA_2$ singular points of $X$. Since $T|_S = \Gamma + \Delta$, we have

\[ (\Gamma \cdot \Delta)_S = (T|_S \cdot \Gamma)_S - (\Gamma^2)_S \ge 1 = (- K_X \cdot \Delta). \]

Thus, by Lemma 2.8, the curve $\Gamma$ is not a maximal center.

Proof. Let $\mathsf {p} \in X$ be a $cA_1$ point. Suppose that $\mathsf {p}$ is a maximal center. Then, by Theorem 3.14, there exists a mobile linear system $\mathcal {M} \subset |\mathcal {O}_X (n)|$ on $X$ such that

\[ \operatorname{mult}_{\mathsf{p}} (D_1 \cdot D_2) > 2 n^2, \]

where $D_1, D_2$ are general members in $\mathcal {M}$. Let $\mathcal {H}$ be the linear subsystem of $|\mathcal {O}_X (1)|$ consisting of divisors vanishing at $\mathsf {p}$. Then the base locus of $\mathcal {H}$ consists of at most $2$ points (including $\mathsf {p}$) and, hence, we can take a divisor $S$ in $\mathcal {H}$ which does not contain any component of the effective $1$-cycle $D_1 \cdot D_2$. Then, we have

\[ 2 n^2 = (S \cdot D_1 \cdot D_2) > 2 n^2. \]

This is a contradiction and $\mathsf {p}$ is not a maximal center.