Notice that we construct the $i$th branch of $\widetilde{X}^{JS}$ and the $i$th linear chain of $\widetilde{X}^{PS}$ in the exactly same way as in the case of cyclic quotient surface singularities. On the other hand, according to Remark <ref>, we may choose the compactified decorated curves of $\widetilde{X}^{JS}$ over the central curve $E_0$ by the images of the lines $M_j$, which are just the curves $\widetilde{C}_j$ in $\widetilde{X}^{PS}$ given in Figure <ref>.

There is a diffeomorphism $\phi \colon \widetilde{X}^{JS} \to \widetilde{X}^{PS}$ such that $\phi(E_0)=E_0$, $\phi(E_{\infty})=E_{\infty}$, $\phi(B_{ij})=B_{ij}$, and $\phi(\widetilde{C}_j) = \widetilde{C}_j$ for all $i,j$. Furthermore, if $\widetilde{C}_{ij}$ denote the compactified decorated curves in $\widetilde{X}^{JS}$, then, after relabeling properly, there are positive integers $n_{ij}$ so that we have

\begin{equation*}
\phi_{\ast}([\widetilde{C}_{ij}]) = n_{ij} l_{\infty} + [A_{i1}] + \dotsb + [A_{ij}]
\end{equation*}

for all $i,j$.

This follows from Proposition <ref>.

Let $\mathcal{X} \to \Delta$ be a smoothing of $X$. Let $\overline{\mathcal{X}}^{JS} \to \Delta$ and $\overline{\mathcal{X}}^{PS} \to \Delta$ be the compatible extension of $\mathcal{X} \to \Delta$ given in Theorem <ref>.

There is a diffeomorphism $\phi_t \colon \overline{X}^{JS}_t \to \overline{X}^{PS}_t$ between general fibers $\overline{X}^{JS}_t$ and $\overline{X}^{PS}_t$ such that $\phi_t(\widetilde{C}_j)=\widetilde{C}_j$ for all $j$ and

\begin{equation*}
(\phi_t)_{\ast}([\widetilde{C}_{ij}]) = n_{ij} l_{\infty} + [A_{i1}] + \dotsb + [A_{ij}]
\end{equation*}

for all $i,j$.

There is a blowing down $\overline{\mathcal{X}}^{PS}_t \to \overline{X}^{P}_t$ between general fibers.

According to Remark <ref>, $\widetilde{X}^{PS}$ can be obtained from $\widetilde{X}^{P}$ by blowing up along $E_{\infty}$, where the exceptional curves are exactly $\widetilde{C}_j$'s. Hence the assertion follows.

In summary,

We have maps between general fibers

\begin{equation*}
\overline{X}^{JS}_t \xrightarrow{\phi} \overline{X}^{PS}_t \xrightarrow{\psi} \overline{X}^{P}_t
\end{equation*}

where $\phi$ is a diffeomorphism and $\psi$ is a blowing down.

§ SMOOTHINGS OF NEGATIVE WEIGHTS BY PINKHAM

Let $X$ be a cyclic quotient surface singularity or a weighted homogeneous surface singularity with the big node. We summarize the theory of smoothings of negative weights by Pinkham [26] according to the singularities that we are interested in. And we introduce matrices that encode combinatorial descriptions of $\mathcal{C}(X)$.

§.§ Smoothings of negative weights

Let $\mathcal{X} \to \Delta$ be a smoothing of $X$ and let $\overline{\mathcal{X}}^{P} \to \Delta$ be its extension given in Theorem <ref>. The extension $\overline{\mathcal{X}}^{P} \to \Delta$ is a smoothing of negative weight of $\overline{X}^{P}$; that is it is a smoothing obtained by adding terms of lower weight to the defining equations of $\overline{X}^{P}$. See Pinkham [26] for details. In particular, the compactifying divisor $\widetilde{E}_{\infty}^{P}$ (cf. Definition <ref>) of $X$ is embedded in a general fiber $\overline{X}^{P}_t$ of the extension $\overline{\mathcal{X}}^{P} \to \Delta$ and it supports an ample divisor on $\overline{X}^{P}_t$.

Conversely, under certain cohomological conditions, Pinkham <cit.> presented a way to construct a negative weight smoothing of $\overline{X}^{P}$ from a curve $E$ in a smooth surface $S$ whose dual graph is the same as that of $\widetilde{E}_{\infty}^{P}$. Here we present a special case of Pinkham's result.

Let $S$ be a smooth rational surface. Suppose that there is a reduced curve $E$ in $S$ that supports an ample divisor, whose dual graph is the same as that of $\widetilde{E}_{\infty}^{P}$. Then there is a unique one-parameter smoothing $\overline{\mathcal{X}}^{P} \to \Delta$ of $\overline{X}^{P}$ such that the interior of its Milnor fiber is diffeomorphic to $S \setminus E$.

By “unique” in the proposition, we mean that it is unique up to isomorphism of deformations.

The cohomological conditions given in Pinkham <cit.> are automatically satisfied if $S$ is a smooth rational surface. For checking this fact, one may refer the proof of Pinkham <cit.> or the proof of Stipsicz–Szabó–Wahl <cit.>. Therefore there is a one-parameter smoothing $\overline{\mathcal{X}}^{P} \to \Delta$ of $\overline{X}^{P}$ such that $Z \cong \overline{X}^{P}_t$ and $E \cong \widetilde{E}^{P}$.

On the other hand, Fowler <cit.> gave a proof of the uniqueness using Looijenga <cit.>. We briefly recall the proof in Fowler <cit.>. There exists a formal versal $\mathbb{C}^{\ast}$-equivariant  deformation $p \colon \mathcal{V} \to B$ of $X$; Pinkham <cit.>. Let $p^{-} \colon \mathcal{V}^{-} \to B^{-}$ be the part of negative weight of the versal deformation of $X$. Then Pinkham <cit.> showed that Then $p^{-} \colon \mathcal{V}^{-} \to B^{-}$ extends to a proper fiat family $\overline{p}^{-} \colon \overline{\mathcal{V}}^{-} \to B^{-}$ of deformaions of $\overline{X}^{P}$. Looijenga <cit.> interpreted the family $\overline{p}^{-} \colon \overline{\mathcal{V}}^{-} \to B^{-}$ as a fine moduli space consisting of the triples $(Z; F; \phi)$, where (i) $Z$ is a projective surface with cyclic quotient singularities along a curve $F \subset Z$, (ii) $F$ is an ample Weil divisor on $Y$ which is isomorphic to the singular compactifying divisor $\overline{E}_{\infty}$, and (iii) if $Y=\Proj(R)$ and $X=\Spec(A)$ then $\phi \colon R/(t) \cong A$ for an element $t$ representing $1$. By contracting the branches on $E_{\infty}$ on $\overline{X}^{P}_t$ to cyclic quotient singularities, we obtain the Looijenga's triple from $(Z,E)$. So $(Z,E)$ determines a unique negative weight smoothings of $X$.

§.§ Homology matrices

A general fiber $\overline{X}^P_t$ ($t \neq 0$) is a smooth rational surface because it contains a smooth rational curve $E_{\infty}$ with positive self-intersection number.

Let $X$ be a cyclic quotient surface singularity. Then there is a sequence of blow-downs from a general fiber $\overline{X}_t^{P}$ ($t \neq 0$) to $\mathbb{CP}^2$ such that (i) no blowing down occur on $E_{\infty}$, and (ii) the image of the compactifying divisor $\widetilde{E}_{\infty}^{P}$ under the sequence of blow-downs consists of two lines in $\mathbb{CP}^2$: One from $E_{\infty}$ and the other from $A_1$.

By Theorem <ref>, there is a diffeomorphism $\phi \colon \overline{X}^{JS}_t \to \overline{X}^{P}_t$ between general fibers such that $\phi(E_{\infty})=E_{\infty}$ and

\begin{equation*}
\phi_{\ast}([\widetilde{C}_j]) = n_j l_{\infty} + [A_1] + \dotsb + [A_j]
\end{equation*}

A general fiber $\overline{X}^{JS}_t$ is obtained by blowing up several times $\mathbb{CP}^2$ at some points on the decorated curves in a general fiber of the compactified picture deformations $(\mathcal{C}, \mathcal{L})$; Theorem <ref>. So each $(-1)$-curves in $\overline{X}^{JS}_t$ intersect only $\widetilde{C}_j$'s. Therefore each $(-1)$-curves in $\overline{X}^{P}_t$ intersect only $A_j$'s, not $E_{\infty}$. Hence there is a sequence of blow-downs $\psi \colon \overline{X}^{P}_t \to \mathbb{CP}^2$ such that no blowing down occur on $E_{\infty}$. Furthermore $\psi(A_1)$ must be a line because it intersects with $E_{\infty}$ at one point and the other $A_j$'s ($j \ge 2$) must be contracted to the same point on $\psi(A_1)$ because they do not intersect with $E_{\infty}$.

We have a similar result for weighted cases.

Let $X$ be a weighted homogeneous surface singularity with the big node. There is a sequence of blow-downs from a general fiber $\overline{X}_t^{PS}$ ($t \neq 0$) to $\mathbb{CP}^2$ such that (i) no blowing down occur on $E_{\infty}$, and (ii) the image of the compactifying divisor $\widetilde{E}_{\infty}^{PS}$ under the sequence of blow-downs consists of lines in $\mathbb{CP}^2$.

By Theorem <ref>, there is a diffeomorphism $\phi \colon \overline{X}^{JS}_t \to \overline{X}^{PS}_t$ such that $\phi(\widetilde{C}_k)=\widetilde{C}_k$ for all $k$ and

\begin{equation*}
(\phi_t)_{\ast}([\widetilde{C}_{ij}]) = n_{ij} l_{\infty} + [A_{i1}] + \dotsb + [A_{ij}]
\end{equation*}

for all $i,j$. Notice that $\overline{X}^{JS}_t$ is obtained by blowing up $\mathbb{CP}^2$ at some points on the images of the decorated curves $\widetilde{C_{ij}}$ and $\widetilde{C}_k$ in a general fiber of the corresponding picture deformation to $\overline{\mathcal{X}}^{JS} \to \Delta$. Therefore there is also a sequence of blow-downs from $\overline{X}^{PS}_t$ to $\mathbb{CP}^2$.

On the other hand, any $(-1)$-curves in $\overline{X}^{JS}_t$ intersect  only $\widetilde{C}_{ij}$ and $\widetilde{C}_k$ and they do not intersect the image of the line at infinity $L_{\infty}$, which is just $E_{\infty}$ in $\overline{X}^{JS}_t$. So any $(-1)$-curves in $\overline{X}^{PS}_t$ intersect only $A_{ij}$'s and $\widetilde{C}_k$'s. Therefore the image of $E_{\infty}$ in $\mathbb{CP}^2$ is a line. Furthermore, each $A_{i1}$ ($i=1,\dotsc,t$) and each $\widetilde{C}_k$ ($k=1,\dotsc,d-t-1$) intersect with $E_{\infty}$ at one point, respectively. So their images in $\mathbb{CP}^2$ must be lines.

We define a matrix that encodes combinatorial and homological properties of the ample embedding of the compactifying divisor $\widetilde{E}_{\infty}$ into a general fiber $\overline{X}^P_t$:

The homology matrix of the smoothing $\overline{\mathcal{X}}^P \to \Delta$ of $\overline{X}^P$ is the matrix that have one row for every irreducible component $A_i$'s (for the cyclic case) or $A_{ij}$'s (for the weighted case) of the compactifying divisor $\widetilde{E}_{\infty}$ and one column for every $(-1)$-curve in $\overline{X}^P_t$ such that the $(i,j)$-entry is defined by the intersection number of the component of $\widetilde{E}_{\infty}$ corresponding to the $i$th row and the $(-1)$-curves corresponding to $j$th column. We denote by $\mathcal{H}(X)$ by the set of all homology matrices of $X$.

Every component of $\Def(X)$ is a smoothing component and every smoothing of $X$ can be extended to that of $\overline{X}^P$. So we have a well-defined map:

The homology matrix map of $X$ is a map

\begin{equation*}
\phi_H \colon \mathcal{C}(X) \to \mathcal{H}(X)
\end{equation*}

defined by assigning the corresponding homology matrix in $\mathcal{H}(X)$ to each irreducible component in $\mathcal{C}(X)$.

This map may be another combinatorial description of $\mathcal{C}(X)$.

Determine when $\phi_H$ is injective.

We prove that $\phi_H$ is injective for certain weighted homogeneous surface singularities in Theorem <ref>.

§.§ Isotopies

We prove that $\phi_H$ is injective for cyclic quotient surface singularities and weighted homogeneous surface singularities with the `enough' big nodes.

For a cyclic quotient surface singularity $X$, the map $\phi_H \colon \mathcal{C}(X) \to \mathcal{H}(X)$ is injective; hence, it is bijective.

Suppose that there are two smoothings $\mathcal{X}_1 \to \Delta$ and $\mathcal{X}_2 \to \Delta$ of $X$ corresponding to the same homology matrix. According to Proposition <ref>, each smoothing $\mathcal{X}_i \to \Delta$ is completely determined by the pair $(S_i, \widetilde{E}_{\infty})$, where $S_i=\overline{X_i}^P_t$ is a smooth rational surface where the divisor $\widetilde{E}_{\infty}$ is amply embedded. Notice that there is an isotopy between $(\mathbb{CP}^2, L_1 \cup M_1)$ and $(\mathbb{CP}^2, L_2 \cup M_2)$. Since $\mathcal{X}_i$'s have the same homology matrix, each pairs $(S_i, \widetilde{E}_{\infty})$ are constructed from the pairs $(\mathbb{CP}^2, L_i \cup M_i)$ via the same sequence of blowing ups. Therefore there is also an isotopy between $(S_1, \widetilde{E}_{\infty})$ and $(S_2, \widetilde{E}_{\infty})$. Therefore there is a path in $\Def(X)$ which connects two points corresponding to each smoothings $\mathcal{X}_1$ and $\mathcal{X}_2$. Therefore they lie on the same component in $\Def(X)$.

We now assume that $X$ is a weighted homogeneous surface singularity with the big node.

Let $\mathcal{X} \to \Delta$ be a smoothing of $X$. The line arrangement of $\mathcal{X} \to \Delta$ is the set of lines in $\mathbb{CP}^2$ that is obtained from $\overline{X}^{PS}_t$ by blowing down as in the above Lemma <ref>

If $X$ is a weighted homogeneous surface singularity with $d \ge t+2$, then the line arrangement of each smoothing $\mathcal{X} \to \Delta$ is one of the following arrangements given in Figure <ref>. Explicitly, let $L_{\infty}$ be the line that is the image of $E_{\infty}$. Then there is one point outside $L_{\infty}$ such that (A) all the other lines passe through the point or such that (B) all the other lines except only one line pass through the point, or (C) there are two points outside $L_{\infty}$ such that the rest of the lines must pass through one of the two points and at least three lines pass through each point.

[-] (0,0)–(3,0);

[-] (0,0.5)–(2,-2);
[-] (0.5,0.5)–(1.833,-2);
[-] (1,0.5)–(1.666,-2);
[-] (1.5,0.5)–(1.5,-2);
[-] (3,0.5)–(1,-2);

[dotted] (2,0.25)–(2.5,0.25);
[-] (0,0)–(3,0);

[-] (0,0.5)–(3,-2);
[-] (0.5,0.5)–(1.833,-2);
[-] (1,0.5)–(1.666,-2);
[-] (1.5,0.5)–(1.5,-2);
[-] (3,0.5)–(1,-2);

[dotted] (2,0.25)–(2.5,0.25);

[-] (0,0)–(3,0);

[-] (0.25,0.5)–(1,-2);
[-] (0.5,0.5)–(0.875,-2);
[-] (1.25,0.5)–(0.5,-2);

[-] (1.75,0.5)–(2.5,-2);
[-] (2,0.5)–(2.375,-2);
[-] (2.75,0.5)–(2,-2);

[dotted] (0.75,0.25)–(1,0.25);
[dotted] (2.25,0.25)–(2.5,0.25);

Line arrangements

Let $L_{\infty}$ be the image of $E_{\infty}$. Let $M_j$ be the line that is the image of $\widetilde{C}_j$ for each $j=1,\dotsc,d-t-1$. Notice that the image of $M_1$ in $\overline{X}^{PS}_t$ is a $(-1)$-curve. So there are two points on $M_1$ outside $L_{\infty}$ where blowing-ups occur. So if $d=t+2$, then we have three possibilities (A), (B), (C). On the other hand, if $d \ge t+3$, then only (A) and (B) can happen.

Let $X$ be a weighted homogeneous surface singularity with $d \ge t+2$. The map $\phi_H \colon \mathcal{C}(X) \to \mathcal{H}(X)$ is injective; hence, it is bijective.

A smoothing $\mathcal{X} \to \Delta$ is determined by the pair $(S, \widetilde{E}_{\infty})$ of a smooth rational surface $S = \overline{X}^P_t$ and an amply embedded divisor $\widetilde{E}_{\infty}$ by Proposition <ref>. By Proposition <ref>, we blow up ($d-t-1$) points on $E_{\infty}$ properly so that we get the pair $(S', \widetilde{E}_{\infty}^{PS})$ where $S' = \overline{X}^{PS}_t$ and $\widetilde{E}_{\infty}^{PS}$ is the compactifying divisor for $\overline{X}^{PS}$ defined in Definition <ref>.

Suppose that there are two smoothings $\mathcal{X}_1 \to \Delta$ and $\mathcal{X}_2 \to \Delta$ of $X$ such that their homology matrices are the same. Then two pairs $(S_1', \widetilde{E}_{\infty}^{PS})$ and $(S_2', \widetilde{E}_{\infty}^{PS})$ are isotopic to each other. This is because they are obtained from $\mathbb{CP}^2$ with the line arrangement of the same type (A), (B), or (C) (given in Figure <ref>) via the same sequence of blowing-ups for they have the same homology matrices. But the line arrangements of the same type in Figure <ref> are isotopic to each other by Starkston <cit.>. Therefore there is a isotopy between $\mathcal{X}_1 \to \Delta$ and $\mathcal{X}_2 \to \Delta$, which implies that they lie on the same component of $\Def(X)$.

§ FROM PICTURE DEFORMATIONS TO SMOOTHINGS OF NEGATIVE WEIGHT

Let $X$ be a cyclic quotient surface singularity or a weighted homogeneous surface singularity with the big node with its usual sandwiched structure. We construct a map from $\mathcal{I}(X)$ to $\mathcal{H}(X)$ that commutes with $\phi_I \colon \mathcal{C}(X) \to \mathcal{I}(X)$ and $\phi_H \colon \mathcal{C}(X) \to \mathcal{H}(X)$; Theorem <ref>.

Let $M$ be a matrix with the rows $M(1), \dotsc, M(e)$.

* The difference matrix $\Delta M$ is a matrix with $e$ rows $\Delta M(1), \dotsc, \Delta M(e)$ defined by

\begin{equation*}
\text{$\Delta M(1) = M(1)$ and $\Delta M(i) = M(i)-M(i-1)$ for $i \ge 2$}.
\end{equation*}

* The positive matrix $M^+$ is the matrix obtained from $M$ by deleting columns that contain entries smaller than zero.

Let $X$ be a cyclic quotient surface singularity and let $\mathcal{X} \to \Delta$ be a smoothing of $X$. If $M$ is the incidence matrix of $\mathcal{X} \to \Delta$, then the positive difference matrix

\begin{equation}\label{equation:Delta^+M-cyclic}
(\Delta M)^+
\end{equation}

is the homology matrix of the smoothing.

Any projective $(-1)$-curves in $\overline{X}^{JS}_t$ must intersect with one of $\widetilde{C}_i$'s, but not with $E_{\infty}$. On the other hand, according to Equation <ref>, we have the following relations on their homologies:

\begin{align}\label{equation:AvsC-cyclic}
[A_1] &= m_1 l_{\infty} + \phi_{\ast}([\widetilde{C}_1]) \\
[A_i] &= m_i l_{\infty} + \phi_{\ast}([\widetilde{C}_i]) - \phi_{\ast}([\widetilde{C}_{i-1}])
\end{align}

where $i \ge 2$ and $m_i \in \mathbb{Z}$. If $E$ is a projective $(-1)$-curve in $\overline{X}^{P}_t$, then its inverse image $\phi^{-1}(E)$ is also a (differentiable) $(-1)$-curve on $\overline{X}^{JS}_t$. So its homology is equal to one of the homologies of the projective $(-1)$-curves in  $\overline{X}^{JS}_t$. Hence, the intersection of $\phi^{-1}(E)$ with $\widetilde{C}_i$'s must be equal to one of the column of the incidence matrix $M$. Since any projective curves intersect positively with another projective curve, the intersection of $E$ with $A_i$'s is equal to one of the columns of the positive difference matrix $(\Delta M)^+$ by Equation (<ref>).

Let $X$ be a weighted homogeneous surface singularity with the big node. The procedure for obtaining homology matrices of $X$ from incidence matrices of $X$ is essentially the same for cyclic cases. Let $M$ be the incidence matrix for a smoothing $\mathcal{X} \to \Delta$ of $X$. For each nonzero entries in the rows corresponding to the decorated curves $\widetilde{C}_1, \dotsc, \widetilde{C}_{d-t-1}$, we delete the columns of $M$ that contain the nonzero entries. After that, we delete all rows corresponding to $\widetilde{C}_k$'s. We denote the submatrix by $M \setminus C$. For each $i=1,\dotsc,t$, we denote by $(M \setminus C)(i)$ the submatrix of $M \setminus C$ consisting of the rows of $M \setminus C$ that corresponds to the decorated curves on the $i$th branch.

Let $X$ be a weighted homogeneous surface singularity with the big node. For the incidence matrix $M$ of a smoothing $\mathcal{X} \to \Delta$ of $X$, the corresponding homology matrix to $\mathcal{X} \to \Delta$ of $X$ is given by

\begin{equation}\label{equation:Delta^+M-weighted}
\begin{bmatrix}
\Delta (M \setminus C)(1) \\
\vdots\\
\Delta (M \setminus C)(t)
\end{bmatrix}^+
\end{equation}

By Corollary <ref>, $\overline{X}^{JS}_t$ is diffeomorphic to $\overline{X}^{PS}_t$. Hence the homology matrix for $\overline{X}^{PS}_t$ is given by

\begin{equation*}
\begin{bmatrix}
\Delta M(1) \\
\vdots\\
\Delta M(t)
\end{bmatrix}^+
\end{equation*}

by Theorem <ref>. Since $\overline{X}^{PS}_t$ is blown down to $\overline{X}^{P}_t$ along the $(-1)$-curves $\widetilde{C}_k$ for $k=1,\dotsc,d-t-1$, the images of any $(-1)$-curves intersecting $\widetilde{C}_k$'s are not $(-1)$-curves anymore. Therefore, by deleting the columns corresponding to the $(-1)$-curves intersecting any $\widetilde{C}_k$'s and the rows of all $\widetilde{C}_k$'s, we obtain the homology matrix for $\overline{X}^{P}_t$.

Let $X$ be a cyclic quotient surface singularity or a weighted homogeneous surface singularity with the big node. Then for any incidence matrix $M$ of $X$, we define the homology matrix corresponding to $M$ (denoted by $\Delta^+ M$) by the corresponding homology matrix defined in Equation (<ref>) for cyclic cases or Equation (<ref>) for weighted cases.

Let $X$ be a cyclic quotient surface singularity or a weighted homogeneous surface singularity with the big node. Then we have a map

\begin{equation*}
\phi_{IH} \colon \mathcal{I}(X) \to \mathcal{H}(X)
\end{equation*}

defined by $\phi_{IH}(M) = \Delta^+M$ so that $\phi_H = \phi_{IH} \circ \phi_{I}$.

If $X$ is a cyclic quotient surface singularity or a weighted homogeneous surface singularity with $d \ge t+2$, then the map $\phi_I \colon \mathcal{C}(X) \to \mathcal{I}(X)$ is injective; hence, it is bijective.

This follows from Theorem <ref> and Theorem <ref>.

As an easy application of Némethi–Popescu-Pampu [20]:

If $X$ is a cyclic quotient surface singularity or a weighted homogeneous surface singularity with $d \ge t+2$, then Minor fibers corresponding to different components of the reduced miniversal deformation space of $X$ are not `strongly' diffeomorphic to each other.

Here we follow the definition of `strong diffeomorphism' presented in Némethi and Popescu-Pampu [21]: If $M_1$ and $M_2$ are two oriented manifolds with boundary, endowed with a fixed isotopy class of orientation-preserving diffeomorphisms $\partial M_1 \to \partial M_2$, then $M_1$ and $M_2$ are strongly diffeomorphic (with respect to that class) if there exists an orientation-preserving diffeomorphism $M_1 \to M_2$ whose restriction to the boundary belongs to the given class.

Némethi–Popescu-Pampu <cit.> showed if the incidence matrices of two picture deformations are different up to permutation of columns, then their associated Milnor fibers are not strongly diffeomorphic. Since $\phi_I$ is injective, we can conclude that there is no pair of smoothing components such that their corresponding Milnor fibers are strongly diffeomorphic to each other.

§ FROM P-RESOLUTIONS TO SMOOTHINGS OF NEGATIVE WEIGHTS

Let $X$ be a cyclic quotient surface singularity or a weighted homogeneous surface singularity with the big node. We define a map from $\mathcal{P}(X)$ to $\mathcal{H}(X)$. For cyclic cases, it is done in PPSU <cit.>. The procedure for weighted cases is identical to that described in PPSU <cit.>. We briefly recall the method given in PPSU <cit.>.

Let $\mathcal{X} \to \Delta$ be a smoothing of $X$. Suppose that there is a P-resolution $Y \to X$ of $X$ that corresponds to $\mathcal{X} \to \Delta$. Since the minimal resolution of $Y$ dominates the minimal resolution of $X$, we have a compactification $\overline{Y}^P$ of $Y$ that dominates $\overline{X}^P$.

There is no local-to-global obstruction of $\overline{Y}^P$. In particular, every smoothing of $X$ can be lifted to that of $\overline{Y}^P$ that is locally trivial along $\overline{Y}^P \setminus X$.

The proof is verbatim to that of Theorem <ref>.

In particular, we have a map $\overline{\mathcal{Y}}^P \to \overline{\mathcal{X}}^P$ over $\Delta$, where $\overline{\mathcal{Y}}^P$ is an extension of $\mathbb{Q}$-Gorenstein smoothings of T-singularities on $Y$ that is locally trivial outside the singularities.

By applying only Iitaka–Kodaira divisorial contractions and usual flips to curves coming from the arms of $\overline{Y}^P$, we can run the semi-stable MMP to $\overline{\mathcal{Y}}^P \to \Delta$ until we obtain a deformation $\mathcal{W} \to \Delta$ whose central fiber $W_0$ is smooth.

The proof is identical to that of PPSU <cit.>.

A flip changes only central fibers. But divisorial contractions are just blow-downs of $(-1)$-curves on each general fibers. Hence:

In Proposition <ref>, a general fiber $\overline{Y}^P_t$ ($t \neq 0$) of the smoothing $\overline{\mathcal{Y}}^P \to \Delta$ is obtained by blowing up several times a general fiber $W_t$ of the smoothing $\mathcal{W} \to \Delta$.

By comparing the central fiber $W_0$ and a general fiber $W_t$, one can get the data of positions of $(-1)$-curves in $W_t$. Then, one can get the data of intersections of $(-1)$-curves with the compactifying divisor $\widetilde{E}_{\infty}$ in $\overline{Y}^P_t$ by tracking the blow-downs $\overline{Y}^P_t \to W_t$. The data of $(-1)$-curves can be encoded as a homology matrix of $\overline{\mathcal{X}}^P$. So we have a map:

Let $X$ be a cyclic quotient surface singularity of a weighted homogeneous surface singularity with the big node. Via the procedure discussed in the above, we can define a map

\begin{equation*}
\phi_{PH} \colon \mathcal{P}(X) \to \mathcal{H}(X)
\end{equation*}

such that $\phi_{PH} = \phi_H \circ \phi_P$.

§ DEFORMATIONS OF CYCLIC QUOTIENT SURFACE SINGULARITIES

Let $X$ be a cyclic quotient surface singularity of type $\frac{1}{n}(1,a)$. Christophersen [4] and Stevens [30] described the equations of each components in $\mathcal{C}(X)$. In this section we study the correspondence between their results and other theories of deformations.

§.§ The equations of the reduced miniversal base space

We recall briefly Christophersen [4] and Stevens [30]. We first define a set $\mathcal{K}(X)$. For any sequence $\underline{x}=(x_1,\dotsc,x_e)$ of positive integers, we define an $s \times s$ matrix $M(\underline{x})$ by $M_{i,i}=x_i$, $M_{i,j}=-1$ if $\abs{i-j}=1$, and $M_{i,j}=0$ otherwise.

A sequence $\underline{k}=(k_1,\dotsc,k_e) \in \mathbb{N}^e$ is called admissible if the matrix $M(\underline{k})$ is positive semi-definite of rank at least $e-1$.

Christophersen [4]
For $e \ge 1$, we define

\begin{equation*}
K_e = \{ (k_1,\dotsc,k_e) \in \mathbb{N}^e \mid \text{$(k_1,\dotsc,k_e)$ is admissible and $[k_1,\dotsc,k_e]=0$}\}.
\end{equation*}

that is, the set of all admissible $e$-tuples which represent a zero Hirzebruch–Jung continued fraction.

Suppose that $n/(n-a)=[a_1,\dotsc,a_e]$.

We define

\begin{equation*}
\mathcal{K}(X) = \{(k_1,\dotsc,k_e) \in K_e \mid \text{$k_i \le a_i$ for all $i$}\}.
\end{equation*}

As a toric variety, one can find a system of equations that define the singularity $(X,p)$. The system includes the equations

\begin{equation*}
\end{equation*}

for $i=1,\dotsc,e$. Then Christophersen [4] and Stevens [30] showed that the deformation component parameterized by $\underline{k} \in \mathcal{K}(X)$ contains a 1-parameter deformation which has a set of equations that contains

\begin{equation*}
\end{equation*}

for all $i=1,\dotsc,e$.

So we may regard that the set $\mathcal{K}(X)$ parameterizes $\mathcal{C}(X)$ via equations. Therefore we can define a bijective map

\begin{equation*}
\phi_K \colon \mathcal{C}(X) \to \mathcal{K}(X).
\end{equation*}

§.§ From smoothings of negative weights to the equations

Let $H$ be a homology matrix of $X$. For each $i$th row (corresponding to $A_i$) of $H$, we define a sequence $\underline{k}$ for $H$ as follows:

For each $i$, we define a positive integer $d_i$ by the number of nonzero entries on the $i$th row of $H$. Let $k_i = a_i - d_i$. We denote the sequence $(k_1,\dotsc,k_e)$ by $\underline{k}(H)$.

We have $\underline{k}(H) \in \mathcal{K}(X)$.

Notice that $d_i$ is equal to the number of $(-1)$-curves in $\overline{X}^P_t$ that intersect with $A_i$. But every $(-1)$-curve intersects only one $A_i$ by Lemma <ref>. Furthermore there is a sequence of blow-downs from $\overline{X}^P_t$ to $\mathbb{CP}^2$, where (i) no blow-down occur on $E_{\infty}$, (ii) the image of $A_1$ is a line, and (iii) all the other $A_j$'s are contracted to a point on $A_1$. Therefore $d_i \le a_i$. If we blow down all $(-1)$-curves intersecting $A_i$'s at once, then the images of $A_i$'s becomes a linear chain $k_1-\dotsb-k_e$. But the chain should be contracted to a point on $A_1$. Therefore we have $\underline{k} \in \mathcal{K}(X)$.

We have a map

\begin{equation*}
\phi_{HK} \colon \mathcal{H}(X) \to \mathcal{K}(X), \quad H \mapsto \underline{k}(H).
\end{equation*}

§.§ From picture deformations to the equations

Let $I$ be an incidence matrix of $X$.

We define a number $d_i$ by

\begin{equation}\label{equation:di}
d_i = \#\{j \mid \text{$I_{pj}=0$ for $p < i$ and $I_{pj}=1$ for $p \ge i$}\}.
\end{equation}

Let $k_i=a_i-d_i$ for $i=1,\dotsc,e$. We denote the sequence $(k_1,\dotsc,k_e)$ by $\underline{k}(I)$.

We have $\underline{k}(I) \in \mathcal{K}(X)$.

This follows immediately from Theorem <ref>, that is, $\phi_{IH}(I)=\Delta^+I$.

The number $d_i$ is defined also in Némethi–Popescu-Pampu <cit.> in a similar way.

§.§ From P-resolutions to the equations

This topic is one of the main results in PPSU [27]. They showed that one can obtain the corresponding sequence $\underline{k}$ from a given P-resolution via the semi-stable MMP.

The idea can be explained using the terminology in this paper. As in Section <ref>, for each P-resolution $Y$ of $X$, we can construct the corresponding smoothings $\overline{\mathcal{X}}^P$ of negative weights of $X$. Then the sequence of $\underline{k}$ for the P-resolution $Y$ is just the image of the homology matrix of $\overline{\mathcal{X}}^P$. In short, $\underline{k} = \phi_{HK}(\phi_{PH}(Y))$.

§.§ Correspondence between four deformation theories

For every $\underline{k} \in \mathcal{K}(X)$ the picture deformation, the P-resolution, and the smoothing of negative weight corresponding to $\underline{k}$ parameterize the same component in $\mathcal{C}(X)$. But in order to show that the equation associated to $\underline{k}$ also parameterize the same component, we need the result of Némethi–Popescu-Pampu [20].

In the paper, they showed that the picture deformation and the equation corresponding to the same $\underline{k}$ parameterize the same component in $\mathcal{C}(X)$. For this, they applied the classification of minimal symplectic fillings of cyclic quotient surface singularities by Lisca [16].

Therefore we have a complete picture of the correspondence between all known deformation theories of cyclic quotient surface singularities, which is given in Figure <ref>.

Let $X$ be a cyclic quotient surface singularities $\frac{1}{19}(1,7)$. Then the correspondence are given as follows.

Incidence matrices     Homology matrices     P-resolutions     Equations
0 & 0 & 0 & 0 & 0 & 1 & 1\\
0 & 0 & 0 & 1 & 1 & 0 & 1 \\
0 & 0 & 1 & 0 & 1 & 0 & 1 \\
1 & 1 & 0 & 0 & 1 & 0 & 1 \\
\end{bmatrix}$
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0
\end{bmatrix}$
0 & 0 & 0 & 0 & 1 & 1 \\
0 & 0 & 1 & 1 & 0 & 1 \\
0 & 1 & 0 & 1 & 0 & 1 \\
1 & 1 & 1 & 0 & 0 & 1 \\
\end{bmatrix}$
0 & 0 & 1 \\
0 & 0 & 0 \\
0 & 1 & 0 \\
1 & 0 & 0
\end{bmatrix}$
0 & 0 & 0 & 1 & 1 \\
0 & 1 & 1 & 0 & 1 \\
1 & 0 & 1 & 0 & 1 \\
1 & 1 & 1 & 1 & 0 \\
\end{bmatrix}$
0 & 0 \\
0 & 1 \\
1 & 0 \\
0 & 0
\end{bmatrix}$

§ A WEIGHTED HOMOGENEOUS SURFACE SINGULARITY ADMITTING QHDS

The singularity with which we are concerned in this section is known as $W(p,q,r)$ (after J. Wahl), whose resolution graph is given by in Figure <ref>. In this section, we prove that Kollár conjecture holds for $W(p,q,r)$. We follow the strategy described in Introduction.

[bullet] (00) at (0,0) [labelAbove=$-4$] ;

[bullet] (-350) at (-3.5,0) [labelBelow=$-(p+3)$] ;
[bullet] (-250) at (-2.5,0) [labelAbove=$-2$] ;
[empty] (-20) at (-2,0) [] ;
[empty] (-150) at (-1.5,0) [] ;
[bullet] (-10) at (-1,0) [labelAbove=$-2$] ;

[-] (-350)–(-250);
[-] (-250)–(-20);
[dotted] (-20)–(-150);
[-] (-150)–(-10);
[-] (-10)–(00);
[decorate, decoration = calligraphic brace,mirror] (-2.5,-0.15)–(-1,-0.15)  node[pos=0.5,below=0.1em,black]$q$;

[bullet] (350) at (3.5,0) [labelBelow=$-(q+3)$] ;
[bullet] (250) at (2.5,0) [labelAbove=$-2$] ;
[empty] (20) at (2,0) [] ;
[empty] (150) at (1.5,0) [] ;
[bullet] (10) at (1,0) [labelAbove=$-2$] ;

[-] (350)–(250);
[-] (250)–(20);
[dotted] (20)–(150);
[-] (150)–(10);
[-] (10)–(00);
[decorate, decoration = calligraphic brace,mirror] (1,-0.15)–(2.5,-0.15)  node[pos=0.5,below=0.1em,black]$r$;

[bullet] (0-35) at (0,-3.5) [label=right:$-(r+3)$] ;
[bullet] (0-25) at (0,-2.5) [label=left:$-2$] ;
[empty] (0-2) at (0,-2) [] ;
[empty] (0-15) at (0,-1.5) [] ;
[bullet] (0-1) at (0,-1) [label=left:$-2$] ;

[-] (0-35)–(0-25);
[-] (0-25)–(0-2);
[dotted] (0-2)–(0-15);
[-] (0-15)–(0-1);
[-] (0-1)–(00);
[decorate, decoration = calligraphic brace,mirror] (0.15,-2.5)–(0.15,-1)  node[pos=0.5,right=0.1em,black]$p$;

$W(p,q,r)$, where $p, q, r \ge 0$

§.§ A sandwiched structure

Let $X=W(p,q,r)$. It is sandwiched for having the big node. We add ($p+2$) $(-1)$-vertices to the $-(p+3)$-vertex, ($q+2$) $(-1)$-vertices to the $-(q+3)$-vertex, and ($r+2$) $(-1)$-vertices to the $-(r+3)$-vertex in the resolution graph, making the resolution graph a sandwiched graph. We then consider small curvetta attached to each $(-1)$-vertices, which we denote by $\widetilde{A}_1, \dotsc, \widetilde{A}_{p+2}$, $\widetilde{B}_1, \dotsc, \widetilde{B}_{q+2}$, and $\widetilde{C}_1, \dotsc, \widetilde{C}_{r+2}$, respectively; Figure <ref>.

The map $\phi_I \colon \mathcal{C}(X) \to \mathcal{I}(X)$ is injective; hence it is bijective.

Since it has three branches and it has the big node, every line arrangement (Definition <ref>) corresponding to $X$ is one of them given in Figure <ref>. Therefore we can prove that $\phi_H \colon \mathcal{C}(X) \to \mathcal{H}(X)$ is injective as we did in the proof of Theorem <ref>. Then it follows that $\phi_I \colon \mathcal{C}(X) \to \mathcal{I}(X)$ is also injective.

[bullet] (00) at (0,0) [labelAbove=$-4$] ;

[bullet] (-505) at (-5,0.5) [label=left:$\widetilde{A}_1$] ;
[bullet] (-5-05) at (-5,-0.5) [label=left:$\widetilde{A}_{p+2}$] ;
[dotted] (-5,0.25)–(-5,-0.25);

[bullet] (-4505) at (-4.5,0.5) [labelAbove=$-1$] ;
[bullet] (-45-05) at (-4.5,-0.5) [labelBelow=$-1$] ;
[dotted] (-4.5,0.25)–(-4.5,-0.25);

[-] (-4505)–(-505);
[-] (-45-05)–(-5-05);

[-] (-4505)–(-350);
[-] (-45-05)–(-350);

[bullet] (-350) at (-3.5,0) [labelBelow=$-(p+3)$] ;
[bullet] (-250) at (-2.5,0) [labelAbove=$-2$] ;
[empty] (-20) at (-2,0) [] ;
[empty] (-150) at (-1.5,0) [] ;
[bullet] (-10) at (-1,0) [labelAbove=$-2$] ;

[-] (-350)–(-250);
[-] (-250)–(-20);
[dotted] (-20)–(-150);
[-] (-150)–(-10);
[-] (-10)–(00);
[decorate, decoration = calligraphic brace,mirror] (-2.5,-0.15)–(-1,-0.15)  node[pos=0.5,below=0.1em,black]$q$;

[bullet] (505) at (5,0.5) [label=right:$\widetilde{B}_1$] ;
[bullet] (5-05) at (5,-0.5) [label=right:$\widetilde{B}_{q+2}$] ;
[dotted] (5,0.25)–(5,-0.25);

[bullet] (4505) at (4.5,0.5) [labelAbove=$-1$] ;
[bullet] (45-05) at (4.5,-0.5) [labelBelow=$-1$] ;
[dotted] (4.5,0.25)–(4.5,-0.25);

[-] (4505)–(505);
[-] (45-05)–(5-05);

[-] (4505)–(350);
[-] (45-05)–(350);

[bullet] (350) at (3.5,0) [labelBelow=$-(q+3)$] ;
[bullet] (250) at (2.5,0) [labelAbove=$-2$] ;
[empty] (20) at (2,0) [] ;
[empty] (150) at (1.5,0) [] ;
[bullet] (10) at (1,0) [labelAbove=$-2$] ;

[-] (350)–(250);
[-] (250)–(20);
[dotted] (20)–(150);
[-] (150)–(10);
[-] (10)–(00);
[decorate, decoration = calligraphic brace,mirror] (1,-0.15)–(2.5,-0.15)  node[pos=0.5,below=0.1em,black]$r$;

[bullet] (-05-5) at (-0.5,-5) [label=below:$\widetilde{C}_1$] ;
[bullet] (05-5) at (0.5,-5) [label=below:$\widetilde{C}_{r+2}$] ;
[dotted] (-0.25,-5)–(0.25,-5);

[bullet] (-05-45) at (-0.5,-4.5) [label=left:$-1$] ;
[bullet] (05-45) at (0.5,-4.5) [label=right:$-1$] ;
[dotted] (-0.25,-4.5)–(0.25,-4.5);

[-] (-05-5)–(-05-45);
[-] (05-5)–(05-45);

[-] (-05-45)–(0-35);
[-] (05-45)–(0-35);

[bullet] (0-35) at (0,-3.5) [label=right:$-(r+3)$] ;
[bullet] (0-25) at (0,-2.5) [label=left:$-2$] ;
[empty] (0-2) at (0,-2) [] ;
[empty] (0-15) at (0,-1.5) [] ;
[bullet] (0-1) at (0,-1) [label=left:$-2$] ;

[-] (0-35)–(0-25);
[-] (0-25)–(0-2);
[dotted] (0-2)–(0-15);
[-] (0-15)–(0-1);
[-] (0-1)–(00);
[decorate, decoration = calligraphic brace,mirror] (0.15,-2.5)–(0.15,-1)  node[pos=0.5,right=0.1em,black]$p$;

A sandwiched structure for $W(p,q,r)$

§.§ The decorated curve and the incidence relations

We denote by $A_i, B_j, C_k$ the decorated curves of $W(p,q,r)$ in $\mathbb{CP}^2$ that are induced from the sandwiched structure given in Figure <ref>.

Notice that all curves $A_i, B_j, C_k$ are smooth. Indeed $A_i$'s are plane curves given by $y=a_i x^{q+2}$, and $B_j$'s and $C_k$'s are those given by $y=b_jx^{r+2}$ and $y=c_kx^{p+2}$, respectively. So we have

\begin{equation}\label{equation:Wpqr-matrix-delta-l}
\begin{gathered}
\delta(A_i)=\delta(b_j)=\delta(C_k)=1 \\
l(A_i)=q+3, \quad l(B_j)=r+3, \quad l(C_k)=p+3
\end{gathered}
\end{equation}

for all $i,j,k$. Furthermore the incidence relations between $A_i, B_j, C_k$ are given as follows.

\begin{equation}\label{equation:Wpqr-matrix-intersection}
\begin{gathered}
A_i \cdot A_{i'}= q+2, \quad B_j \cdot B_{j'} = r+2, \quad C_k \cdot C_{k'} = p+2 \\
A_i \cdot B_j = 1, \quad B_j \cdot C_k = 1, \quad C_k \cdot A_i = 1
\end{gathered}
\end{equation}

for all $i, i', j, j', k, k'$.

§.§ Submatrices of combinatorial incidence matrices

We find all combinatorial incidence matrices for the decorated curve of $W(p,q,r)$.

First of all, since $\delta(A_i)=\delta(B_j)=\delta(C_k)=1$, all entries of any incidence matrices are $1$. One the other hand, $l(A_i)=q+3$ and $A_i \cdot A_{i'}=q+2$. So the submatrix consisting of the rows corresponding to $A_i$'s are one of the followings:

Notice that $A(II)$ can occur only if $q+2-p \ge 0$. Furthermore, if $p=0$, then $A(I)=A(II)$. So $A(II)$ is considered only when $p \ge 1$.

Similarly, those corresponding to $B_j$'s and $C_k's$ are as follows:

where $B(II)$ and $C(II)$ can occur only if $r+2-q \ge 0$, $q \ge 1$, and $p+2-r \ge 0$, $r \ge 1$, respectively.

§.§ Combinations of the submatrices

We will explain how the columns of the above submatrices are combined to produce combinatorial incidence matrices. Notice that we should have $A_i \cdot B_j = 1$, $B_j \cdot C_k = 1$, $C_k \cdot A_i = 1$ from Equation (<ref>). That is, the inner product of any pair of the $A_i$-rows ($B_j$-rows, or $C_k$-rows) and the $B_j$-rows($C_k$-rows, or $A_i$-rows, respectively) is equal to $1$.

We begin with analyzing all possible combinations of the columns of the submatrices $A$ and $B$ such that $A_i \cdot B_j = 1$.

At first, suppose that $A(I)$ appears in a combinatorial incidence matrix $M$. We divide cases according to which columns of $B(I)$ or $B(II)$ can appear under the first column of $A(I)$.

Case 1: A column of $B(I)$ or $B(II)$ consisting of only $1$ under the first column of $A(I)$.

The submatrix corresponding to $A$-rows and $B$-rows looks like

Then the inner products `$A_i \cdot B_j =1$' occur on the first column of the incidence matrix $M$ for any $i, j$. So there must be no $1$ below the other columns of $A(I)$. Therefore the combinatorial incidence matrix $M$ contains the following submatrix

where $B'$ is the submatrix obtained from $B(I)$ or $B(II)$ by deleting one column consisting of only $1$'s.

Case 2: A column of $B(I)$ with only one $1$ under the first column of $A(I)$.

We may assume that the submatrix corresponding to $A$-rows and $B$-rows looks like

Then the inner product `$A_i \cdot B_1=1$' occur on the first entry $1$ of $B_1$ for all $i$. Therefore there is no $1$ on the $B_1$-row until the last column of $A$. So we have

Hence any columns of $B(I)$ consisting of only $1$'s cannot appear under any columns of $A(I)$. So in order to satisfy the conditions `$A_i \cdot B_j=1$', the submatrix corresponding to $A_i$-rows and $B_j$-rows must be of the form

\begin{equation}\label{equation:Wpqr-AB-2}
\includegraphics{figure-for-arXiv-7}
%%A_1 & 1 & \Cdots & 1 & 1 &  & 0  &0  & \Cdots & 0\\
%%\Vdots & \Vdots &  & \Vdots &  &  \Ddots &  & \Vdots  & & \Vdots\\
%%A_{p+2} & 1 & \Cdots & 1 & 0 &  & 1 & 0 & \Cdots & 0\\
%%B_1 & 1  &  & 0 & 0 & \Cdots & 0 &  1& \Cdots & 1\\
%%\Vdots &  & \Ddots &  & \Vdots &  & \Vdots  & \Vdots &  & \Vdots\\
%%B_{q+2} & 0  &  & 1 & 0 & \Cdots  & 0  & 1 & \Cdots & 1
\end{equation}

Case 3. A column of $B(II)$ with only one $0$ under the first column of $A(I)$.

We may assume that the submatrix corresponding to $A$-rows and $B$-rows looks like

As before, because of the inner product condition `$A_i \cdot B_j=1$', the entries of $B_j$-rows for $j=1,\dots,q+1$ must be $0$ until the last column of $A(I)$. So we have

But, since $q \ge 1$, there is no way to complete the $B_{q+2}$-row so that the inner product condition $A_i \cdot B_{q+2}=1$ satisfies for any $i$. Therefore Case 3 cannot occur.

Case 4. No columns of $B(I)$ and $B(II)$ under the first column of $A(I)$.

That is, we may assume that

Then there must be the columns of $B(I)$ or $B(II)$ consisting of only $1$'s under the $(p+2)$-column of $A(I)$ in order to satisfy the inner product condition `$A_i \cdot B_j=1$. That is, we have

\begin{equation}\label{equation:Wpqr-AB-3}
\begin{aligned}
\includegraphics{figure-for-arXiv-11}
\end{aligned}
%%A_1 & 1 & \Cdots & 1 & 1 &  & 0 & 0 & \Cdots & 0\\
%%\Vdots & \Vdots &  & \Vdots &  &  \Ddots & & \Vdots &  & \Vdots\\
%%A_{p+2} & 1 & \Cdots & 1 & 0 &  & 1 & 0 & \Cdots & 0\\
%%B_1 & 0 & \Cdots  & 0 & 1 & \Cdots & 1 &  &  & \\
%%\Vdots & \Vdots &  & \Vdots & \Vdots & & \Vdots &  & B' &\\
%%B_{q+2} & 0 & \Cdots  & 0 & 1 & \Cdots & 1 &  &  &\\
\end{equation}

where $B'$ is the submatrix obtained from $B(I)$ or $B(II)$ by deleting $(p+2)$ columns consisting of only $1$'s. Notice that we must assume that

\begin{equation}\label{equation:Wpqr-AB-inequality}
\text{$r+2 \ge p+2$ for $B(I)$ or $r+2-q \ge p+2$ for $B(II)$}
\end{equation}

because we need at most $(p+2)$ columns consisting of only $1$'s in $B(I)$ or $B(II)$.

Next, assume that the submatrix $A(II)$ appears in the incidence matrix $M$. As before, we divide the cases as follows.

Case 5. A column of $B(I)$ or $B(II)$ consisting of only $1$'s under the first column of $A(II)$.

By the similar arguments as in Case 1, we have

where $B'$ is the submatrix obtained from $B(I)$ or $B(II)$ by deleting one column consisting of only $1$'s.

Case 6. A column of $B(I)$ with only one $1$ under the first column of $A(II)$.

As in Case 2, the entries of the $B_1$-row except the first one must be $0$. So we have

Since we assume that $p \ge 1$, there must be a $B_j$-row whose entries under the $(q+2-p)$-columns of $A(II)$ are all $0$. We may assume that the $B_{q+2}$-row is of this type. Then there must be only one $1$ in the entries of the $B_{q+2}$-row under the $(p+2)$-columns of $A(II)$ in order to satisfy the inner product condition `$A_i \cdot B_{q+2}=1$'. However no matter where $1$ is placed, there must be a row of $A(II)$ that does not satisfy the inner product condition. So Case 6 cannot occur.

Case 7. A column of $B(II)$ with only one $0$ under the first column of $A(II)$.

Because of the exactly same reason in Case 3, this case cannot happen.

Case 8. No columns of $B(I)$ or $B(II)$ under the first column of $A(II)$.

We will show that this last Case 8 cannot occur as well. We may assume that all entries under the $(q+2-p)$-columns of $A(II)$ are $0$. That is,

Notice that, under the $(p+2)$-columns of $A(II)$, there must be a non-zero column of $B(I)$ or $B(II)$. So we may assume that the first entry in the $B_1$-row under the first column of the $(p+2)$-columns of $A(II)$ is $1$. Then the other entries of the $B_1$-row should be zero because $A_i\cdot B_1 =1$ and $p \ge 1$. Then we have

But, $A_{p+2} \cdot B_1 = 0$, which violates the inner product conditions. Therefore Case 8 cannot occur.

One can analyze combinations of the submatrices $B, C$ and $C, A$ in a similar way. In summary,

The inner products occur either on the columns of the submatrices $A, B, C$ consisting of only $1$'s or on the identity submatrix of $A(I), B(I), C(I)$ and the submatrix consisting of only $1$'s of $A, B, C$ under the suitable conditions on the number of columns as in Equation (<ref>).

§.§ All possible combinatorial incidence matrices

Suppose that the inner product occurs only on the columns of the submatrices whose entries consisting only $1$. Then there are only two possibilities:

Notice that $A', A''$, $B', B''$, $C', C''$ in the figure are the submatrices obtained from $A, B, C$ by deleting one or two columns consisting of only $1$'s, respectively. So if any of the submatrices $A(II)$, $B(II)$, or $C(II)$ are encountered in the above incidence matrices, then $M(1)$ can occur under the conditions

\begin{equation}\label{equation:condition-M(1)-with-II}
\text{$p \ge 1$, $q+2-p \ge 1$, or $q \ge 1$, $r+2-q \ge 1$, or $r \ge 1$, $p+2-r \ge 1$,}
\end{equation}

respectively, and $M(2)$ can occur under the conditions

\begin{equation}\label{equation:condition-M(2)-with-II}
\text{$p \ge 1$, $q+2-p \ge 2$, or $q \ge 1$, $r+2-q \ge 2$, or $r \ge 1$, $p+2-r \ge 2$,}
\end{equation}

Suppose that the inner product happens on the identity submatrix of $A(I)$ and the submatrix of $B$ consisting of only $1$'s as in Equation (<ref>). Then the inner products on $A, C$ and $B, C$ must occur on the columns consisting of only $1$'s by Lemma <ref>. Therefore we have

where $B'$ and $C'$ are the remained parts of $B$ and $C$, respectively. Notice that the above configuration can happen under the condition

\begin{equation*}
l := \begin{cases}
r-p \ge 1 & \text{for $B(I)$} \\
r-p-q \ge 1 & \text{for $B(II)$}
\end{cases}
\end{equation*}

We claim that the above configuration can happen only for $C(I)$: The submatrix $C(II)$ must have at least two columns consisting of only $1$'s. So $p+2-r \ge 2$, which contradicts, however, to the above condition $l \ge 1$. Therefore we have the following incidence matrix

under the condition

\begin{equation}\label{equation:condition-M(3)}
l := \begin{cases}
r-p \ge 1 & \text{if $B=B(I)$} \\
r-p-q \ge 1, q \ge 1 & \text{if $B=B(II)$}
\end{cases}
\end{equation}

Similarly, we have more incidence matrices

under the condition

\begin{equation}\label{equation:condition-M(4)}
l := \begin{cases}
p-q \ge 1 & \text{if $C=C(I)$} \\
p-q-r \ge 1, r \ge 1 & \text{if $C=C(II)$}
\end{cases}
\end{equation}

under the condition

\begin{equation}\label{equation:condition-M(5)}
l := \begin{cases}
q-r \ge 1 & \text{if $A=A(I)$} \\
q-r-p \ge 1, p \ge 1 & \text{if $A=A(II)$}
\end{cases}
\end{equation}

Finally, suppose that the inner product happens on the identity submatrix of $A(I)$ and the submatrix of $B$ consisting of only $1$'s as in Equation (<ref>). Then there is only one possible combinatorial incidence matrix

In addition, if $p=q=r$, then we have one more combinatorial matrix (which Jonathan Wahl informed us):

There are seven types of combinatorial incidence matrices $M(1), \dotsc, M(6)$ for $W(p,q,r)$ under the suitable conditions on $p,q,r$ described in Equation (<ref>) for $M(1)$, (<ref>) for $M(2)$, (<ref>) for $M(3)$, (<ref>) for $M(4)$, (<ref>) for $M(5)$.

§ KOLLÁR CONJECTURE FOR $W(P,Q,R)$

In this section we prove:

Kollár conjecture holds for the singularities $W(p,q,r)$.

We divide the proof according to the types of the combinatorial incidence matrices of $W(p,q,r).$

§.§ P-resolutions for $M(1)$ and $M(2)$

We first construct P-resolutions corresponding to each of the combinatorial incidence matrices of type $M(1)$ or $M(2)$.

Each combinatorial incidence matrices of type $M(1)$ or $M(2)$ correspond to one of the following M-resolution of $W(p,q,r)$.

* The minimal resolution itself. It corresponds to the (combinatorial) incidence matrix $M(1)$ with $A(I)$, $B(I)$, and $C(I)$.

* A M-resolution obtained by contracting the central $(-4)$-curve. It corresponds to the (combinatorial) incidence matrix $M(2)$ with $A(I)$, $B(I)$, and $C(I)$.

* A M-resolution obtained by contracting the $(-p-3)$-curve and the $(p-1)$ $(-2)$-curves that follow it (that is, the Wahl configuration $[p+3,2,\dotsc,2]$) from the $A$-arm (assuming $p \ge 1$ and $q \ge p-1$). It corresponds to the (combinatorial) incidence matrix $M(1)$ with $A(II)$, but $B(I)$ and $C(I)$.

* A M-resolution obtained by contracting the Wahl configuration $[q+3,2,\dotsc,2]$ from the $B$-arm (assuming $q \ge 1$ and $r \ge q-1$). It corresponds to the (combinatorial) incidence matrix $M(1)$ with $A(I), B(II), C(I)$.

* A M-resolution obtained by contracting the Wahl configuration $[r+3,2,\dotsc,2]$ from the $C$-arm (assuming $r \ge 1$ and $p \ge r-1$). It corresponds to the (combinatorial) incidence matrix $M(1)$ with $A(I), B(I), C(II)$.

* Any M-resolutions obtained by combining (3), (4), or (5) under the necessary hypotheses on $p, q, r$. They correspond to the (combinatorial) incidence matrix $M(1)$ whose submatrices $A, B, C$ are determined by combining as in (3), (4), (5).

* Any M-resolutions obtained by combining (3), (4), or (5) with (2) under the stronger conditions  $q \ge p$, $p \ge r$, and $r \ge q$, respectively (in order to ensure that the contractions can be performed independently). They correspond to the (combinatorial) incidence matrix $M(2)$ whose submatrices $A, B, C$ are determined by combining as in (3), (4), (5).

For P-resolutions of type $M(1)$, there are no T-singularities. We need only divisorial contractions. As we see in Section <ref>, it is easy to track how $(-1)$-curves intersect the decorated curves in a general fiber. For P-resolutions of type $M(2)$, we've already covered the case where the singularity $[4]$ is located in the central node in Section <ref>. Also we have dealt with cases in which P-resolutions possess only Wahl singularities of type $[a+4,2,\dotsc,2]$ in Section <ref>. Therefore, we can prove the proposition in a similar way.

§.§ P-resolutions for $M(3)$, $M(4)$, and $M(5)$

If $r-p-1 \ge 0$ and $q \ge 1$, then we can construct a P-resolution of $W(p,q,r)$ whose dual graph is given in Figure <ref>. It has two T-singularities:

\begin{equation*}
\text{$[p+3,\underbrace{2,\cdots,2}_{q-1}, 3, \underbrace{2,\dotsc,2}_{p}]$ and $[p+5,\underbrace{2,\dotsc,2}_{p+1}]$}
\end{equation*}

If $r-p-1 \ge 0$ but $q=0$, then we can construct a P-resolution of $W(p,q,r)$ as in Figure <ref> which has two T-singularities:

\begin{equation*}
\text{$[p+4,\underbrace{2,\dotsc,2}_{p}]$ and $[p+5,\underbrace{2,\dotsc,2}_{p+1}]$}
\end{equation*}

Moreover, if $r-p-1 \ge q$ and $q \ge 1$, we can contract the Wahl configuration $2-\dotsb-2-(q+3)$ in the $B$-arm to obtain another P-resolution of $W(p,q,r)$ as shown in Figure <ref> which has one more T-singularity  in addition to the two singularities:

\begin{equation*}
\end{equation*}

Observe that it is possible to verify that the canonical divisors of the aforementioned P-resolutions are ample by applying the algorithm for calculating discrepancies described, for instance, in Urzúa–Vilches <cit.>.

[rectangle] (00) at (0,0) [labelAbove=$-p-5$] ;

[rectangle] (-80) at (-8,0) [labelAbove=$-(p+3)$] ;

[rectangle] (-70) at (-7,0) [labelAbove=$-2$] ;
[empty] (-650) at (-6.5,0) [] ;
[empty] (-60) at (-6,0) [] ;
[rectangle] (-550) at (-5.5,0) [labelAbove=$-2$] ;

[decorate, decoration = calligraphic brace,mirror] (-7,-0.15)–(-5.5,-0.15)  node[pos=0.5,below=0.1em,black]$q-1$;

[rectangle] (-450) at (-4.5,0) [labelAbove=$-3$] ;

[rectangle] (-350) at (-3.5,0) [labelAbove=$-2$] ;
[empty] (-30) at (-3,0) [] ;
[empty] (-250) at (-2.5,0) [] ;
[rectangle] (-20) at (-2,0) [labelAbove=$-2$] ;

[decorate, decoration = calligraphic brace,mirror] (-3.5,-0.15)–(-2,-0.15)  node[pos=0.5,below=0.1em,black]$p$;

[bullet] (-10) at (-1,0) [labelAbove=$-1$] ;

[-] (-80)–(-70);

[-] (-70)–(-650);
[dotted] (-650)–(-60);
[-] (-60)–(-550);
[-] (-550)–(-450);

[-] (-450)–(-350);

[-] (-350)–(-30);
[dotted] (-30)–(-250);
[-] (-250)–(-20);

[-] (-10)–(-20);
[-] (-10)–(00);

[rectangle] (10) at (1,0) [labelAbove=$-2$] ;
[empty] (150) at (1.5,0) [] ;
[empty] (20) at (2,0) [] ;
[rectangle] (250) at (2.5,0) [labelAbove=$-2$] ;

[decorate, decoration = calligraphic brace,mirror] (1,-0.15)–(2.5,-0.15)  node[pos=0.5,below=0.1em,black]$p+1$;

[bullet] (350) at (3.5,0) [label=right:$-2$] ;
[empty] (35-05) at (3.5,-0.5) [] ;
[empty] (35-1) at (3.5,-1) [] ;
[bullet] (35-15) at (3.5,-1.5) [label=right:$-2$] ;

[decorate, decoration = calligraphic brace, mirror] (3.35,0)–(3.35,-1.5)  node[pos=0.5,left=0.1em,black]$r-p+1$;

[bullet] (35-25) at (3.5,-2.5) [label=right:$-(q+3)$] ;

[-] (00)–(10);

[-] (10)–(150);
[dotted] (150)–(20);
[-] (20)–(250);

[-] (250)–(350);
[-] (350)–(35-05);
[dotted] (35-05)–(35-1);
[-] (35-1)–(35-15);

[-] (35-15)–(35-25);

[bullet] (0-35) at (0,-3.5) [label=left:$-(r+3)$] ;

[bullet] (0-25) at (0,-2.5) [label=left:$-2$] ;
[empty] (0-2) at (0,-2) [] ;
[empty] (0-15) at (0,-1.5) [] ;
[bullet] (0-1) at (0,-1) [label=left:$-2$] ;

[decorate, decoration = calligraphic brace,mirror] (0.15,-2.5)–(0.15,-1)  node[pos=0.5,right=0.1em,black]$p$;

[-] (0-35)–(0-25);
[-] (0-25)–(0-2);
[dotted] (0-2)–(0-15);
[-] (0-15)–(0-1);
[-] (0-1)–(00);

The P-resolution for $M(3)$ with $B(I)$ for $r-p \ge 1$, $q \ge 1$

[rectangle] (00) at (0,0) [labelAbove=$-p-5$] ;

[rectangle] (-450) at (-4.5,0) [labelAbove=$-(p+4)$] ;

[rectangle] (-350) at (-3.5,0) [labelAbove=$-2$] ;
[empty] (-30) at (-3,0) [] ;
[empty] (-250) at (-2.5,0) [] ;
[rectangle] (-20) at (-2,0) [labelAbove=$-2$] ;

[bullet] (-10) at (-10) [labelAbove=$-1$] ;

[-] (-450)–(-350);
[-] (-350)–(-30);
[dotted] (-30)–(-250);
[-] (-250)–(-20);
[-] (-20)–(-10);
[-] (-10)–(00);
[decorate, decoration = calligraphic brace,mirror] (-3.5,-0.15)–(-2,-0.15)  node[pos=0.5,below=0.1em,black]$p$;

[bullet] (60) at (6,0) [labelAbove=$-3$] ;

[bullet] (50) at (5,0) [labelAbove=$-2$] ;
[empty] (450) at (4.5,0) [] ;
[empty] (40) at (4,0) [] ;
[bullet] (350) at (3.5,0) [labelAbove=$-2$] ;

[rectangle] (250) at (2.5,0) [labelAbove=$-2$] ;
[empty] (20) at (2,0) [] ;
[empty] (150) at (1.5,0) [] ;
[rectangle] (10) at (1,0) [labelAbove=$-2$] ;

[-] (60)–(50);
[-] (50)–(450);
[dotted] (450)–(40);
[-] (40)–(350);

[decorate, decoration = calligraphic brace,mirror] (3.5,-0.15)–(5,-0.15)  node[pos=0.5,below=0.1em,black]$r-p-1$;

[-] (350)–(250);
[-] (250)–(20);
[dotted] (20)–(150);
[-] (150)–(10);
[-] (10)–(00);
[decorate, decoration = calligraphic brace,mirror] (1,-0.15)–(2.5,-0.15)  node[pos=0.5,below=0.1em,black]$p+1$;

[bullet] (0-35) at (0,-3.5) [label=left:$-(r+3)$] ;
[bullet] (0-25) at (0,-2.5) [label=left:$-2$] ;
[empty] (0-2) at (0,-2) [] ;
[empty] (0-15) at (0,-1.5) [] ;
[bullet] (0-1) at (0,-1) [label=left:$-2$] ;

[-] (0-35)–(0-25);
[-] (0-25)–(0-2);
[dotted] (0-2)–(0-15);
[-] (0-15)–(0-1);
[-] (0-1)–(00);
[decorate, decoration = calligraphic brace,mirror] (0.15,-2.5)–(0.15,-1)  node[pos=0.5,right=0.1em,black]$p$;

The P-resolution for $M(3)$ with $B(I)$ for $r-p \ge 1$, $q=0$

[rectangle] (00) at (0,0) [labelAbove=$-p-5$] ;

[rectangle] (-80) at (-8,0) [labelAbove=$-(p+3)$] ;

[rectangle] (-70) at (-7,0) [labelAbove=$-2$] ;
[empty] (-650) at (-6.5,0) [] ;
[empty] (-60) at (-6,0) [] ;
[rectangle] (-550) at (-5.5,0) [labelAbove=$-2$] ;

[decorate, decoration = calligraphic brace,mirror] (-7,-0.15)–(-5.5,-0.15)  node[pos=0.5,below=0.1em,black]$q-1$;

[rectangle] (-450) at (-4.5,0) [labelAbove=$-3$] ;

[rectangle] (-350) at (-3.5,0) [labelAbove=$-2$] ;
[empty] (-30) at (-3,0) [] ;
[empty] (-250) at (-2.5,0) [] ;
[rectangle] (-20) at (-2,0) [labelAbove=$-2$] ;

[decorate, decoration = calligraphic brace,mirror] (-3.5,-0.15)–(-2,-0.15)  node[pos=0.5,below=0.1em,black]$p$;

[bullet] (-10) at (-1,0) [labelAbove=$-1$] ;

[-] (-80)–(-70);

[-] (-70)–(-650);
[dotted] (-650)–(-60);
[-] (-60)–(-550);
[-] (-550)–(-450);

[-] (-450)–(-350);

[-] (-350)–(-30);
[dotted] (-30)–(-250);
[-] (-250)–(-20);

[-] (-10)–(-20);
[-] (-10)–(00);

[rectangle] (10) at (1,0) [labelAbove=$-2$] ;
[empty] (150) at (1.5,0) [] ;
[empty] (20) at (2,0) [] ;
[rectangle] (250) at (2.5,0) [labelAbove=$-2$] ;

[decorate, decoration = calligraphic brace,mirror] (1,-0.15)–(2.5,-0.15)  node[pos=0.5,below=0.1em,black]$p+1$;

[bullet] (350) at (3.5,0) [labelAbove=$-2$] ;
[empty] (40) at (4,0) [] ;
[empty] (450) at (4.5,0) [] ;
[bullet] (50) at (5,0) [labelAbove=$-2$] ;

[decorate, decoration = calligraphic brace, mirror] (3.5,-0.15)–(5,-0.15)  node[pos=0.5,below=0.1em,black]$r-p-q$;

[rectangle] (60) at (6,0) [label=right:$-2$] ;
[empty] (6-05) at (6,-0.5) [] ;
[empty] (6-1) at (6,-1) [] ;
[rectangle] (6-15) at (6,-1.5) [label=right:$-2$] ;

[decorate, decoration = calligraphic brace, mirror] (5.85,0)–(5.85,-1.5)  node[pos=0.5,left=0.1em,black]$q-1$;

[rectangle] (6-25) at (6,-2.5) [label=right:$-(q+3)$] ;

[-] (00)–(10);

[-] (10)–(150);
[dotted] (150)–(20);
[-] (20)–(250);

[-] (250)–(350);
[-] (350)–(40);
[dotted] (40)–(450);
[-] (450)–(50);

[-] (50)–(60);

[-] (60)–(6-05);
[dotted] (6-05)–(6-1);
[-] (6-1)–(6-15);

[-] (6-15)–(6-25);

[bullet] (0-35) at (0,-3.5) [label=left:$-(r+3)$] ;

[bullet] (0-25) at (0,-2.5) [label=left:$-2$] ;
[empty] (0-2) at (0,-2) [] ;
[empty] (0-15) at (0,-1.5) [] ;
[bullet] (0-1) at (0,-1) [label=left:$-2$] ;

[decorate, decoration = calligraphic brace,mirror] (0.15,-2.5)–(0.15,-1)  node[pos=0.5,right=0.1em,black]$p$;

[-] (0-35)–(0-25);
[-] (0-25)–(0-2);
[dotted] (0-2)–(0-15);
[-] (0-15)–(0-1);
[-] (0-1)–(00);

The P-resolution for $M(3)$ with $B(II)$ for $r-p-q \ge 1$, $q \ge 1$

Every (combinatorial) incidence matrix of type $M(3)$ corresponds to one of the P-resolutions given in Figure <ref>, <ref>, and <ref> according to the choice of $B$. Similarly, any (combinatorial) incidence matrices of type $M(4)$ or $M(5)$ correspond to the P-resolutions in the same figures; however, their subscripts are modified as follows: $(p,q,r)\rightarrow (q,r,p)$ for $M(4)$ and $(p,q,r) \rightarrow (r,p,q)$ for $M(5)$.

Let $M_3$ be the (combinatorial) incidence matrix in Figure <ref>. We will prove the proposition for $M_3$. The other cases are proven in a similar fashion.

We take the crepant M-resolution of Figure <ref>. Notice that the crepant M-resolution of the T-singularity $[p+3,2,\dotsc,2,3,2,\dotsc,2]$ in Figure <ref> is given by

\begin{equation*}
\end{equation*}

There are $q$ $(-1)$-curves in the above crepant M-resolution.

Let $Z$ be the compactified crepant M-resolution of Figure <ref> and let $\mathcal{Z} \to \Delta$ be a $\mathbb{Q}$-Gorenstein smoothing of the central fiber $Z_0=Z$.

At first, applying the MMP to the B-arm and the C-arm, we obtain the following gray part of the incidence matrix $M_3$:

After the MMP, we have a new deformation, denoted again by $\mathcal{Z} \to \Delta$ for simplicity, whose central fiber contains the following configuration

\begin{equation}\label{equation:Wpqr-M(3)-after-BC-arms}
\begin{aligned}
\begin{tikzpicture}[scale=0.85]

\node at (-1.75,0) [] {(A-arm)};
\node[empty] (-10) at (-1,0) [] {};

\node[bullet] (00) at (0,0) [labelAbove={$-p-3$},label=below left:{$F$}] {};

\node[bullet] (105) at (1,0.5) [label=right:{$\widetilde{B}_1'$}] {};
\node[bullet] (1-05) at (1,-0.5) [label=right:{$\widetilde{B}_{q+2}'$}] {};

\node[bullet] (-05-1) at (-0.5,-1) [label=below:{$\widetilde{C}_1$}] {};
\node[bullet] (05-1) at (0.5,-1) [label=below:{$\widetilde{C}_{r+2}$}] {};

\draw[-] (-10)--(00);
\draw[-] (00)--(105);
\draw[-] (00)--(1-05);
\draw[-] (00)--(-05-1);
\draw[-] (00)--(05-1);

\draw[dotted] (1,0.35)--(1,-0.35);
\draw[dotted] (0.35,-1)--(-0.35,-1);
\end{tikzpicture}
\end{aligned}
\end{equation}

Here we have degenerations of $\widetilde{B}_j$ induced by the usual flip:

\begin{equation*}
\widetilde{B}_j \rightsquigarrow \widetilde{B}_j' + F
\end{equation*}

for all $j=1,\dotsc,q+2$.

Next, we apply the usual flips to $\mathcal{Z}$ along with the $(-1)$-curves that intersect with $\widetilde{A}_1, \dotsc, \widetilde{A}_{p+1}$ in this order. Then the leftmost Wahl singularity $[p+4,2,\dotsc,2]$ of $Z$ is resolved so that we have a new deformation, denoted again by $\mathcal{Z} \to \Delta$ for simplicity, whose central fiber contains

\begin{equation*}
\begin{tikzpicture}[scale=0.75]
\node[bullet] (01) at (0,1) [label=left:{$\widetilde{A}_1'$}] {};
\node[bullet] (0-05) at (0,-0.5) [label=left:{$\widetilde{A}_{p+1}'$}] {};
\node[bullet] (0-1) at (0,-1) [label=left:{$\widetilde{A}_{p+2}$}] {};

\node[bullet] (05-05) at (0.5,-0.5) [labelBelow={\color{red}{$-1$}},label=right:{\color{red}{$E$}}] {};

\node[bullet] (10) at (1,0) [labelAbove={$-3$},label=below:{$G$}] {};

\node[bullet] (20) at (2,0) [labelAbove={$-2$},label=below:{$G_{p}$}] {};

\node[empty] (250) at (2.5,0) []{};
\node[empty] (30) at (3,0) []{};

\node[bullet] (350) at (3.5,0) [labelAbove={$-2$},label=below:{$G_1$}] {};

\node[bullet] (450) at (4.5,0) [labelAbove={$-1$},label=below:{$E'$}] {};

\node[empty] (50) at (5,0) []{};
\node[empty] (550) at (5.5,0) []{};

\draw[dotted] (0,0.85)--(0,-0.35);

\draw[-] (01)--(10);
\draw[-] (0-05)--(10);
\draw[-] (0-1)--(05-05)--(10);

\draw[-] (10)--(20);

\draw[-] (20)--(250);
\draw[dotted] (250)--(30);
\draw[-] (30)--(350);

\draw[-] (350)--(450);

\draw[-] (450)--(50);
\draw[dotted] (50)--(550);
\end{tikzpicture}
\end{equation*}

The degenerations occur as follows:

\begin{align*}
\widetilde{A}_1 &\rightsquigarrow \widetilde{A}_1' + G + G_p + \dotsb + G_1 \\
\widetilde{A}_2 &\rightsquigarrow \widetilde{A}_2' + G + G_p + \dotsb + G_2 \\
\vdots \\
\widetilde{A}_p &\rightsquigarrow \widetilde{A}_1' + G + G_p \\
\widetilde{A}_{p+1} &\rightsquigarrow \widetilde{A}_1' + G
\end{align*}

Therefore the red $(-1)$-curve $E$ induces the following gray column of the incidence matrix $M_3$:

We then apply a divisorial contraction to $E$. If there is another Wahl singularity $[p+4,2,\dotsc,2]$ on the $(-1)$-curve $E'$, we apply the usual flips beginning with $E'$ to the flipped images of $G_1, \dotsc, G_p$, and $G$ in succession. The sequence of flips then resolves the Wahl singularity, resulting in a new configuration:

\begin{equation*}
\begin{tikzpicture}[scale=0.75]
\node[bullet] (01) at (0,1) [label=left:{$\widetilde{A}_1'$}] {};
\node[bullet] (0-05) at (0,-0.5) [label=left:{$\widetilde{A}_{p+1}'$}] {};
\node[bullet] (0-1) at (0,-1) [label=left:{$\widetilde{A}_{p+2}$}] {};

\node[bullet] (10) at (1,0) [labelAbove={\color{red}{$-1$}},label=below:{\color{red}{$E$}}] {};

\node[bullet] (20) at (2,0) [labelAbove={$-3$},label=below:{$H$}] {};

\node[bullet] (30) at (3,0) [labelAbove={$-2$},label=below:{$H_{p}$}] {};

\node[empty] (350) at (3.5,0) []{};
\node[empty] (40) at (4,0) []{};

\node[bullet] (450) at (4.5,0) [labelAbove={$-2$},label=below:{$H_1$}] {};

\node[bullet] (550) at (5.5,0) [labelAbove={$-1$},label=below:{$E'$}] {};

\node[empty] (60) at (6,0) []{};
\node[empty] (650) at (6.5,0) []{};

\draw[dotted] (0,0.85)--(0,-0.35);

\draw[-] (01)--(10);
\draw[-] (0-05)--(10);
\draw[-] (0-1)--(10);

\draw[-] (10)--(20);
\draw[-] (20)--(30);

\draw[-] (30)--(350);
\draw[dotted] (350)--(40);
\draw[-] (40)--(450);

\draw[-] (450)--(550);

\draw[-] (550)--(60);
\draw[dotted] (60)--(650);
\end{tikzpicture}
\end{equation*}

The degenerations are given by

\begin{align*}
\widetilde{A}_1 &\rightsquigarrow \widetilde{A}_1' + E + H + H_p + \dotsb + H_2 + H_1 \\
\widetilde{A}_2 &\rightsquigarrow \widetilde{A}_2' + E + H + H_p + \dotsb + H_2 \\
\vdots \\
\widetilde{A}_p &\rightsquigarrow \widetilde{A}_1' + E + H + H_p \\
\widetilde{A}_{p+1} &\rightsquigarrow \widetilde{A}_1' + E + H
\end{align*}

So the red $(-1)$-curve $E$ induces the second gray column of $M_3$:

Then we apply a divisorial contraction to the $(-1)$-curve $E$.

Repeating the above process until there is no Wahl singularity $[p+4,2,\dotsc,2]$, we obtain the following $(q+1)$ gray columns of $M_3$:

After that, we have the configuration (combined with Equation (<ref>)):

\begin{equation*}
\begin{tikzpicture}[scale=0.75]

\node[bullet] (-4505) at (-4.5,0.5) [label=left:{$\widetilde{A}_1'$}] {};
\node[bullet] (-45-05) at (-4.5,-0.5) [label=left:{$\widetilde{A}_{p+1}'$}] {};
\node[bullet] (-45-1) at (-4.5,-1) [label=left:{$\widetilde{A}_{p+2}$}] {};

\node[bullet] (-350) at (-3.5,0) [labelAbove={$-2$},label=below:{\phantom{$-$}\ $H_{p+1}$}] {};
\node[empty] (-30) at (-3,0) []{};
\node[empty] (-250) at (-2.5,0) []{};
\node[bullet] (-20) at (-2,0) [labelAbove={$-2$},label=below:{$H_1$}] {};

\node[bullet] (-10) at (-1,0) [labelAbove={\color{red}{$-1$}},label=below:{\color{red}{$E$}}] {};

\draw[-] (00)--(-10);
\draw[-] (-10)--(-20);
\draw[-] (-20)--(-250);
\draw[dotted] (-250)--(-30);
\draw[-] (-30)--(-350);
\draw[-] (-350)--(-4505);
\draw[-] (-350)--(-45-05);
\draw[-] (-350)--(-45-1);
\draw[dotted] (-4.5,0.35)--(-4.5,-0.35);

\node[bullet] (00) at (0,0) [labelAbove={$-p-3$},label=below left:{$F$}] {};

\node[bullet] (105) at (1,0.5) [label=right:{$\widetilde{B}_1'$}] {};
\node[bullet] (1-05) at (1,-0.5) [label=right:{$\widetilde{B}_{q+2}'$}] {};

\node[bullet] (-05-1) at (-0.5,-1) [label=below:{$\widetilde{C}_1$}] {};
\node[bullet] (05-1) at (0.5,-1) [label=below:{$\widetilde{C}_{r+2}$}] {};

\draw[-] (-10)--(00);
\draw[-] (00)--(105);
\draw[-] (00)--(1-05);
\draw[-] (00)--(-05-1);
\draw[-] (00)--(05-1);

\draw[dotted] (1,0.35)--(1,-0.35);
\draw[dotted] (0.35,-1)--(-0.35,-1);

\end{tikzpicture}
\end{equation*}

and the degenerations are given as follows:

\begin{align*}
\widetilde{A}_1 &\rightsquigarrow \widetilde{A}_1' + H_{p+1} + \dotsb + H_2 + H_1 \\
\widetilde{A}_2 &\rightsquigarrow \widetilde{A}_2' + H_{p+1} + \dotsb + H_2 \\
\vdots \\
\widetilde{A}_{p+1} &\rightsquigarrow \widetilde{A}_1' + H_{p+1} \\
\widetilde{B}_1 &\rightsquigarrow \widetilde{B}_1' + F \\
\vdots \\
\widetilde{B}_{q+2} &\rightsquigarrow \widetilde{B}_{q+2}' + F
\end{align*}

The red $(-1)$-curve $E$ induces the column $[1\ 0 \dotsb 0]^T$ of $M_3$. After divisorially contracting it, then the curve $H_1$ becomes another $(-1)$-curve, which gives us the column $[0\ 1 \ 0 \dotsb 0]^T$ of $M_3$. Repeating this process until the curve $F$ becomes a $(-1)$-curve $F'$, we obtain all columns of $M_3$ except the first one; but it is induced by the last $(-1)$-curve $F'$.

In Figure <ref>, we present another P-resolution for $M(3)$ with $B=B(I)$ for $r-p \ge 1$, $q \ge 1$. But we need one more condition $p \ge 1$ in order to guarantee that the canonical divisor is ample. One can also construct a P-resolution for $M(3)$ with $B(II)$ in a similar fashion.

font =

[rectangle] (00) at (0,0) [label=above:$-q-5$] ;

[rectangle] (-350) at (-3.5,0) [label=below:$-(p+3)\phantom{-}$] ;
[rectangle] (-250) at (-2.5,0) [label=above:$-2$] ;
[empty] (-20) at (-2,0) [] ;
[empty] (-150) at (-1.5,0) [] ;
[rectangle] (-10) at (-1,0) [label=above:$-2$] ;

[-] (-350)–(-250);
[-] (-250)–(-20);
[dotted] (-20)–(-150);
[-] (-150)–(-10);
[-] (-10)–(00);
[decorate, decoration = calligraphic brace,mirror] (-2.5,-0.15)–(-1,-0.15)  node[pos=0.5,below=0.1em,black]$q$ $(-2)$'s;

[rectangle] (10) at (1,0) [label=above:$-2$] ;
[empty] (150) at (1.5,0) [] ;
[empty] (20) at (2,0) [] ;
[rectangle] (250) at (2.5,0) [label=above:$-2$] ;

[bullet] (350) at (3.5,0) [label=above:$-1$] ;

[rectangle] (450) at (4.5,0) [label=above:$-p-3$] ;

[rectangle] (550) at (5.5,0) [label=above:$-2$] ;
[empty] (60) at (6,0) [] ;
[empty] (650) at (6.5,0) [] ;
[rectangle] (70) at (7,0) [label=above:$-2$] ;

[rectangle] (80) at (8,0) [label=above:$-3$] ;

[rectangle] (90) at (9,0) [label=above:$-2$] ;
[empty] (950) at (9.5,0) [] ;
[empty] (100) at (10,0) [] ;
[rectangle] (1050) at (10.5,0) [label=above:$-2$] ;

[bullet] (105-1) at (10.5,-1) [label=left:$-2$] ;
[empty] (105-15) at (10.5,-1.5) [] ;
[empty] (105-2) at (10.5,-2) [] ;
[bullet] (105-25) at (10.5,-2.5) [label=left:$-2$] ;

[bullet] (105-35) at (10.5,-3.5) [label=right:$-(q+3)\phantom{-}$] ;

[-] (00)–(10);

[-] (10)–(150);
[dotted] (150)–(20);
[-] (20)–(250);
[decorate, decoration = calligraphic brace,mirror] (1,-0.15)–(2.5,-0.15)  node[pos=0.5,below=0.1em,black]$p+1$;

[-] (250)–(350);
[-] (350)–(450);

[-] (450)–(550);
[-] (550)–(60);
[dotted] (60)–(650);
[-] (650)–(70);
[decorate, decoration = calligraphic brace,mirror] (5.5,-0.15)–(7,-0.15)  node[pos=0.5,below=0.1em,black]$q-1$;

[-] (70)–(80);

[-] (80)–(90);
[-] (90)–(950);
[dotted] (950)–(100);
[-] (100)–(1050);
[decorate, decoration = calligraphic brace,mirror] (9,-0.15)–(10.5,-0.15)  node[pos=0.5,below=0.1em,black]$p$;

[-] (1050)–(105-1);
[-] (105-1)–(105-15);
[dotted] (105-15)–(105-2);
[-] (105-2)–(105-25);
[decorate, decoration = calligraphic brace, mirror] (10.65,-2.5)–(10.65,-1)  node[pos=0.5,right=0.1em,black]$r-p-1$;

[-] (105-25)–(105-35);

[bullet] (0-35) at (0,-3.5) [label=right:$-(r+3)\phantom{-}$] ;
[bullet] (0-25) at (0,-2.5) [label=left:$-2$] ;
[empty] (0-2) at (0,-2) [] ;
[empty] (0-15) at (0,-1.5) [] ;
[bullet] (0-1) at (0,-1) [label=left:$-2$] ;

[-] (0-35)–(0-25);
[-] (0-25)–(0-2);
[dotted] (0-2)–(0-15);
[-] (0-15)–(0-1);
[-] (0-1)–(00);
[decorate, decoration = calligraphic brace,mirror] (0.15,-2.5)–(0.15,-1)  node[pos=0.5,right=0.1em,black]$p$;

Another P-resolution for $M(3)$

§.§ The P-modifications for the incidence matrices $M(6)$ and $M(7)$

A smoothing of $W(p,q,r)$ whose Milnor fiber is a rational homology disk occurs on a one-dimensional smoothing component (called a $\mathbb{Q}$HD component) of the deformation space of $W(p,q,r)$ and that such a smoothing can be chosen to be $\mathbb{Q}$-Gorenstein; Wahl <cit.>. According to Fowler [6] (refer also Wahl [38]), there is only one $\mathbb{Q}$HD component of $W(p,q,r)$ for any $p,q,r$ except $p=q=r$, whose corresponding ample embedding of the compactifying divisor of $W(p,q,r)$ is given in Figure <ref>. If $p=q=r$, there is another $\mathbb{Q}$HD component, whose corresponding embedding is given in Figure <ref>.

[bullet] (00) at (0,0) [labelBelow=$+1$] ;

[bullet] (-10) at (-1,0) [labelBelow=$-(p+2)$] ;
[bullet] (-20) at (-2,0) [] ;

[empty] (-250) at (-2.5,0) [] ;
[empty] (-30) at (-3,0) [] ;

[bullet] (-350) at (-3.5,0) [] ;

[-] (00)–(-10);
[-] (-10)–(-20);
[-] (-20)–(-250);
[dotted] (-250)–(-30);
[-] (-30)–(-350);

[decorate, decoration = calligraphic brace,mirror] (-2,0.15)–(-3.5,0.15)  node[pos=0.5,above]$r+1$;

[bullet] (01) at (0,1) [label=left:$-(q+2)$] ;
[bullet] (02) at (0,2) [] ;

[empty] (025) at (0,2.5) [] ;
[empty] (03) at (0,3) [] ;

[bullet] (035) at (0,3.5) [] ;

[-] (00)–(01);
[-] (01)–(02);
[-] (02)–(025);
[dotted] (025)–(03);
[-] (03)–(035);

[decorate, decoration = calligraphic brace,mirror] (0.15,2)–(0.15,3.5)  node[pos=0.5,right]$p+1$;

[bullet] (10) at (1,0) [labelAbove=$-(r+2)$] ;
[bullet] (20) at (2,0) [] ;

[empty] (250) at (2.5,0) [] ;
[empty] (30) at (3,0) [] ;

[bullet] (350) at (3.5,0) [] ;

[-] (00)–(10);
[-] (10)–(20);
[-] (20)–(250);
[dotted] (250)–(30);
[-] (30)–(350);

[decorate, decoration = calligraphic brace,mirror] (2,-0.15)–(3.5,-0.15)  node[pos=0.5,below]$q+1$;

[circle] (-135) at (-1,3.5) [labelAbove=$-1$] ;
[circle] (-35-1) at (-3.5,-1) [labelBelow=$-1$] ;
[circle] (351) at (3.5,1) [labelAbove=$-1$] ;

[-] (035)–(-135);
[-] (-135)–(-10);

[-] (-350)–(-35-1);
[-] (-35-1)–(1,-1);
[-] (1,-1)–(10);

[-] (350)–(351);
[-] (351)–(01);

The $\mathbb{Q}$HD embedding of the compactifying divisor of $W(p,q,r)$

[bullet] (00) at (0,0) [labelBelow=$+1$] ;

[bullet] (-10) at (-1,0) [labelAbove=$-(p+2)$] ;
[bullet] (-20) at (-2,0) [] ;

[empty] (-250) at (-2.5,0) [] ;
[empty] (-30) at (-3,0) [] ;

[bullet] (-350) at (-3.5,0) [] ;

[-] (00)–(-10);
[-] (-10)–(-20);
[-] (-20)–(-250);
[dotted] (-250)–(-30);
[-] (-30)–(-350);

[decorate, decoration = calligraphic brace] (-2,-0.15)–(-3.5,-0.15)  node[pos=0.5,below]$p+1$;

[bullet] (01) at (0,1) [label=right:$-(p+2)$] ;
[bullet] (02) at (0,2) [] ;

[empty] (025) at (0,2.5) [] ;
[empty] (03) at (0,3) [] ;

[bullet] (035) at (0,3.5) [] ;

[-] (00)–(01);
[-] (01)–(02);
[-] (02)–(025);
[dotted] (025)–(03);
[-] (03)–(035);

[decorate, decoration = calligraphic brace] (-0.15,2)–(-0.15,3.5)  node[pos=0.5,left]$p+1$;

[bullet] (10) at (1,0) [labelBelow=$-(p+2)$] ;
[bullet] (20) at (2,0) [] ;

[empty] (250) at (2.5,0) [] ;
[empty] (30) at (3,0) [] ;

[bullet] (350) at (3.5,0) [] ;

[-] (00)–(10);
[-] (10)–(20);
[-] (20)–(250);
[dotted] (250)–(30);
[-] (30)–(350);

[decorate, decoration = calligraphic brace] (2,0.15)–(3.5,0.15)  node[pos=0.5,above]$p+1$;

[circle] (135) at (1,3.5) [labelAbove=$-1$] ;
[circle] (-351) at (-3.5,1) [labelAbove=$-1$] ;
[circle] (35-1) at (3.5,-1) [labelBelow=$-1$] ;

[-] (035)–(135);
[-] (135)–(10);

[-] (-350)–(-351);
[-] (-351)–(0,1);

[-] (350)–(35-1);
[-] (35-1)–(-1,-1);
[-] (-1,-1)–(-10);

Another $\mathbb{Q}$HD embedding of the compactifying divisor of $W(p,q,r)$ for $p=q=r$

There is a P-modification of $W(p,q,r)$ that corresponds to the (combinatorial) incidence matrix $M(6)$. In case $p=q=r$, there is another P-modification that corresponds to $M(7)$.

For any $p, q, r$ except $p=q=r$, $M(6)$ is the only combinatorial incidence matrix whose Milnor number is zero. Therefore the $\mathbb{Q}$-Gorenstein smoothing occurring on the unique $\mathbb{Q}$HD component must be the P-modification corresponding to $M(6)$.

Suppose that $p=q=r$. There are two $\mathbb{Q}$HD components. One can check that the homology matrices $\Delta^+ M(6)$ and $\Delta^+ M(7)$ correspond to the smoothings of negative weights given in Figure <ref> and Figure <ref>, respectively. Therefore, the $\mathbb{Q}$-Gorenstein smoothings of $W(p,p,p)$ associated to the smoothings of negative weights in given in Figure <ref> and Figure <ref> are the P-modifications correspond to $M(6)$ and $M(7)$, respectively.

One can obtain $M(6)$ from the rational homology disk smoothing of $W(p,q,r)$ via the semi-stable MMP. The key point is that there is a flip along a $(-1)$-curve intersecting the $(-p-3)$-curve. In Figure <ref>, we describe how the central fiber is changed after applying a flip along the flipping curve $E^-$. In a similar fashion, one can describe the flip for $M(7)$ in case $p=q=r$.

[rectangle] (00) at (0,0) [labelAbove=$-4$] ;

[bullet] (-450) at (-4.5,0) [labelAbove=$-1$,label=below:$E^-$];

[rectangle] (-350) at (-3.5,0) [labelAbove=$-(p+3)$] ;
[rectangle] (-250) at (-2.5,0) [labelAbove=$-2$] ;
[empty] (-20) at (-2,0) [] ;
[empty] (-150) at (-1.5,0) [] ;
[rectangle] (-10) at (-1,0) [labelAbove=$-2$] ;

[-] (-450)–(-350);
[-] (-350)–(-250);
[-] (-250)–(-20);
[dotted] (-20)–(-150);
[-] (-150)–(-10);
[-] (-10)–(00);
[decorate, decoration = calligraphic brace,mirror] (-2.5,-0.15)–(-1,-0.15)  node[pos=0.5,below=0.1em,black]$q$;

[rectangle] (350) at (3.5,0) [labelAbove=$-(q+3)$] ;
[rectangle] (250) at (2.5,0) [labelAbove=$-2$] ;
[empty] (20) at (2,0) [] ;
[empty] (150) at (1.5,0) [] ;
[rectangle] (10) at (1,0) [labelAbove=$-2$] ;

[-] (350)–(250);
[-] (250)–(20);
[dotted] (20)–(150);
[-] (150)–(10);
[-] (10)–(00);
[decorate, decoration = calligraphic brace,mirror] (1,-0.15)–(2.5,-0.15)  node[pos=0.5,below=0.1em,black]$r$;

[rectangle] (0-35) at (0,-3.5) [label=left:$-(r+3)$] ;
[rectangle] (0-25) at (0,-2.5) [label=left:$-2$] ;
[empty] (0-2) at (0,-2) [] ;
[empty] (0-15) at (0,-1.5) [] ;
[rectangle] (0-1) at (0,-1) [label=left:$-2$] ;

[-] (0-35)–(0-25);
[-] (0-25)–(0-2);
[dotted] (0-2)–(0-15);
[-] (0-15)–(0-1);
[-] (0-1)–(00);
[decorate, decoration = calligraphic brace,mirror] (0.15,-2.5)–(0.15,-1)  node[pos=0.5,right=0.1em,black]$p$;

[->] (-5.5,0)–(-4.5,0);
at (-5,0) [label=above:flip] ;

[rectangle] (00) at (0,0) [labelAbove=$-4$] ;

[rectangle] (-350) at (-3.5,0) [labelAbove=$-(p+2)$] ;
[rectangle] (-250) at (-2.5,0) [labelAbove=$-2$] ;
[empty] (-20) at (-2,0) [] ;
[empty] (-150) at (-1.5,0) [] ;
[rectangle] (-10) at (-1,0) [labelAbove=$-2$] ;

[-] (-350)–(-250);
[-] (-250)–(-20);
[dotted] (-20)–(-150);
[-] (-150)–(-10);
[-] (-10)–(00);
[decorate, decoration = calligraphic brace,mirror] (-2.5,-0.15)–(-1,-0.15)  node[pos=0.5,below=0.1em,black]$q$;

[rectangle] (350) at (3.5,0) [labelBelow=$-(q+3)$] ;
[rectangle] (250) at (2.5,0) [labelAbove=$-2$] ;
[empty] (20) at (2,0) [] ;
[empty] (150) at (1.5,0) [] ;
[rectangle] (10) at (1,0) [labelAbove=$-2$] ;

[-] (350)–(250);
[-] (250)–(20);
[dotted] (20)–(150);
[-] (150)–(10);
[-] (10)–(00);
[decorate, decoration = calligraphic brace,mirror] (1,-0.15)–(2.5,-0.15)  node[pos=0.5,below=0.1em,black]$r$;

[rectangle] (0-8) at (0,-8) [label=left:$-(r+4)$] ;
[rectangle] (0-7) at (0,-7) [label=left:$-2$] ;
[empty] (0-65) at (0,-6.5) [] ;
[empty] (0-6) at (0,-6) [] ;
[rectangle] (0-55) at (0,-5.5) [label=left:$-2$] ;

[bullet] (0-45) at (0,-4.5) [label=left:$-1$,label=right:$E^+$] ;

[rectangle] (0-35) at (0,-3.5) [label=left:$-(r+3)$] ;
[rectangle] (0-25) at (0,-2.5) [label=left:$-2$] ;
[empty] (0-2) at (0,-2) [] ;
[empty] (0-15) at (0,-1.5) [] ;
[rectangle] (0-1) at (0,-1) [label=left:$-2$] ;

[-] (0-7)–(0-8);
[-] (0-65)–(0-7);
[dotted] (0-6)–(0-65);
[-] (0-55)–(0-6);
[-] (0-45)–(0-55);

[decorate, decoration = calligraphic brace,mirror] (0.15,-7)–(0.15,-5.5)  node[pos=0.5,right=0.1em,black]$r$;

[-] (0-35)–(0-45);

[-] (0-35)–(0-25);
[-] (0-25)–(0-2);
[dotted] (0-2)–(0-15);
[-] (0-15)–(0-1);
[-] (0-1)–(00);
[decorate, decoration = calligraphic brace,mirror] (0.15,-2.5)–(0.15,-1)  node[pos=0.5,right=0.1em,black]$p-1$;

A flip of the rational homology disk smoothing of $W(p,q,r)$

With Jaekwan Jeon, the authors prove Kollár conjecture for all weighted homogeneous surface singularities admitting rational homology disk smoothings in JPS [9] in a similar fashion. Due to the lengthy arguments, however, we decide to write a second paper.

§ MISCELLANEOUS EXAMPLES

We show through several examples that the results obtained in this paper can be applied to various cases.

§.§ An example of using a non-usual flip

Let $(X,p)$ be a weighted homogeneous surface singularity with the dual graph

\begin{equation}\label{equation:WHSS-non-usual-flip}
\begin{aligned}
\begin{tikzpicture}[scale=0.75]
\node[bullet] (-10) at (-1,0) [labelAbove={$-3$}] {};
\node[bullet] (00) at (0,0) [labelAbove={$-5$}] {};
\node[bullet] (10) at (1,0) [labelAbove={$-2$}] {};

\node[bullet] (0-1) at (0,-1) [labelBelow={$-2$}] {};

\draw[-] (-10)--(00)--(10);
\draw[-] (0-1)--(00);
\end{tikzpicture}
\end{aligned}
\end{equation}

We may choose its sandwiched structure as in Figure <ref>(a) in a non-usual way. The decorated curve $(C,l)$ consists of four plane curves $C_1, C_2, C_3, C_4$ whose equations are given by $y^2=a_ix^3$ for $i=1, 2, 3, 4$ and of the decorations given by  $l_1=l_2=l_3=5$ and $l_4=6$. Notice that $C_i \cdot C_j = 6$ for all $i, j$. One can verify that $(X,p)$ has four incidence matrices $A_1, \dotsc, A_4$ and four M-resolutions $M_1, \dotsc, M_4$, which are given in Figure <ref>.

[bullet] (0-3) at (0,-3) [label=below:$\widetilde{C}_4$] ;

[bullet] (0-2) at (0,-2) [label=left:$-1$] ;

[bullet] (0-1) at (0,-1) [label=left:$-2$] ;

[bullet] (-10) at (-1,0) [labelAbove=$-3$] ;
[bullet] (00) at (0,0) [labelAbove=$-5$] ;
[bullet] (10) at (1,0) [labelAbove=$-2$] ;

[bullet] (-11) at (-1,1) [label=left:$-1$] ;
[bullet] (01) at (0,1) [label=left:$-1$] ;
[bullet] (11) at (1,1) [label=left:$-1$] ;

[bullet] (-12) at (-1,2) [label=above:$\widetilde{C}_1$] ;
[bullet] (02) at (0,2) [label=above:$\widetilde{C}_2$] ;
[bullet] (12) at (1,2) [label=above:$\widetilde{C}_3$] ;

[-] (0-3)–(0-2)–(0-1)–(00);

[-] (-10)–(00)–(10);

[-] (00)–(-11);
[-] (00)–(01);
[-] (00)–(11);

[-] (-11)–(-12);
[-] (01)–(02);
[-] (11)–(12);

[] at (0,-4.5) [] (a) A sandwiched structure;

[bullet] (0-3) at (0,-3) [label=below:$\widetilde{C}_4$] ;

[bullet] (0-2) at (0,-2) [label=left:$-1$] ;

[bullet] (0-1) at (0,-1) [label=left:$-2$] ;

[rectangle] (-10) at (-1,0) [labelAbove=$-3$] ;
[rectangle] (00) at (0,0) [labelAbove=$-5$] ;
[rectangle] (10) at (1,0) [labelAbove=$-2$] ;

[bullet] (-11) at (-1,1) [label=left:$-1$,label=above right:$E^-$] ;
[bullet] (01) at (0,1) [label=left:$-1$] ;
[bullet] (11) at (1,1) [label=left:$-1$] ;

[bullet] (-12) at (-1,2) [label=above:$\widetilde{C}_1$] ;
[bullet] (02) at (0,2) [label=above:$\widetilde{C}_2$] ;
[bullet] (12) at (1,2) [label=above:$\widetilde{C}_3$] ;

[-] (0-3)–(0-2)–(0-1)–(00);

[-] (-10)–(00)–(10);

[-] (00)–(-11);
[-] (00)–(01);
[-] (00)–(11);

[-] (-11)–(-12);
[-] (01)–(02);
[-] (11)–(12);

[] at (0,-4.5) [] (b) The compactification $Z$;

An example of using a non-usual flip

\[\begin{aligned}
A_1 &= \begin{bmatrix}
2 & 1 & 1 & 1 & 0 & 0 & 0 \\
2 & 1 & 1 & 0 & 1 & 0 & 0 \\
2 & 1 & 1 & 0 & 0 & 1 & 0 \\
2 & 1 & 1 & 0 & 0 & 0 & 1
\end{bmatrix} &
A_2 &= \begin{bmatrix}
2 & 1 & 1 & 1 & 0 & 0 \\
2 & 1 & 1 & 0 & 1 & 0 \\
2 & 1 & 0 & 1 & 1 & 0 \\
2 & 0 & 1 & 1 & 1 & 1
\end{bmatrix}\\
A_3 &= \begin{bmatrix}
2 & 1 & 1 & 1 & 0 & 0 \\
2 & 1 & 1 & 0 & 1 & 0 \\
2 & 1 & 1 & 0 & 1 & 1 \\
1 & 2 & 1 & 1 & 1 & 1
\end{bmatrix} &
A_4 &= \begin{bmatrix}
2 & 1 & 1 & 1 & 0 \\
1 & 2 & 1 & 1 & 0 \\
1 & 1 & 2 & 1 & 0 \\
1 & 1 & 1 & 2 & 1
\end{bmatrix}
\end{aligned}\]

[empty] (-11) at (-1,1) []$M_1$;

[bullet] (-10) at (-1,0) [labelAbove=$-3$] ;
[bullet] (00) at (0,0) [labelAbove=$-5$] ;
[bullet] (10) at (1,0) [labelAbove=$-2$] ;

[bullet] (0-1) at (0,-1) [labelBelow=$-2$] ;

[-] (-10)–(00)–(10);
[-] (0-1)–(00);

[empty] (-11) at (-1,1) []$M_2$;

[bullet] (-10) at (-1,0) [labelAbove=$-3$] ;
[rectangle] (00) at (0,0) [labelAbove=$-5$] ;
[rectangle] (10) at (1,0) [labelAbove=$-2$] ;

[bullet] (0-1) at (0,-1) [labelBelow=$-2$] ;

[-] (-10)–(00)–(10);
[-] (0-1)–(00);

[empty] (-11) at (-1,1) []$M_3$;

[bullet] (-10) at (-1,0) [labelAbove=$-3$] ;
[rectangle] (00) at (0,0) [labelAbove=$-5$] ;
[bullet] (10) at (1,0) [labelAbove=$-2$] ;

[rectangle] (0-1) at (0,-1) [labelBelow=$-2$] ;

[-] (-10)–(00)–(10);
[-] (0-1)–(00);

[empty] (-11) at (-1,1) []$M_4$;

[rectangle] (-10) at (-1,0) [labelAbove=$-3$] ;
[rectangle] (00) at (0,0) [labelAbove=$-5$] ;
[rectangle] (10) at (1,0) [labelAbove=$-2$] ;

[bullet] (0-1) at (0,-1) [labelBelow=$-2$] ;

[-] (-10)–(00)–(10);
[-] (0-1)–(00);

Four incidence matrices and four M-resolutions

One can check that the M-resolutions $M_1, M_2, M_3$ correspond to the incidence matrices $A_1, A_2, A_3$. We will show that the M-resolution $M_4$ corresponds to the incidence matrix $A_4$. Nonetheless, $M_4$ requires a non-usual flip.

The mk1A $[3,\overline{5},2]$ is of flipping type and its flips is $[4]-1-[5,2]$. If $C$ is an irreducible curve intersecting with the flipping $(-1)$-curve $E^-$ of $[3,\overline{5},2]$, then, after the flip, the curve $C$ in a general fiber is degenerated to $C'+2E^+$ in the central fiber; that is,

\begin{equation*}
C \rightsquigarrow C'+2E^+
\end{equation*}

By Proposition <ref>, the mk1A $[3,\overline{5},2]$ is converted to the mk2A $[3,5,2]-1-[4]$. One can check that this mk2A is an initial one by Proposition <ref>. By the same proposition, one can show that the flip of $[3,\overline{5},2]$ is the M-resolution $[4]-1-[5,2]$.

It remains to show that the degeneration coefficient $\beta$ (defined in Proposition <ref> is equal $2$.

As in Equation (<ref>), we have

\begin{equation*}
K_Z \cdot C = K_{Z_t} \cdot = K_{Z^+_t} \cdot C = K_{Z^+} \cdot (C' + \beta E^+)
\end{equation*}

Let $\pi \colon \widetilde{Z} \to Z$ and $\pi^+ \colon \widetilde{Z}^+ \to Z^+$ be the minimal resolutions of the singularities in $Z$ and $Z^+$, respectively. At first we have

\begin{equation*}
K_Z \cdot C = K_{\widetilde{Z}} \cdot C.
\end{equation*}

Next, we have

\begin{align*}
K_{Z^+} \cdot C' &= (\pi^+)^{\ast}(K_{Z^+}) \cdot C' \\
K_{Z^+} \cdot E^+ &= (\pi^+)^{\ast}(K_{Z^+}) \cdot E^+
\end{align*}

From Figure <ref>(c), we have

\begin{equation*}
(\pi^+)^{\ast}(K_{Z^+}) = K_{\widetilde{Z}^+} + \frac{1}{2} E_1 + \frac{2}{3} E_2 + \frac{1}{3} E_3
\end{equation*}

\begin{equation*}
K_{Z^+} \cdot (C' + \beta E^+) = K_{\widetilde{C}^+} \cdot C' - \frac{2}{3} + \beta \left(-1 + \frac{2}{3} + \frac{1}{2}  \right)
\end{equation*}

On the other hand, the birational morphism $\widetilde{Z} \dashrightarrow \widetilde{Z}^+$ is a composition of a blow-down followed by one blow-up. In particular, the morphism $C \to C'$ is just a blow-up. So

\begin{equation*}
K_{\widetilde{Z}} \cdot C = K_{\widetilde{Z}^+} \cdot C' - 1
\end{equation*}

Therefore, combining the above equations, we have $\beta=2$.

Let $Z$ the compactified M-resolution of $M_4$, which is given in Figure <ref>(b). In Figure <ref>, we present the procedure for identifying the corresponding incidence matrix from $Z$ via the semi-stable MMP. However we draw only central fibers for simplicity. Each step in the figure is fairly straightforward. However, since there is a problem with multiplicities during degenerations caused by flips, we will explain each steps briefly.

In Step 1, the red $(-1)$-curve induces the column $[0\ 0\ 0\ 1]^T$ of $A_4$. We can apply the divisorial contraction to it. In Step 2, we apply a flip to the red $(-1)$-curve. By Lemma <ref>, we have Step 3. Notice that the flip induces the following degeneration

\begin{equation*}
\widetilde{C}_1 \rightsquigarrow \widetilde{C}_1' + 2(1)
\end{equation*}

where $(1)$ denotes the $(-1)$-curve in Step 3 and $2$ denotes its degeneration coefficient $\beta$.

We apply a usual flip to the red $(-1)$-curve in Step 3. In Step 4, the new degenerations occur:

\begin{align*}
\widetilde{C}_1 &\rightsquigarrow \widetilde{C}_1' + 2(1) + (2) \\
\widetilde{C}_2 &\rightsquigarrow \widetilde{C}_2' + (2)
\end{align*}

After applying a usual flip again, we get Step 5 where $\widetilde{C}_3$ is also degenerated:

\begin{align*}
\widetilde{C}_1 &\rightsquigarrow \widetilde{C}_1' + 2(1) + 2(3) + (2) \\
\widetilde{C}_2 &\rightsquigarrow \widetilde{C}_2' + (3) + (2) \\
\widetilde{C}_3 &\rightsquigarrow \widetilde{C}_3' + (3)
\end{align*}

Then the red $(-1)$-curve induces the column $[2\ 1\ 1\ 1]^T$ of the incidence matrix $A_4$.

Applying a divisorial contraction to the red $(-1)$-curve, we get the red $(-1)$-curve in Step 6, which can be flipped. In Step 7, we get a non-singular central fiber. Over there the degenerations are as follows.

\begin{align*}
\widetilde{C}_1 &\rightsquigarrow \widetilde{C}_1' + (3) + 2(1) + (2) \\
\widetilde{C}_2 &\rightsquigarrow \widetilde{C}_2' + (3) + (2) + (2) \\
\widetilde{C}_3 &\rightsquigarrow \widetilde{C}_3' + (3) + (1)
\end{align*}

Here the red $(-1)$-curve induces the column $[1\ 2\ 1\ 1]^T$ of $A_4$.

Step 8 is obtained by applying a divisorial contraction. Here we draw the curves as lines because we need how they are intersect with each other explicitly. Since the degenerations are given by

\begin{align*}
\widetilde{C}_1 &\rightsquigarrow \widetilde{C}_1' + (2) + (1) \\
\widetilde{C}_2 &\rightsquigarrow \widetilde{C}_2' + (2) + (1) \\
\widetilde{C}_3 &\rightsquigarrow \widetilde{C}_3' + (2)
\end{align*}

the red $(-1)$-curve in Step 8 induces the column $[1 \ 1 \ 2 \ 1]^T$ of $A_4$.

Finally, we again contract the $(-1)$-curve, then we have Step 9 with degenerations:

\begin{align*}
\widetilde{C}_1 &\rightsquigarrow \widetilde{C}_1' + (1) \\
\widetilde{C}_2 &\rightsquigarrow \widetilde{C}_2' + (1) \\
\widetilde{C}_3 &\rightsquigarrow \widetilde{C}_3' + (1)
\end{align*}

Notice that all curves $\widetilde{C}_i$ intersects with the red $(-1)$-curve twice at the same point. Therefore the $(-1)$-curves induces the column $[1\ 1\ 1\ 2]^T$ of $A_4$. That's all. A final divisorial contraction gives us the picture deformation corresponding to the incidence matrix $A_4$.

The procedure for identifying the incidence matrix

Kollár conjecture holds for a weighted homogeneous surface singularity whose dual graph is given in Equation (<ref>).

There are four components in $\mathcal{C}(X)$ because the map $\phi_I \colon \mathcal{C}(X) \to \mathcal{I}(X)$ is injective by Theorem <ref> (if we choose an usual sandwiched structure of $X$). In the above we find exactly four P-resolutions corresponding to each components of $\mathcal{C}(X)$. Therefore Kollár conjecture holds as asserted.

§.§ An example with a non-P-resolution

We investigate Example 6.3.6 in Kollár [12]. Let $(X,p)$ be a rational surface singularity with the following dual resolution graph:

\begin{equation}\label{equation:non-T-graph}
\begin{aligned}
\begin{tikzpicture}[scale=0.75]
\node[bullet] (01) at (0,1) [labelAbove={$-2$}] {};
\node[bullet] (00) at (0,0) [label=below left:{$-3$}] {};
\node[bullet] (0-1) at (0,-1) [labelBelow={$-2$}] {};

\node[bullet] (-20) at (-2,0) [labelBelow={$-2$}] {};
\node[bullet] (-10) at (-1,0) [labelBelow={$-2$}] {};
\node[bullet] (10) at (1,0) [labelBelow={$-4$}] {};

\draw[-] (01)--(00)--(0-1);
\draw[-] (-20)--(-10)--(00)--(10);
\end{tikzpicture}
\end{aligned}
\end{equation}

It is a sandwiched surface singularity. We attach $(-1)$-vertices on the $(-2)$-vertices in North and South, respectively. Consider small curvetta, say $C_1$ and $C_2$, passing through the $(-1)$-curves, respectively. Then we have a decorated curve $(C=C_1 \cup C_2, l=\{l_1=8, l_2=8\})$ of $(X,p)$. Indeed the decorated curve can be realized as plane curve singularities, for example, $C_1=y^3-x^4$ and $C_2=y^3-2x^4$. Notice that $\delta(C_i)=3$ for $i=1,2$ and $C_1 \cdot C_2 = 12$.

Using the necessary conditions on incidence matrices in Equation (<ref>), one can show that there are exactly six combinatorial incidence matrices:

\begin{align*}
A_1 &=\begin{bmatrix}
1 & 0 & 1 & 0 & 1 & 1 & 1 & 3 \\
0 & 1 & 0 & 1 & 1 & 1 & 1 & 3
\end{bmatrix} &
A_2 &=\begin{bmatrix}
1 & 0 & 1 & 0 & 2 & 2 & 2\\
0 & 1 & 0 & 1 & 2 & 2 & 2
\end{bmatrix} \\
A_3 &=\begin{bmatrix}
1 & 0 & 2 & 1 & 2 & 2 \\
0 & 1 & 1 & 2 & 2 & 2
\end{bmatrix} &
A_4 &=\begin{bmatrix}
0 & 1 & 1 & 2 & 2 & 3 \\
1 & 1 & 1 & 1 & 1 & 3
\end{bmatrix} \\
A_5 &=\begin{bmatrix}
1 & 1 & 1 & 1 & 1 & 3 \\
0 & 1 & 1 & 2 & 2 & 3
\end{bmatrix} &
A_6 &=\begin{bmatrix}
2 & 2 & 2 & 1 & 1\\
2 & 1 & 1 & 2 & 2
\end{bmatrix}
\end{align*}

On the other hand, there are five P-resolutions dominated by its minimal resolution described in the below and their corresponding M-resolutions are given in Figure <ref>:

* The minimal Du Val resolution.

* Contract all $(-2)$-curves and the $(-4)$-curve.

* Contract any of the three configurations $2-3-4$ and the $(-2)$-curve on the left if necessary.

[empty] (-21) at (-2,1) []$M_1$;

[bullet] (01) at (0,1) [labelAbove=$-2$] ;
[bullet] (00) at (0,0) [label=below left:$-3$] ;
[bullet] (0-1) at (0,-1) [labelBelow=$-2$] ;

[bullet] (-20) at (-2,0) [labelBelow=$-2$] ;
[bullet] (-10) at (-1,0) [labelBelow=$-2$] ;
[bullet] (10) at (1,0) [labelBelow=$-4$] ;

[-] (01)–(00)–(0-1);
[-] (-20)–(-10)–(00)–(10);

[empty] (-21) at (-2,1) []$M_2$;

[bullet] (01) at (0,1) [labelAbove=$-2$] ;
[bullet] (00) at (0,0) [label=below left:$-3$] ;
[bullet] (0-1) at (0,-1) [labelBelow=$-2$] ;

[bullet] (-20) at (-2,0) [labelBelow=$-2$] ;
[bullet] (-10) at (-1,0) [labelBelow=$-2$] ;
[rectangle] (10) at (1,0) [labelBelow=$-4$] ;

[-] (01)–(00)–(0-1);
[-] (-20)–(-10)–(00)–(10);

[empty] (-21) at (-2,1) []$M_3$;

[bullet] (01) at (0,1) [labelAbove=$-2$] ;
[rectangle] (00) at (0,0) [label=below left:$-5$] ;
[bullet] (0-1) at (0,-1) [labelBelow=$-2$] ;

[bullet] (-20) at (-2,0) [labelBelow=$-2$] ;
[rectangle] (-10) at (-1,0) [labelBelow=$-2$] ;
[bullet] (10) at (1,0) [labelBelow=$-1$] ;
[rectangle] (20) at (2,0) [labelBelow=$-2$] ;
[rectangle] (30) at (3,0) [labelBelow=$-5$] ;

[-] (01)–(00)–(0-1);
[-] (-20)–(-10)–(00)–(10)–(20)–(30);

[empty] (-21) at (-2,1) []$M_4$;

[rectangle] (01) at (0,1) [labelAbove=$-2$] ;
[rectangle] (00) at (0,0) [label=below left:$-5$] ;
[bullet] (0-1) at (0,-1) [labelBelow=$-2$] ;

[bullet] (-20) at (-2,0) [labelBelow=$-2$] ;
[bullet] (-10) at (-1,0) [labelBelow=$-2$] ;
[bullet] (10) at (1,0) [labelBelow=$-1$] ;
[rectangle] (20) at (2,0) [labelBelow=$-2$] ;
[rectangle] (30) at (3,0) [labelBelow=$-5$] ;

[-] (01)–(00)–(0-1);
[-] (-20)–(-10)–(00)–(10)–(20)–(30);

[empty] (-21) at (-2,1) []$M_5$;

[bullet] (01) at (0,1) [labelAbove=$-2$] ;
[rectangle] (00) at (0,0) [label=below left:$-5$] ;
[rectangle] (0-1) at (0,-1) [labelBelow=$-2$] ;

[bullet] (-20) at (-2,0) [labelBelow=$-2$] ;
[bullet] (-10) at (-1,0) [labelBelow=$-2$] ;
[bullet] (10) at (1,0) [labelBelow=$-1$] ;
[rectangle] (20) at (2,0) [labelBelow=$-2$] ;
[rectangle] (30) at (3,0) [labelBelow=$-5$] ;

[-] (01)–(00)–(0-1);
[-] (-20)–(-10)–(00)–(10)–(20)–(30);

Five M-resolutions

There is also a one-parameter family of P-modifications given in Kollár <cit.>. We first blow up any point $x$ on the $(-3)$-curve that is not intersection points. We have a $(-1)$-curve and the following configuration.

\begin{equation}\label{equation:M6-configuration}
\begin{aligned}
\begin{tikzpicture}[scale=0.75]
\tikzset{label distance=-0.15em}
\tikzset{font=\scriptsize}
\node[bullet] (01) at (0,1) [labelAbove={$-2$}] {};
\node[bullet] (00) at (0,0) [label=below left:{$-4$}] {};
\node[bullet] (0-1) at (0,-1) [labelBelow={$-2$}] {};

\node[bullet] (-20) at (-2,0) [labelBelow={$-2$}] {};
\node[bullet] (-10) at (-1,0) [labelBelow={$-2$}] {};
\node[bullet] (10) at (1,0) [labelBelow={$-4$}] {};

\draw[-] (01)--(00)--(0-1);
\draw[-] (-20)--(-10)--(00)--(10);
\end{tikzpicture}
\end{aligned}
\end{equation}

Any singularity with the dual resolution graph as in Equation (<ref>) has a $\mathbb{Q}$-Gorenstein component.

We briefly summary the proof in Kollár <cit.>. Any singularity with the dual graph (<ref>) is a quotient of the elliptic singularity $\{x^2z + xy^2 + z^5 + ay^2z^2 =0 \} \subset \mathbb{C}^3$ by the $\mathbb{Z}_5$-action with weight $(1,2,3)$. Then the $\mathbb{Q}$-Gorenstein smoothing is given by

\begin{equation*}
\{x^2z + xy^2 + z^5 + ay^2z^2 + t=0 \} / \mathbb{Z}_5. \qedhere
\end{equation*}

We contract all the other curves except the $(-1)$-curve, which induces the following P-modification $M_6$.

\begin{equation*}
\begin{tikzpicture}[scale=0.75]
\tikzset{label distance=-0.15em}
\tikzset{font=\scriptsize}
\node[empty] (-21) at (-2,1) []{$M_6$};

\node[rectangle] (01) at (0,1) [labelAbove={$-2$}] {};
\node[rectangle] (00) at (0,0) [label=below left:{$-4$}] {};
\node[rectangle] (0-1) at (0,-1) [labelBelow={$-2$}] {};

\node[rectangle] (-20) at (-2,0) [labelBelow={$-2$}] {};
\node[rectangle] (-10) at (-1,0) [labelBelow={$-2$}] {};
\node[rectangle] (10) at (1,0) [labelBelow={$-4$}] {};

\node[bullet] (0505) at (0.5,0.5) [labelAbove={$-1$}] {};

\draw[-] (01)--(00)--(0-1);
\draw[-] (-20)--(-10)--(00)--(10);
\draw[-] (00)--(0505);
\end{tikzpicture}
\end{equation*}

§.§ The correspondence

It is not difficult to show that, for each $i=1,\dotsc,5$, the M-resolution $M_i$ corresponds to the incidence matrix $A_i$. On the other hand, we can also apply the semi-stable MMP machinery to the M-resolution $M_6$ even though it has a singularity which is not a T-singularity; Proposition <ref>.

The Milnor number of the $\mathbb{Q}$-Gorenstein smoothing in Lemma <ref> is $3$.

At first, let $H=\{f(x,y,z)=x^2z + xy^2 + z^5 + ay^2z^2 =0 \}$. Since it is a hypersurface singularity, its Milnor number is given by the length of the Jacobian algebra

\begin{equation*}
\begin{split}
J &= \mathbb{C}[[x,y,z]]/\langle \partial f/\partial x, \partial f/\partial y, \partial f/\partial z \rangle \\
&= \mathbb{C}[[x,y,z]]/\langle 2xz+y^2, 2xy+2ayz^2, x^2+5z^4+2ay^2z\rangle.
\end{split}
\end{equation*}

Then one can check that its length is $19$.

Let $H_t=\{f(x,y,z)=x^2z + xy^2 + z^5 + ay^2z^2 +t =0 \}$ for a fixed $t \neq 0$. If $M$ is the Milnor fiber of the $\mathbb{Q}$-Gorenstein smoothing in Lemma <ref>, then there is a unramified connected covering $H_t \to M$ of degree $5$. Then, since $b_1(M)=b_1(H_t)=0$, we have

\begin{equation*}
\end{equation*}

Therefore $b_2(M)=3$.

Let $(E^{-} \subset \mathcal{Z}) \to (Q \in \mathcal{Y})$ be an extremal neighborhood given by the following dual graph

\begin{equation*}
\begin{tikzpicture}[scale=0.75]
\tikzset{label distance=-0.15em}
\tikzset{font=\scriptsize}

%\node[bullet] (03) at (0,3) [label=left:{$C$}] {};
\node[bullet] (02) at (0,2) [label=left:{$-1$},label=right:{$E^{-}$}] {};
\node[rectangle] (01) at (0,1) [label=left:{$-2$}] {};
\node[rectangle] (00) at (0,0) [label=below left:{$-4$}] {};
\node[rectangle] (0-1) at (0,-1) [labelBelow={$-2$}] {};

\node[rectangle] (-20) at (-2,0) [labelBelow={$-2$}] {};
\node[rectangle] (-10) at (-1,0) [labelBelow={$-2$}] {};
\node[rectangle] (10) at (1,0) [labelBelow={$-4$}] {};

\draw[-] (02)--(01)--(00)--(0-1);
\draw[-] (-20)--(-10)--(00)--(10);
\end{tikzpicture}
\end{equation*}

Then there is a flip $(E^+ \subset \mathcal{Z}^+)$ given by the dual graph

\begin{equation}\label{equation:M6-flip}
\begin{aligned}
\begin{tikzpicture}[scale=0.75]
\tikzset{label distance=-0.15em}
\tikzset{font=\scriptsize}

%\node[bullet] (03) at (0,3) [label=left:{$C$}] {};
%\node[bullet] (02) at (0,2) [label=left:{$-1$},label=right:{$E^{-}$}] {};
%\node[rectangle] (01) at (0,1) [label=left:{$-2$}] {};
\node[rectangle] (00) at (0,0) [label=below left:{$-3$}] {};
\node[rectangle] (0-1) at (0,-1) [labelBelow={$-2$}] {};

\node[rectangle] (-20) at (-2,0) [labelBelow={$-2$}] {};
\node[bullet] (-10) at (-1,0) [labelBelow={$-2$},labelAbove={$E^+$}] {};
\node[rectangle] (10) at (1,0) [labelBelow={$-4$}] {};

\draw[-] (00)--(0-1);
\draw[-] (-20)--(-10)--(00)--(10);
\end{tikzpicture}
\end{aligned}
\end{equation}

The singularity $Q \in Y$ has a dual graph

\begin{equation*}
\begin{tikzpicture}[scale=0.75]
\tikzset{label distance=-0.15em}
\tikzset{font=\scriptsize}

%\node[bullet] (03) at (0,3) [label=left:{$C$}] {};
%\node[bullet] (02) at (0,2) [label=left:{$-1$},label=right:{$E^{-}$}] {};
%\node[rectangle] (01) at (0,1) [label=left:{$-2$}] {};
\node[bullet] (00) at (0,0) [label=below left:{$-3$}] {};
\node[bullet] (0-1) at (0,-1) [labelBelow={$-2$}] {};

\node[bullet] (-20) at (-2,0) [labelBelow={$-2$}] {};
\node[bullet] (-10) at (-1,0) [labelBelow={$-2$}] {};
\node[bullet] (10) at (1,0) [labelBelow={$-4$}] {};

\draw[-] (00)--(0-1);
\draw[-] (-20)--(-10)--(00)--(10);
\end{tikzpicture}
\end{equation*}

and the extremal neighborhood $(E^{-} \subset \mathcal{Z})$ induces a smoothing $\mathcal{Y} \to \Delta$ of $Q \in Y$ whose Milnor number is $3$ by Lemma <ref>.

Notice that $Q \in Y$ is a sandwiched surface singularity. If we choose its sandwiched $(-1)$-curves by one attached to the $(-3)$-curve and the other one attached to the $(-2)$-curve in south position, then we have four incidence matrices

\begin{align*}
A_1' &=\begin{bmatrix}
0 & 1 & 0 & 1 & 1 & 1 & 3 \\
1 & 0 & 1 & 1 & 1 & 1 & 3
\end{bmatrix} &
A_2' &=\begin{bmatrix}
0 & 1 & 0 & 2 & 2 & 2\\
1 & 0 & 1 & 2 & 2 & 2
\end{bmatrix} \\
A_3' &=\begin{bmatrix}
0 & 2 & 1 & 2 & 2 \\
1 & 1 & 2 & 2 & 2
\end{bmatrix} &
A_5' &=\begin{bmatrix}
1 & 1 & 1 & 1 & 3 \\
1 & 1 & 2 & 2 & 3
\end{bmatrix}
\end{align*}

Here $A_3'$ and $A_5'$ correspond to one-parameter smoothings of $Q \in Y$ with Milnor number $3$. But one can check via the MMP machinery in the paper that the following $P$-resolutions $P_3'$ and $P_5'$ correspond to $A_3'$ and $A_5'$, respectively:

\begin{equation*}
\begin{tikzpicture}[scale=0.75]
\tikzset{label distance=-0.15em}
\tikzset{font=\scriptsize}

\begin{scope}[shift={(0,0)}]
\node[empty] (-21) at (-2,0.5) []{$P_3'$};

\node[rectangle] (00) at (0,0) [label=below left:{$-3$}] {};
\node[bullet] (0-1) at (0,-1) [labelBelow={$-2$}] {};

\node[bullet] (-20) at (-2,0) [labelBelow={$-2$}] {};
\node[rectangle] (-10) at (-1,0) [labelBelow={$-2$}] {};
\node[rectangle] (10) at (1,0) [labelBelow={$-4$}] {};

\draw[-] (00)--(0-1);
\draw[-] (-20)--(-10)--(00)--(10);
\end{scope}

\begin{scope}[shift={(5,0)}]
\node[empty] (-21) at (-2,0.5) []{$P_5'$};

\node[rectangle] (00) at (0,0) [label=below left:{$-3$}] {};
\node[rectangle] (0-1) at (0,-1) [labelBelow={$-2$}] {};

\node[rectangle] (-20) at (-2,0) [labelBelow={$-2$}] {};
\node[bullet] (-10) at (-1,0) [labelBelow={$-2$}] {};
\node[rectangle] (10) at (1,0) [labelBelow={$-4$}] {};

\draw[-] (00)--(0-1);
\draw[-] (-20)--(-10)--(00)--(10);
\end{scope}
\end{tikzpicture}
\end{equation*}

According to Kollár–Mori <cit.>, the exceptional curve $E^+$ is irreducible. Therefore the flip of the extremal neighborhood $(E^{-} \subset \mathcal{Z}) \to (Q \in \mathcal{Y})$ is given by Equation (<ref>) as asserted.

Let $C$ be an irreducible curve contained in the central fiber of the extremal neighborhood $(E^{-} \subset \mathcal{Z})$ in Lemma <ref> such that $C \cdot E^{-}=1$ and $C$ does not pass through the singular point. Let $C'$ be the proper transform of $C$ after the flip along $E^{-}$. Then the curve $C$ in the general fiber $Z^+_t$ ($t \neq 0$) degenerates to $C'+2E^{+}$ in the central fiber $Z^+_0$.

Suppose that $C$ in a general fiber $Z^+_t$ degenerates to $C'+\beta E^+$ in the central fiber $Z^+_0$. By Proposition <ref>, we have

\begin{equation*}
K_{Z_0} \cdot C = K_{Z_t} \cdot C = K_{Z^+_t} \cdot {C} = K_{Z^+_0} \cdot (C'+\beta E^+) = K_{Z^+_0} \cdot C' + \beta K_{Z^+_0} \cdot E^+.
\end{equation*}

Let $\pi \colon \widetilde{Z}_0 \to Z_0$ and $\pi^+ \colon \widetilde{Z}^+_0 \to Z^+_0$ be the minimal resolutions, respectively. Let $\widetilde{C}$ and $\widetilde{C}'$ be the proper transforms of $C$ and $C'$, respectively. Then we have

\begin{equation*}
K_{Z_0} \cdot C = K_{\widetilde{Z}_0} \cdot \widetilde{C}.
\end{equation*}

On the other hand, notice that

\begin{equation*}
(\pi^+)^{\ast} K_{Z^+_0} = K_{\widetilde{Z}^+_0} + \frac{1}{3} E_1 + \frac{2}{3} E_2 + \frac{2}{3} E_3
\end{equation*}

where $[E_1, E_2, E_3]$ are the exceptional divisors $[2,3,4]$. So we have

\begin{equation*}
K_{Z^+_0} \cdot C' + \beta K_{Z^+_0} \cdot E^+ = K_{\widetilde{Z}^+_0} \cdot \widetilde{C}' + \frac{2}{3} + \frac{2}{3} \beta.
\end{equation*}

But $\widetilde{Z}_0 \to \widetilde{Z}^+_0$ is a composition of two blowing-ups. Hence

\begin{equation*}
K_{\widetilde{Z}^+_0} \widetilde{C}' = K_{Z_0} \cdot C = K_{\widetilde{Z}_0} \cdot \widetilde{C} - 2
\end{equation*}

Therefore $\beta=2$.

The 6th P-modification $M_6$ corresponds to the 6th incidence matrix $A_6$.

A P-modification $M_6$ with a non-T-singularity

We present in Figure <ref> the process of the semi-stable minimal model program for identifying the picture deformation corresponding to the P-modification $M_6$. We briefly explain each steps. In Step 1, we apply the non-usual flip described in Lemma <ref>. Then we have Step 2. By Lemma <ref>, we have a degeneration

\begin{equation*}
\widetilde{C}_1 \rightsquigarrow \widetilde{C}_1' + 2(2)
\end{equation*}

where $(2)$ denotes the $(-2)$-curve in Step 2. We illustrate the degeneration of $\widetilde{C}_1$ by enclosing them with blue lines. Then, we compute the crepant M-resolution of Step 2. In Step 3, we perform a usual flip on $E^-$. Then $\widetilde{C}_2$ is degenerated, as shown by the orange lines in Step 4. We apply again a usual flip to $E^-$ in Step 4. Then we resolve all T-singularities. Each $(-1)$-curves will now generate columns of $A_6$, which is easily calculable.

Kollár conjecture for this singularity if we can prove that $\phi_I \colon \mathcal{C}(X) \to \mathcal{I}(X)$ is injective. But we don't know how to prove it yet.

§.§ An example with a nonnormal P-modification

We investigate Example 6.6.3 in Kollár [12]. Let $(X,p)$ be a rational surface singularity whose dual graph of the minimal resolution is given by

\begin{equation*}
\begin{tikzpicture}[scale=0.75]
\tikzset{label distance=-0.15em}
\tikzset{font=\scriptsize}
\node[bullet] (01) at (0,1) [labelAbove={$-2$}] {};
\node[bullet] (00) at (0,0) [label=below left:{$-4$}] {};
\node[bullet] (0-1) at (0,-1) [labelBelow={$-2$}] {};

\node[bullet] (-10) at (-1,0) [labelBelow={$-3$}] {};
\node[bullet] (10) at (1,0) [labelBelow={$-2$}] {};
\node[bullet] (20) at (2,0) [labelBelow={$-2$}] {};

\draw[-] (01)--(00)--(0-1);
\draw[-] (-10)--(00)--(10)--(20);
\end{tikzpicture}
\end{equation*}

It is a sandwiched surface singularity and its sandwiched structure is given in Figure <ref>.

[bullet] (01) at (0,1) [label=left:$-2$] ;
[bullet] (00) at (0,0) [label=below left:$-4$] ;
[bullet] (0-1) at (0,-1) [label=left:$-2$] ;

[bullet] (-10) at (-1,0) [labelBelow=$-3$] ;
[bullet] (10) at (1,0) [labelBelow=$-2$] ;
[bullet] (20) at (2,0) [labelBelow=$-2$] ;

[bullet] (30) at (3,0) [labelBelow=$-1$] ;
[bullet] (40) at (4,0) [labelBelow=$C_1$] ;

[bullet] (02) at (0,2) [label=left:$-1$] ;
[bullet] (03) at (0,3) [label=left:$C_2$] ;

[bullet] (0-2) at (0,-2) [label=left:$-1$] ;
[bullet] (0-3) at (0,-3) [label=left:$C_3$] ;

[bullet] (-20) at (-2,0) [labelBelow=$-1$] ;
[bullet] (-30) at (-3,0) [labelBelow=$C_4$] ;

[-] (03)–(02)–(01)–(00)–(0-1)–(0-2)–(0-3);
[-] (-30)–(-20)–(-10)–(00)–(10)–(20)–(30)–(40);

A sandwiched structure

The decorated curve is given by $(C_1 \cup C_2 \cup C_3 \cup C_4, \{5,4,4,2\})$ and they satisfy the following incidence relations: $C_i \cdot C_j =2$ for $i \neq j$ with $i, j \le 3$ and $C_k \cdot C_4=1$ for all $k \le 3$:

There are six picture deformations of $(X,p)$. In Figure <ref>, we present the decorated curves in a general fiber for each picture deformations. On the other hand, there are also six P-modifications: Fives are P-resolutions, but the sixth is a nonnormal P-modification. We present the dual graphs of five P-resolutions in Figure <ref>.

Six picture deformations

[empty] (-11) at (-1,1) []$\fbox{1}$;

[rectangle] (01) at (0,1) [labelAbove=$-2$] ;
[bullet] (00) at (0,0) [label=below left:$-4$] ;
[rectangle] (0-1) at (0,-1) [labelBelow=$-2$] ;

[bullet] (-10) at (-1,0) [labelBelow=$-3$] ;
[rectangle] (10) at (1,0) [labelBelow=$-2$] ;
[rectangle] (20) at (2,0) [labelBelow=$-2$] ;

[-] (01)–(00)–(0-1);
[-] (-10)–(00)–(10)–(20);

[empty] (-11) at (-1,1) []$\fbox{2}$;

[bullet] (01) at (0,1) [labelAbove=$-2$] ;
[rectangle] (00) at (0,0) [label=below left:$-4$] ;
[bullet] (0-1) at (0,-1) [labelBelow=$-2$] ;

[bullet] (-10) at (-1,0) [labelBelow=$-3$] ;
[bullet] (10) at (1,0) [labelBelow=$-2$] ;
[rectangle] (20) at (2,0) [labelBelow=$-2$] ;

[-] (01)–(00)–(0-1);
[-] (-10)–(00)–(10)–(20);

[empty] (-11) at (-1,1) []$\fbox{3}$;

[rectangle] (01) at (0,1) [labelAbove=$-2$] ;
[rectangle] (00) at (0,0) [label=below left:$-5$] ;
[bullet] (0-1) at (0,-1) [labelBelow=$-2$] ;

[rectangle] (-20) at (-2,0) [labelBelow=$-4$] ;
[bullet] (-10) at (-1,0) [labelBelow=$-1$] ;
[bullet] (10) at (1,0) [labelBelow=$-2$] ;
[rectangle] (20) at (2,0) [labelBelow=$-2$] ;

[-] (01)–(00)–(0-1);
[-] (-20)–(-10)–(00)–(10)–(20);

[empty] (-21) at (-2,1) []$\fbox{4}$;

[bullet] (01) at (0,1) [labelAbove=$-2$] ;
[rectangle] (00) at (0,0) [label=below left:$-5$] ;
[rectangle] (0-1) at (0,-1) [labelBelow=$-2$] ;

[rectangle] (-20) at (-2,0) [labelBelow=$-4$] ;
[bullet] (-10) at (-1,0) [labelBelow=$-1$] ;
[bullet] (10) at (1,0) [labelBelow=$-2$] ;
[rectangle] (20) at (2,0) [labelBelow=$-2$] ;

[-] (01)–(00)–(0-1);
[-] (-20)–(-10)–(00)–(10)–(20);

[empty] (-21) at (-2,1) []$\fbox{5}$;

[bullet] (01) at (0,1) [labelAbove=$-2$] ;
[rectangle] (00) at (0,0) [label=below left:$-5$] ;
[bullet] (0-1) at (0,-1) [labelBelow=$-2$] ;

[rectangle] (-20) at (-2,0) [labelBelow=$-4$] ;
[bullet] (-10) at (-1,0) [labelBelow=$-1$] ;
[rectangle] (10) at (1,0) [labelBelow=$-2$] ;
[bullet] (20) at (2,0) [labelBelow=$-2$] ;

[-] (01)–(00)–(0-1);
[-] (-20)–(-10)–(00)–(10)–(20);

Five P-resolutions

It is not difficult to show that the $i$-th P-resolution corresponds to the $i$-th picture deformation for $i=1,\dotsc,5$ by applying the MMP machinery.

§.§.§ A nonnormal P-modification

Kollár <cit.> also present a nonnormal P-modification of $(X,p)$. We briefly recall the construction. There are four distinguished points on the central $(-4)$-curve $C$ corresponding to the four intersection points with another exceptional curves. We denote the intersection points by $N, E, S, W$ according to the direction of the exceptional curves given in the above resolution graph. There is a unique involution $\tau$ on $C$ such that $\tau(N)=S$ and $\tau(E)=W$. We first contract all curves except $C$. Then we have a normal surface $C'' \subset X''$. Then for each $x \in C''$ we identify $x$ and $\tau(x)$ to obtain a nonnormal surface germ $g \colon C' \subset X' \to (0 \in X)$. Along $C'$ we have generically normal crossing points. There are also two pinch points corresponding to branch points of the involution $\tau$ and two singular points of the form

\begin{equation*}
\text{$(xy=0) \subset \mathbb{C}^3/\mathbb{Z}_2(1,-1,1)$ and $(xy=0) \subset \mathbb{C}^3/\mathbb{Z}_3(1,-1,1)$}
\end{equation*}

Kollár <cit.> show that the singularities on $X'$ have $\mathbb{Q}$-Gorenstein smoothings and $X'$ is indeed a (nonnormal) P-modification of $X$.

The picture deformation #6 seems to correspond to the non-normal P-modification. But it is not clear whether one can see the correspondence via the semistable minimal model program.

[1] Behnke, Kurt; Christophersen, Jan Arthur. $M$-resolutions and deformations of quotient surface singularities. Amer. J. Math. 116 (1994),  no. 4, 881–903.

[2] Bhupal, Mohan; Stipsicz, András I. Weighted homogeneous singularities and rational homology disk smoothings. Amer. J. Math. 133 (2011), no. 5, 1259–1297.

[3] Burns, D. Matthew Jr.; Wahl, Jonathan. Local contributions to global deformations of surfaces. Invent. Math. 26 (1974), 67–88.

[4] Christophersen, J. Arthur. On the components and discriminant of the versal base space of cyclic quotient singularities. Singularity theory and its applications, Part I (Coventry, 1988/1989), 81–92, Lecture Notes in Math. 1462, Springer, 1991

[5] Flenner, Hubert; Zaidenberg, Mikhail. $\mathbb{Q}$-acyclic surfaces and their deformations. Contemp. Math. 162 (1994), 143–208.

[6] Fowler, Jacob R. Rational homology disk smoothing components of weighted homogeneous surface singularities. Thesis (Ph.D.)—The University of North Carolina at Chapel Hill. 2013

[7] Hacking, Paul; Tevelev, Jenia; Urzúa, Giancarlo. Flipping surfaces. J. Algebraic Geom. 26 (2017), no. 2, 279–345.

[8] Hartshorne, Robin. Algebraic geometry. Graduate Texts in Mathematics no. 52, Springer-Verlag, 1977.

[9] Jeon, Jaekwan; Heesang Park; Shin, Dongsoo. Deformations of weighted homogeneous surface singularities admitting rational homology disk smoothings. Preprint.

[10] Jeon, Jaekwan; Shin, Dongsoo. Deformations of weighted homogeneous surface singularities with big nodes. Preprint 2022.

[11] de Jong, Theo; van Straten, Duco. Deformation theory of sandwiched surface singularities. Duke Math. J. 95 (1998), no. 3, 451–522.

[12] Kollár, János. Flips, flops, minimal models, etc. Surveys in differential geometry (Cambridge, MA, 1990), 113–199, Lehigh Univ., Bethlehem, PA, 1991.

[13] Kollár, János; Mori, Shigefumi. Classification of three-dimensional flips, J. Amer. Math. Soc. 5 (1992), no. 3, 533–703.

[14] Kollár, János ; Shepherd-Barron, Nick I. Threefolds and deformations of surface singularities. Invent. Math. 91 (1988), no. 2, 299–338.

[15] Lee, Yongnam; Park, Jongil. A simply connected surface of general type with $p_g=0$ and $K^2=2$. Invent. Math. 170 (2007), no. 3, 483–505.

[16] Lisca, Paolo; On symplectic fillings of lens spaces. Trans. Amer. Math. Soc. 360 (2008),  no. 2, 765–799.

[17] Looijenga, E. The smoothing components of a triangle singularity. II. Math. Ann. 269 (1984), no. 3, 357–387.

[18] Möhring, Konrad. On Sandwiched Singularities. Dissertation, Mainz 2004.

[19] Mori, Shigefumi. Flip theorem and the existence of minimal models for 3-folds. J. Amer. Math. Soc. 1 (1988), no. 1, 117–253.

[20] Némethi, András; Popescu-Pampu, Patrick. On the Milnor fibres of cyclic quotient singularities. Proc. Lond. Math. Soc. 101 (2010), no. 2, 554–588.

[21] Némethi, András; Popescu-Pampu, Patrick. On the Milnor fibers of sandwiched singularities. Int. Math. Res. Not. IMRN 2010, no. 6, 1041–1061.

[22] Orlik, Peter; Wagreich, Philip. Isolated singularities of algebraic surfaces with $\mathbb{C}^{\ast}$-action. Ann. of Math. (2) 93 (1971), 205–228.

[23] Orlik, Peter; Wagreich, Philip. Algebraic surfaces with $k^{\ast}$-action. Acta Math. 138 (1977), no. 1–2, 43–81.

[24] Pinkham, Henry. Deformations of algebraic varieties with $\mathbb{G}_m$ action. Asterisque 20 (1974), 1–131.

[25] Pinkham, Henry. Normal surface singularities with $\mathbb{C}^{\ast}$ action. Math. Ann. 227 (1977), no. 2, 183–193.

[26] Pinkham, Henry. Deformations of normal surface singularities with $\mathbb{C}^{\ast}$ action. Math. Ann. 232 (1978), no. 1, 65–84.

[27] Park, Heesang; Park, Jongil; Shin, Dongsoo; Urzúa, Giancarlo. Milnor fibers and symplectic fillings of quotient surface singularities. Adv. Math. 329 (2018), 1156–1230.

[28] Spivakovsky, Mark. Sandwiched surface singularities and desingularization of surfaces by normalized Nash transformations. Ann. Math. 131 (1990), 411–491.

[29] Starkston, Laura. Symplectic fillings of Seifert fibered spaces. Trans. Amer. Math. Soc. 367 (2015), no. 8, 5971–6016.

[30] Stevens, Jan. On the versal deformation of cyclic quotient singularities. Singularity theory and its applications, Part I (Coventry, 1988/1989), 302–319, Lecture Notes in Math. 1462, Springer 1991.

[31] Stevens, Jan. Deformations of singularities. Lecture Notes in Mathematics 1811. Springer 2003.

[32] Stipsicz, András I.; Szabó, Zoltán; Wahl, Jonathan. Rational blowdowns and smoothings of surface singularities. J. Topol. 1 (2008), no. 2, 477–517.

[33] Urzúa, Giancarlo. Identifying neighbors of stable surfaces. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 16 (2016), no. 4, 1093–1122.

[34] Urzúa, Giancarlo. $\mathbb{Q}$-Gorenstein smoothings of surfaces and degenerations of curves. Rend. Semin. Mat. Univ. Padova 136 (2016), 111–136.

[35] Urzúa, Giancarlo ; Vilches, Nicolás. On wormholes in the moduli space of surfaces. arXiv:2102.02177

[36] Wahl, Jonathan. Smoothings of normal surface singularities. Topology 20 (1981), no. 3, 219–246.

[37] Wahl, Jonathan. Log-terminal Smoothings of
Graded Normal Surface Singularities. Michigan Math. J. 62 (2013), 475–489.

[38] Wahl, Jonathan. Complex surface singularities with rational homology disk smoothings. arXiv:2006.14696.