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Classification of weighted dual graphs consisting of $-2$-curves and exactly one $-3$-curve
S. S.-T. Yauab, Qiwei Zhua, Huaiqing Zuoa
a Department of Mathematical Sciences, Tsinghua University, Beijing, P. R. China
b Yanqi Lake Beijing Institute of Mathematical Sciences and Applications,
Huairou, P. R. China
Аннотация:
Let $(V, p)$ be a normal surface singularity. Let $\pi\colon (M, A)\to (V, p)$ be a minimal good resolution of $V$. The weighted dual graphs $\Gamma$ associated to $A$ completely describes the topology and differentiable structure of the embedding of $A$ in $M$. In this paper, we classify all the weighted dual graphs of $A=\bigcup_{i=1}^n A_i$ such that one of the curves $A_i$ is $-3$-curve, and the rest are all $-2$-curves. This is a natural generalization of Artin's classification of rational triple points. Moreover, we compute the fundamental cycles of maximal graphs (see Section 5) which can be used to determine whether the singularities are rational, minimally elliptic or weakly elliptic. We also give the formulas for computing arithmetic and geometric genera of star-shaped graphs.
Bibliography: 28 titles.
Ключевые слова:
normal singularities, topological classification, weighted dual graph.
Поступило в редакцию: 22.03.2022
Исправленный вариант: 21.09.2022
§ 1. Introduction Let $p$ be a normal singularity of the 2-dimensional Stein space $V$. Let $\pi\colon M\to V$ be a resolution of $V$ such that the irreducible components $A_i$, $1\leqslant i\leqslant n$, of $A=\pi^{-1}(p)$ are nonsingular and have only normal crossings. Associated to $A$ is a weighted dual graph $\Gamma$ (e.g., see [1] or [2]) which, along with the genera of the $A_i$, fully describes the topology and differentiable structure of $A$ in $M$ [3]. On a nonsingular surface $M$, a $-k$-curve means a nonsingular rational curve with self- intersection $-k$. M. Artin has studied the rational singularities (those for which $R^1\pi_*(\mathcal{O})=0$). He has shown that all hypersurface rational singularities have multiplicity two and the graphs associated with those singularities belong to the graphs $A_k$, $k\geqslant 1$, $D_k$, $k\geqslant 4$, $E_6$, $E_7$, and $E_8$ arising from the classification of simple Lie groups (in the latter discussion we abuse the notation $A_i$ for weighted dual graph as well as exceptional curves). He has also shown that the existence of a fundamental cycle (see Definiton 2.1) is equivalent to the negative definiteness of $(A_i\cdot A_j)$. Rational triple points are classified into nine classes according to the dual graphs in [4]. These nine classes of graphs consist of $-2$-curves and exactly one $-3$-curve. Here simple means that only finite many isomorphism classes occur in the versal deformation. The rational double points and rational triple points are simple. Stevens [5] conjectured that the simple normal surface singularities are exactly those rational singularities whose resolution graphs can be obtained from the graph of a rational double or triple point by making some number of vertex weights more negative. He showed that no other rational singularities can be simple. He proved simpleness for some special classes of singularities, namely, rational quadruple points or sandwiched singularities in [5]. For the classification of certain classes of rational singularities, interested readers can refer to the recent papers [6]–[9]. In [10], Laufer examined a class of elliptic singularities which satisfy a minimality condition. These minimally elliptic singularities have a theory much like the theory for rational singularities. Laufer [10] also listed all dual graphs corresponding to the minimally elliptic hypersurface singularities. These singularities are exactly Gorenstein singularities with geometric genus equals to $1$. Such a list is extremely useful for researchers in this field. For the classification of Gorenstein singularities with geometric genus greater than 1, the interested readers can refer to [3], [11]–[18]. In [19] (respectively, [20]), the authors of this paper generalized Laufer’s list of dual graphs of minimally elliptic hypersurface singularities. They classified all weighted dual graphs of the simplest Gorenstein non-complete intersection (respectively, complete intersection) singularities of dimension two. These singularities are exactly those minimal elliptic singularities with fundamental cycle self-intersection number $-5$ (respectively, $-4$). In this paper, the strategy of classification is different from [19]. We shall give a complete classification of singularities whose weighted dual graphs are $A=\bigcup_{i=1}^nA_i$ such that all $A_i$ are $-2$-curves except one $-3$-curve $A_j$. Thus, we generalize Artin’s list of dual graphs of rational triple points. The classification statements regarding topological types of normal surface singularities are important for some potential applications in our future work. In fact, these classes of singularities are the most interesting ones in the study of symmetries of surface singularities [21], since there is a natural homomorphism from the automorphism group of the singularities to the automorphism group of the central curve (i.e., the $-3$-curve). Furthermore, we compute the fundamental cycles of maximal graphs. As an application, one can deduce whether these graphs are rational, minimally elliptic or weakly elliptic. Moreover, we discuss the arithmetic and geometric genera of the star-shaped graphs. The main results of this paper are as follows. Main results We give a criteria to determine whether a weighted dual graph is negative-definite (cf. Corollary 3.7). Assuming that all the exceptional curves $A_i$ are $-2$-curves except one $-3$-curve $A_j$, then the weighted dual graph $\Gamma$ must be one of the three cases: Tree graph, Loop graph or Multiple edges graph (cf. Theorem 4.13 for last two cases). The complete classifications are listed in Section 4 (cf. Theorem 4.1, Theorem 4.3, Theorem 4.5, Theorem 4.10, and Theorem 4.13); the fundamental cycles of maximal graphs are computed and listed in Section 5. Furthermore, when the graph is star-shaped, its arithmetic and geometric genus formulas are obtained in Section 5 (cf. Corollary 6.8, Theorem 6.12). Remark 1.1. Our new results also include Theorem 3.5 and Corollary 3.7 which can be of independent interest. These two results shed new light on the classification of more complicated weighted dual graphs of singularities. Acknowledgments The authors would like to thank the anonymous referee for his/her careful reading of the manuscript, as well as his/her valuable comments and suggestions which helped to improve the paper.
§ 2. Preliminaries2.1. Riemann–Roch and fundamental cycle Let $\pi\colon M\to V$ be a resolution of the normal two-dimensional Stein space $V$. We assume that $p$ is the only singularity of $V$. Let $\pi^{-1}(p)=A=\bigcup A_i$, $1\leqslant i\leqslant n$, be the decomposition of the exceptional set $A$ into irreducible components. A cycle $D=\sum d_iA_i$, $1\leqslant i\leqslant n$, is an integral combination of the $A_i$, with $d_i$ an integer. There is a natural partial ordering denoted by $\geqslant $, between cycles defined by comparing the coefficients: $\sum_i m_i A_i \geqslant \sum_i n_i A_i$ if $m_i \geqslant n_i$ for all $i$. If $D_{1} \geqslant D_{2}$ but $D_{1} \neq D_{2}$, then we write $D_{1}>D_{2}$. We let $\operatorname{supp}D=\bigcup A_i$, $d_i\ne 0$, denote the support of $D$. Let $\mathcal{O}$ be the sheaf of germs of holomorphic functions on $M$. Let $\mathcal{O}(-D)$ be the sheaf of germs of holomorphic functions on $M$ which vanish to order $d_i$ on $A_i$. Let $\mathcal{O}_D$ denote $\mathcal{O/\mathcal{O}}(-D)$. Define
$$
\begin{equation}
\chi (D):=\dim H^0 (M,\mathcal{O}_D)-\dim H^1(M,\mathcal{O}_D).
\end{equation}
\tag{2.1}
$$
$$
\begin{equation}
\chi (D)=-\frac12 (D^2+D\cdot K),
\end{equation}
\tag{2.2}
$$
$$
\begin{equation}
A_i\cdot K=-A^2_i+2g_i-2+2\delta_i,
\end{equation}
\tag{2.3}
$$
$$
\begin{equation}
\chi (B+C) =\chi (B)+\chi (C) - B\cdot C.
\end{equation}
\tag{2.4}
$$
Definition 2.1. Associated to $\pi$ is a unique fundamental cycle $Z$ [4; pp. 131, 132] such that $Z>0$, $A_i\cdot Z\leqslant 0$ for all $A_i$ and such that $Z$ is minimal with respect to those two properties. The fundamental cycle $Z$ may be computed from the intersection as follows via a computation sequence for $Z$ in the sense of Laufer [23; Proposition 4.1, p. 607]:
$$
\begin{equation*}
\begin{gathered} \, Z_0=0,\quad Z_1 = A_{i_1},\quad Z_2=Z_1+A_{i_2},\quad \dots,\quad Z_j=Z_{j-1}+A_{i_j},\quad \dots, \\ Z_\ell = Z_{\ell-1}+A_{i_\ell} =Z, \end{gathered}
\end{equation*}
\notag
$$
The $\mathcal{O}(-Z_{j-1})/\mathcal{O}(-Z_j)$ represents the sheaf of germs of sections of a line bundle over $A_{i_j}$ of Chern class $-A_{i_j}\cdot Z_{j-1}$. So
$$
\begin{equation*}
H^0(M,\mathcal{O}(-Z_{j-1})/\mathcal{O} (-Z_j))=0
\end{equation*}
\notag
$$
Consider the exact sequence:
$$
\begin{equation}
0\to \mathcal{O}(-Z_{j-1})\big/\mathcal{O}(-Z_j)\to \mathcal{O}_{Z_j}\to \mathcal{O}_{Z_{j-1}}\to 0.
\end{equation}
\tag{2.5}
$$
$$
\begin{equation}
H^0(M,\mathcal{O}_{Z_k})=\mathbb{C}, \qquad 1\leqslant k\leqslant \ell,
\end{equation}
\tag{2.6}
$$
$$
\begin{equation}
\dim H^1(M,\mathcal{O}_{Z_k})=\sum_{1\leqslant j\leqslant k}\dim H^1\bigl(M,\mathcal{O}(-Z_{j-1}))/\mathcal{O}(-Z_j)\bigr).
\end{equation}
\tag{2.7}
$$
Lemma 2.2 [10]. Let $Z_k$ be part of a computation sequence for $Z$ and such that $\chi (Z_k)=0$. Then $\dim H^1(M,\mathcal{O}_D)\leqslant 1$ for all cycles $D$ such that $0\leqslant D\leqslant Z_k$. Also $\chi (D)\geqslant 0$. 2.2. The canonical cycle Definition 2.3. The rational cycle $Z_{K}$ is called the canonical cycle if $Z_{K} \cdot A_i=-K A_i$ for all $i$, i.e.,
$$
\begin{equation*}
Z_{K} \cdot A_i=A_i^{2}-2 \delta_i-2 g_i+2\quad \text{for all } i,
\end{equation*}
\notag
$$
Definition 2.4. If the coefficients of $Z_{K}$ are integers, then the singularity is called numerical Gorenstein. 2.3. Minimally elliptic and weakly elliptic singularities Based on Lemma 2.2, Laufer introduced the following definition of minimally elliptic cycle. We shall recall some of the properties of minimally elliptic singularities which we need for our classification problem. Definition 2.5. A cycle $E>0$ is minimally elliptic if $\chi (E)=0$ and $\chi (D)>0$ for all cycles $D$ such that $0<D<E$. Wagreich [17] defined the singularity $p$ to be elliptic if $\chi (D)\geqslant 0$ for all cycles $D\geqslant 0$ and $\chi (F)=0$ for some cycles $F>0$. He proved that this definition is independent of the resolution. It is easy to see that under this hypothesis, $\chi (Z)=0$. The converse is also true [10]. Henceforth, we shall adopt the following definition. Definition 2.6. The singularity $(V,p)$ is said to be weakly elliptic if $\chi (Z)=0$. The following proposition holds for weakly elliptic singularities. Proposition 2.7 [10]. Suppose that $\chi (D)\geqslant 0$ for all cycles $D>0$. Let $B=\sum b_iA_i$ and $C=\sum c_iA_i$, $1\leqslant i\leqslant n$, be any cycles such that $B,C>0$ and $\chi (B)= \chi (C)=0$. Let $G=\sum\min(b_i,c_i)A_i$, $1\leqslant i\leqslant n$. Then $G>0$ and $\chi (G)=0$. In particular, there exists a unique minimally elliptic cycle $E$. Theorem 2.8 [10]. Let $\pi\colon M\to V$ be the minimal resolution of the normal two dimensional variety $V$ with one singular point $p$. Let $Z$ be the fundamental cycle on the exceptional set $A=\pi^{-1}(p)$. Then the following statements are equivalent: (1) $Z$ is a minimally elliptic cycle; (2) $Z=Z_K$; (3) $\chi (Z)=0$ and any connected proper subvariety of $A$ is the exceptional set for a rational singularity. In [10], Laufer introduced the notion of minimally elliptic singularity. Definition 2.9. Let $(V,p)$ be a normal two-dimensional singularity. $(V,p)$ is said to be minimally elliptic if the minimal resolution $\pi\colon M\to V$ of a neighborhood of $p$ satisfies one of the conditions of Theorem 2.8. 2.4. Classification of weighted dual graphs In this section, we recall two beautiful results given by Artin in [4]. Let $(V,p)$ be a normal 2-dimensional singularity, $\pi\colon M\to V$ be the minimal resolution, and $Z$ be the fundamental cycle. Definition 2.10. The singularity $(V,p)$ is said to be rational if $\chi (Z)=1$. If $(V,p)$ is a rational singularity, then $\pi$ is also a minimal good resolution, i.e., exceptional set with nonsingular $A_i$ and normal crossings. Moreover, each $A_i$ is a rational curve and $A_i^2=-2$. Theorem 2.11 [4]. If $(V,p)$ is a hypersurface rational singularity, then $(V,p)$ is a rational double point. Moreover, the set of weighted dual graphs of hypersurface rational singularities consists of the following graphs: To each such weighted dual graph is associated an intersection matrix whose $(i,j)$th entry is $A_i \cdot A_j$. These graphs (1)–(5) in Theorem 2.11 are called ADE graphs in the literature. This theorem completely classifies the weighted dual graphs with all $A_i^2=-2$. In general, according to [24] and [4], to classify the weighted dual graphs, we need to classify the corresponding negative-definite matrices. Proposition 2.12 [4]. Let $\{A_i\}_{i=1, \dots, n}$ be a connected bunch of complete curves on a regular two-dimensional scheme. (i) Suppose that $(A_i \cdot A_j)$ is negative-definite, then there exist positive cycles $Z=\sum r_i A_i$ such that $(Z \cdot A_i) \leqslant 0$ for all $i$. (ii) Conversely, if there exists a positive cycle $Z=\sum r_i A_i$ such that $(Z \cdot A_i) \leqslant 0$ for all $i$, then $(A_i \cdot A_j)$ is negative semi-definite. If in addition $(Z^{2})<0$, then $(A_i \cdot A_j)$ is negative-definite. § 3. A general determinant formula and a method for classification In this section, we will give some key results, which will be helpful for classification of weighted dual graphs. The weighted dual graph consists of $-2$-curves and exactly one $-3$-curve, i.e., all $A_i$’s are nonsingular rational curves with $A_j^2=-3$ for some $j$ and $A_i^2=-2$ for all the other $i$’s such that $i\neq j$. In a dual graph, the $*$ represents the $-3$-curve. We will call it the $-3$-point or the $-3$-cycle later. The others are the points corresponding to the $-2$-curves, denoted by  , we will call them $-2$-points or $-2$-cycles later. By Theorem 2.11, if all $A_i$ have $A_i^2=-2$, then the graph must be an ADE graph. Recall that a tree graph is a connected graph without loops. ADE graphs are all tree graphs. However, if there exists one $A_j^2=-3$, then the following two cases are allowed:
and
Here,  denotes  with $n$ vertices and $n+1$ edges. We first begin with tree weighted dual graphs. We abuse the notation of weighted dual graph and the corresponding matrices in the following discussion, if without any confusion. Henceforth, whether $A_k$ is a weighted dual graph or a matrix should be clear from the context. For example, $A_n$ could either denote the weighted dual graph:
or the matrix
$$
\begin{equation*}
\begin{pmatrix} -2 & 1 & 0 & 0 & 0 & 0 \\ 1 & -2 & 1 & 0 & 0 & 0 \\ 0 & 1&\ddots& \ddots& 0 & 0 \\ 0 & 0 &\ddots &\ddots & 1 & 0 \\ 0 & 0 & 0 & 1 & -2 & 1 \\ 0 & 0 & 0 & 0 & 1 & -2 \end{pmatrix}.
\end{equation*}
\notag
$$
We use $\Gamma$ to denote the weighted dual tree graph of $A=\bigcup_{i=1}^nA_i$ such that all $A_i$ are $-2$-curves except one $-3$-curve $A_j$. After removing the point corresponding to the $-3$-curve, the remaining connected graphs are denoted by $\Gamma_1,\dots,\Gamma_m$. Lemma 3.1. With the notations as above, we have $m\leqslant 5$, and $\Gamma_i$ must be $ADE$ for any $1\leqslant i\leqslant m$. Proof. Notice that the first column of the following matrix is ($-1/2$) the sum of the other columns, hence the matrix
$$
\begin{equation*}
\begin{pmatrix} -3 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & -2 & 0 & 0 & 0 & 0 & 0\\ 1 & 0 & -2 & 0 & 0 & 0 & 0\\ 1 & 0 & 0 & -2 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & -2 & 0 & 0\\ 1 & 0 & 0 & 0 & 0 & -2 & 0\\ 1 & 0 & 0 & 0 & 0 & 0 & -2 \end{pmatrix}
\end{equation*}
\notag
$$
In the following, we illustrate a method for computing determinant of general tree graphs. Let us begin with the easiest case. Example 3.2 (determinant formula for $m=2$ in special case). Let $m=2$ and $\Gamma_i$ be $k_i-A_{n_i}$ (see definition of $k-A_n$ in Theorem 4.1), $i=1,2$. Then
$$
\begin{equation*}
\begin{aligned} \, \det(\Gamma) &=(-1)^{n_1+n_2}\bigl(-3\cdot |{\det(A_{n_1}) \det(A_{n_2})}| \\ &\qquad+|{\det(A_{n_1})\det(A_{n_2-k_2-1})\det(A_{k_2})}| \\ &\qquad +|{\det(A_{n_2})\det(A_{n_1-k_1-1})\det(A_{k_1})}|\bigr), \end{aligned}
\end{equation*}
\notag
$$
We explain what this formula means by using a weighted dual graph:
The formula tells us that the Laplacian expansion of $\det(\Gamma)$ is equivalent to “removing points and edges” on the graph $\Gamma$ in some sense as follows:
is divided into
We take $k_1=k_2=1$, $n_1=n_2=3$ as an example to show how this happens. Note that the matrix $\Gamma$ is
$$
\begin{equation*}
\begin{pmatrix} -2&1&0&0&0&0&0\\ 1&-2&1&1&0&0&0\\ 0&1&-2&0&0&0&0\\ 0&1&0&-3&0&1&0\\ 0&0&0&0&-2&1&0\\ 0&0&0&1&1&-2&1\\ 0&0&0&0&0&1&-2 \end{pmatrix}.
\end{equation*}
\notag
$$
$$
\begin{equation*}
\begin{aligned} \, \det(\Gamma) &=(-3)\det(A_3)\det(A_3) \\ &\qquad+1\cdot \det \begin{pmatrix} -2&0&0&0&0&0\\ 1&1&1&0&0&0\\ 0&-2&0&0&0&0\\ 0&0&0&-2&1&0\\ 0&0&1&1&-2&1\\ 0&0&0&0&1&-2 \end{pmatrix} \\ &\qquad+1\cdot \det \begin{pmatrix} -2&1&0&0&0&0\\ 1&-2&1&1&0&0\\ 0&1&-2&0&0&0\\ 0&0&0&0&-2&0\\ 0&0&0&1&1&-1\\ 0&0&0&0&0&-2 \end{pmatrix}. \end{aligned}
\end{equation*}
\notag
$$
$$
\begin{equation*}
\begin{pmatrix} -2&0&0&0&0&0\\ 1&1&1&0&0&0\\ 0&-2&0&0&0&0\\ 0&0&0&-2&1&0\\ 0&0&1&1&-2&1\\ 0&0&0&0&1&-2 \end{pmatrix}
\end{equation*}
\notag
$$
$$
\begin{equation*}
\begin{aligned} \, &\det \begin{pmatrix} (-2)&0&0&0&0&0\\ 1&1&(1)&0&0&0\\ 0&(-2)&0&0&0&0\\ 0&0&0&-2&1&0\\ 0&0&\langle 1\rangle&1&-2&1\\ 0&0&0&0&1&-2 \end{pmatrix} \\ &\qquad=(-1)\cdot(-2)\cdot(-2)\cdot(1)\cdot \det(A_3). \end{aligned}
\end{equation*}
\notag
$$
$$
\begin{equation*}
\begin{pmatrix} -2 &\dots &0& & & & & & & \\ \vdots&\ddots&\vdots& & & & & & & \\ 0&\dots&-2 & & & & & & & \\ & & 1& 1 & 0 &\dots &(1) & & & \\ & & & -2 & \dots &0 &0 & & & \\ & & &\vdots& \ddots &\vdots& \vdots& & & \\ & & & 0 & \dots & -2 & 0 & & & \\ & & & & & & \vdots & -2 &\dots&0\\ & & & & & & \langle 1\rangle & \vdots &\ddots&\vdots\\ & & & & & & & 0 &\dots&-2 \end{pmatrix},
\end{equation*}
\notag
$$
Remark 3.3. One may want to know how to determine the sign of each term. An easy way to see it is that the connection of $-3$ and $\Gamma_i$ contributes to “positive” part in the determinant. Hence, besides the $(-3)|{\det(A_{n_1})\det(A_{n_2})}|$ term, the other terms are positive. Now we try to generalize the formula of $\Gamma_i$ to general graphs. Notation 3.4. For a tree graph with a central curve $E$, we denote the subgraphs connected to $E$ as $\Gamma_1,\dots,\Gamma_s$, the points connected to $E$ as $E_1,\dots,E_s$, and the subgraphs connected to $E_i$ as $G_{i,1},\dots,G_{i,r_i}$:
Theorem 3.5 (general determinant formula). Let the weighted dual graph $\Gamma$ be as above. Then
$$
\begin{equation*}
\det(\Gamma)=\biggl(\prod_{i=1}^{s}\det(\Gamma_i)\biggr) \biggl(E^2+\sum_{j=1}^{s}\frac{(-1)^{n_j}\prod_{l=1}^{r_j}\det(G_{j,l})}{\det(\Gamma_j)}\biggr).
\end{equation*}
\notag
$$
Proof. We prove it by induction on $s$. Assume the formula holds for $s\leqslant k-1$, let us proceed to show it is true for $k$. Let the weighted dual graph be as in Notation 3.4 with $k$ subgraphs connected to $E$, i.e., the weighted dual graph is
Let $n_i$ be the number of points of $\Gamma_i$. The intersection matrix can be represented as
$$
\begin{equation*}
\begin{pmatrix} &\Gamma' & 1 & 0 & 1 & \dots & 1\\ &1 & E^2 & 1 & 0 & \dots & 0\\ &0 & 1 & E_k^2 & 1 & \dots & 1\\ &0 & 0 & 1 & G_{k,1} & \dots & 0 & \\ &0 & 0 & \vdots & \vdots & \ddots & 0 \\ &0 & 0 & 1 & 0 & \dots & G_{k,r_k} \end{pmatrix}.
\end{equation*}
\notag
$$
For simplicity, we use $\left(\begin{smallmatrix} \Gamma' & 1 \\ 1 & E^2 \end{smallmatrix}\right)$ to denote
$$
\begin{equation*}
\begin{pmatrix} E^2 & 1 & \dots & 1\\ 1& \Gamma_{1} & \dots & 0 & \\ \vdots & \vdots & \ddots & \vdots \\ 1 & 0 & \dots & \Gamma_{k-1} \end{pmatrix}.
\end{equation*}
\notag
$$
Using the Laplacian expansion on $E$, we obtain that
$$
\begin{equation*}
\begin{aligned} \, \det(\Gamma) &=\det\begin{pmatrix} \Gamma' & 1 \\ 1 & E^2 \end{pmatrix} \det(\Gamma_k)+(-1)^{(n_k)}\det\begin{pmatrix} &\Gamma' & 1 & 1 & \dots & 1\\ &0 & 1 & 1 & \dots & 1\\ &0 & 0 & G_{k,1} & \dots & 0 & \\ &0 & 0 & \vdots & \ddots & 0 \\ &0 & 0 & 0 & \dots & G_{k,r_k} \end{pmatrix} \\ &=\det\begin{pmatrix} \Gamma' & 1 \\ 1 & E^2 \end{pmatrix} \det(\Gamma_k) +(-1)^{(n_k)}\det(\Gamma')\prod_{l=1}^{r_k}\det(G_{k,l}). \end{aligned}
\end{equation*}
\notag
$$
By the induction hypothesis, we have
$$
\begin{equation*}
\det\begin{pmatrix} \Gamma' & 1 \\ 1 & E^2 \end{pmatrix}=\biggl(\prod_{i=1}^{k-1}\det(\Gamma_i)\biggr)\biggl((E^2) +\sum_{j=1}^{k-1}\frac{(-1)^{n_j}\prod_{l=1}^{r_j}\det(G_{j,l})}{\det(\Gamma_j)}\biggr).
\end{equation*}
\notag
$$
Thus,
$$
\begin{equation*}
\begin{aligned} \, \det(\Gamma) &=\biggl(\prod_{i=1}^{k-1}\det(\Gamma_i)\biggr)\biggl((E^2) +\sum_{j=1}^{k-1}\frac{(-1)^{n_j}\prod_{l=1}^{r_j}\det(G_{j,l})}{\det(\Gamma_j)}\biggr) \det(\Gamma_k) \\ &\qquad + (-1)^{(n_k)}\prod_{i=1}^{k-1}\det(\Gamma_i)\prod_{l=1}^{r_k}\det(G_{k,l}) \\ &=\biggl(\prod_{i=1}^{k}\det(\Gamma_i)\biggr)\biggl((E^2) +\sum_{j=1}^{s}\frac{(-1)^{n_j}\prod_{l=1}^{r_j}\det(G_{j,l})}{\det(\Gamma_j)}\biggr). \end{aligned}
\end{equation*}
\notag
$$
$\Box$ Notice that the selection of $E$ can be arbitrary in a weighted dual graph. Nevertheless, if we choose a suitable $E$ (e.g., the $E$ such that $\Gamma_i$ is negative-definite for $i=1,\dots,s$), then determining the negative-definiteness of det($\Gamma$) is reduced to checking the negative-definiteness of the determinants of the subgraphs containing $E$. Remark 3.6. Theorem 3.5 tells us that the “removing points and edges” method in Example 3.2 also holds for arbitrary tree graphs. Corollary 3.7 (criteria for negative definiteness). With the assumptions in Theorem 3.5, let us further assume that each $\Gamma_i$ is negative-definite, for $i=1,\dots,s$. Then the weighted dual graph is negative-definite if and only if
$$
\begin{equation*}
E^2+\sum_{j=1}^{s}\frac{\prod_{k=1}^{r_j}|{\det(G_{j,k})}|}{|{\det(\Gamma_j)}|}<0.
\end{equation*}
\notag
$$
Proof. Since $\Gamma_i$ is negative-definite, we have
$$
\begin{equation*}
\det(\Gamma_i)=(-1)^{n_i}|{\det(\Gamma_i)}|,\qquad \prod_{k=1}^{r_j}\det(G_{j,k})=(-1)^{n_k-1}\prod_{k=1}^{r_j}|{\det(G_{j,k})}|.
\end{equation*}
\notag
$$
$$
\begin{equation*}
\det(\Gamma)=(-1)^{\sum_{i=1}^s n_i}\biggl(\biggl|\prod_{i=1}^{s}\det(\Gamma_i)\biggr|\biggr)\biggl((E^2) +\sum_{j=1}^{s}\biggl|\frac{\prod_{k=1}^{r_j}\det(G_{j,k})}{\det(\Gamma_j)}\biggr|\biggr).
\end{equation*}
\notag
$$
$$
\begin{equation*}
(-1)^{(1+\sum_{i=1}^{s}n_i)}\det(\Gamma)>0.
\end{equation*}
\notag
$$
$$
\begin{equation*}
E^2+\sum_{j=1}^{s}\frac{\prod_{k=1}^{r_j}|{\det(G_{j,k})}|}{|{\det(\Gamma_j)}|}<0.
\end{equation*}
\notag
$$
$$
\begin{equation*}
\begin{aligned} \, (-1)^n\Delta_n &=(-1)^n\det(\Gamma) \\ &=(-1)^n(-1)^{\sum_{i=1}^s n_i}\biggl(\biggl|\prod_{i=1}^{s}\det(\Gamma_i)\biggr|\biggr)\biggl((E^2) +\sum_{j=1}^{s}\biggl|\frac{\prod_{k=1}^{r_j}\det(G_{j,k})}{\det(\Gamma_j)}\biggr|\biggr) \\ &=(-1)\biggl(\biggl|\prod_{i=1}^{s}\det(\Gamma_i)\biggr|\biggr)\biggr((E^2) +\sum_{j=1}^{s}\biggl|\frac{\prod_{k=1}^{r_j}\det(G_{j,k})}{\det(\Gamma_j)}\biggr|\biggr)>0. \end{aligned}
\end{equation*}
\notag
$$
Remark 3.8. The condition for $\Gamma_i$ to be negative-definite is natural, because Proposition 2.12 and the proof of Lemma 3.1 imply that any subgraph of a negative- definite weighted dual graph is negative-definite. This criteria can also be used to classify other graphs. For example, the weighted dual graphs consist of exactly one $-k$-curve, for $k\geqslant 4$. The classification will be investigated in our subsequent paper. § 4. Classification of weighted dual graphs consists of $-2$-curves and exactly one $-3$-curve In the following, we shall give the classification of the weighted dual graphs $\Gamma$ for different values of $m$. Case 1. $m=0$. The weighted dual graph has just one $-3$ point:
Case 2. $m=1$. Theorem 4.1. When $m=1$, $\Gamma_1$ must be one of the following: Here, we use the notation $k-A_n$ to denote the following graph: $\Gamma_1=A_n$ and $\Gamma$ is
with $n\geqslant 2k+1$. Here, $0-A_n$ (or $A_n$) means that the $-3$-curve connects with $A_n$ at the left or right end point. Similarly, the notation $1-D_n$ means $\Gamma_1=D_n$, with $n\geqslant 5$ and $\Gamma$ is
$0-D_n$ (or $D_n$) means that the $-3$-curve connects with the longest branch of $D_n$. In the later discussions, without the emphasis on the boundness of $n$, it means $n\geqslant1$ can be arbitrary. $D_n'$ is
$E_{k}$, $k=6, 7, 8$, are
$E_6'$ is
$E_{7}'$ is
and $D_4''$ is
Proof. This is clear by Corollary 3.7. $\Box$ Remark 4.2. Note the $k-A_n$ (also $k-D_n$) are terminologies to identify the different connection ways of $\Gamma_1$ and $-3$-point. $D_n'$ is exactly $0-D_n$ when $n=4$, so we require in (3) that $n\geqslant 5$. $1-A_3$ can be regarded as a limiting case of $D_n$. To classify all the tree graphs $\Gamma$, we need to compute the determinant of the corresponding matrices. One remembers that $\det(A_n)=(-1)^n(n+1)$, $\det(D_n)=4\cdot(-1)^n$, $\det(E_6)=3$, $\det(E_7)=-2$, $\det(E_8)=1$. Case 3. $m=2$. Theorem 4.3. When $m=2$, then $\Gamma_1+\Gamma_2$ must be one of the following: - (1) $(k_1-A_{n_1})+(k_2-A_{n_2})$:
$$
\begin{equation*}
\frac{(k_1+1)^2}{n_1+1}+\frac{(k_2+1)^2}{n_2+1}>k_1+k_2-1;
\end{equation*}
\notag
$$
- (2) $(k_1-D_{n_1})+(k_2-A_{n_2})$: $k_1=1$, $k_2=0$; or $k_1=0$, $k_2=0,1$; or $k_1=0$, $k_2=2$, $n_2\leqslant 7$;
- (3) $D_{n_1}+D_{n_2}$;
- (4) $D_{n_1}'+A_{n_2}$: $n_1\leqslant 8$; or $n_1=9$, $n_2=1,2$;
- (5) $E_{n_1}+A_{n_2}$: $n_1=6,7,8$;
- (6) $E_6+(1-A_{n_2})$: $3\leqslant n_2\leqslant 10$;
- (7) $E_6'+A_{n_2}$;
- (8) $E_7'+A_{n_2}$;
- (9) $D_4''+A_{n_2}$;
- (10) $E_{n_1}+D_{n_2}$: $n_1=6,7$;
- (11) $D_{n_1}'+D_{n_2}$: $5\leqslant n_1\leqslant 7$;
- (12) $ E_{n_1}+E_6$: $n_1=6,7$;
- (13) $E_{n_1}+D_{n_2}'$: $n_1=6$, $n_2=5,6$; or $n_1=7$, $n_2=5$.
Proof. We first prove (1). Plugging $|{\det(A_n)}|=n+1$ into the determinant formula in Example 3.2, we obtain that
$$
\begin{equation*}
\begin{aligned} \, \det(\Gamma) &=(-1)^{n_1+n_2}\bigl(-3\cdot (n_1+1)(n_2+1) \\ &\qquad+(n_1-k_1)(k_1+1)(n_2+1)+(n_2-k_2)(k_2+1)(n_1+1)\bigr). \end{aligned}
\end{equation*}
\notag
$$
$$
\begin{equation*}
-3\cdot |{\det(E_7)\det(A_{n_2})}|+|{\det(D_6)\det(A_{n_2})}|+|{\det(E_7)\det(A_{n_2-1})}|<0,
\end{equation*}
\notag
$$
$$
\begin{equation*}
-3\cdot 2(n_2+1)+4(n_2+1)+2n_2<0,
\end{equation*}
\notag
$$
Proof. By Theorem 4.1, $n_1\geqslant 2k_1+1$, $n_2\geqslant 2k_2+1$. Thus,
$$
\begin{equation*}
\frac{(k_1+1)^2}{2k_1+2}+\frac{(k_2+1)^2}{2k_2+2}\geqslant \frac{(k_1+1)^2}{n_1+1}+\frac{(k_2+1)^2}{n_2+1}>k_1+k_2-1.
\end{equation*}
\notag
$$
$$
\begin{equation*}
\frac{4}{n_1+1}+\frac{4}{n_2+1}>1.
\end{equation*}
\notag
$$
$$
\begin{equation*}
n_1<3+\frac{16}{n_2-3},
\end{equation*}
\notag
$$
Case 4. $m=3$. Theorem 4.5. When $m=3$, then $\Gamma_1+\Gamma_2+\Gamma_3$ must be one of the following: Proof. Computations of the above cases are similar to each other. We take (4) as an example. Plugging $\det(E_6)$ and $\det(A_n)$ into the determinant formula in Theorem 3.5, then the negative-definiteness of $\Gamma$ implies that
$$
\begin{equation*}
\begin{aligned} \, 0&>\bigl(-3\cdot |{\det(E_{6}) \det(A_{n_2})\det(A_{n_3})}| +|{\det(E_{5})\det(A_{n_2})\det(A_{n_3})}| \\ &\qquad+|{\det(E_{6})\det(A_{n_2-1})\det(A_{n_3})}| +|{\det(E_{6})\det(A_{n_2})\det(A_{n_3-1})}|\bigr). \end{aligned}
\end{equation*}
\notag
$$
$$
\begin{equation*}
\begin{aligned} \, 0 &> -3\cdot 3 \cdot (n_2+1)(n_3+1) \\ &\qquad+ 4(n_2+1)(n_3+1)+3n_2(n_3+1)+3n_3(n_2+1), \end{aligned}
\end{equation*}
\notag
$$
$$
\begin{equation*}
(n_2-2)(n_3-2)<9.
\end{equation*}
\notag
$$
Lemma 4.7. The inequality
$$
\begin{equation*}
\sum_{i=2}^{3}\frac{(k_i+1)^2}{n_i+1}>\sum_{i=2}^{3}k_i
\end{equation*}
\notag
$$
Lemma 4.8. The inequality
$$
\begin{equation*}
(n_2-2)(n_3-2)<9
\end{equation*}
\notag
$$
Lemma 4.9. The inequality
$$
\begin{equation*}
(n_2-1)(n_3-1)<4
\end{equation*}
\notag
$$
Case 5. $m=4$. Theorem 4.10. When $m=4$, then $\Gamma_1+\Gamma_2+\Gamma_3+\Gamma_4$ must be one of the following: Proof. Let us only illustrate (3) in detail. Plugging $\det(A_n)$ and $\det(D_n)$ into the determinant formula in Theorem 3.5, then the negative-definiteness of $\Gamma$ implies that
$$
\begin{equation*}
\sum_{i=2}^{4}\frac{1}{n_i+1}>1.
\end{equation*}
\notag
$$
$$
\begin{equation*}
\frac{1}{n_4+1}>\frac{1}{3},
\end{equation*}
\notag
$$
Lemma 4.12. The inequality
$$
\begin{equation*}
\sum_{i=2}^{3}\frac{1}{n_i+1}>\frac{1}{2}
\end{equation*}
\notag
$$
Case 6. $m=5$. Theorem 4.13. When $m=5$, then $\Gamma_1+\Gamma_2+\Gamma_3+\Gamma_4+\Gamma_5$ must be one of the following: (1) $A_{n_1}+A_{n_2}+A_{1}+A_{1}+A_{1}$: $n_2=1$; (2) $n_2=2$, $n_1=2,3,4$. Proof. The determinant formula tells us that
$$
\begin{equation*}
\sum_{i=1}^{5}\frac{1}{n_i+1}>2.
\end{equation*}
\notag
$$
$$
\begin{equation*}
\sum_{i=1}^{2}\frac{1}{n_i+1}>\frac{1}{2}.
\end{equation*}
\notag
$$
Case 7. The weighted dual graph is not tree. Now we turn to loop case and multiple edges case, these are stated below (intersection multiplicity $>1$ means that $A_j\cdot A_i>1$, here $A_j$ is $-3$ cycle and $A_i$ is a $-2$ cycle connected with $A_j$). Theorem 4.14. The non-tree graph must be one of the following. (1) Loop cases:
where $m\geqslant 0$ ($m=0$ means no points appear outside the loop);
here,  denotes  with $n$ vertices and $n+1$ edges. (2) Multiple edges cases:
where $m\geqslant 0$ ($m=0$ denotes  ),
Proof. For (1), by directly computing the determinant, we can show these graphs are not allowed:
Since these are not negative-definite.
The following two graphs have determinant $0$:
As for
one can show that $\sum 1\cdot A_i$ is the fundamental cycle, hence it is negative-definite.
As for
the fundamental cycle is (the underlined number represents the $-3$-point)
For example, when $n=1$, the graph and corresponding fundamental cycle are as follows:
For (2), it is easy to see that these two multiple edge cases are exactly the degeneration of the two loop cases by taking $n=1$ and $n=0$, respectively. Multiplicity greater than three are not allowed in our classification. The fundamental cycles of multiple edge cases are
$\Box$
§ 5. The fundamental cycle of weighted dual graph In section 4, we have obtained all weighted dual graphs consisting of $-2$-curves and exactly one $-3$-curve. It is interesting to know what classes the corresponding singularities belong to. A connected weighted dual graph $\Gamma$ is called maximal if for any other connected weighted dual graph $\Gamma'\supset \Gamma$, we have $\Gamma=\Gamma'$. In this section, we compute the fundamental cycle of maximal graphs. For example, we only show the case of $n=11$ for $D_{n}'$. By computing fundamental cycles, we show that some singularities are not weakly elliptic. When $m=0$, the weighted dual graph has just one point which is $-3$-curve $*$, it is easy to see that $\chi(Z)=1$, hence it is a rational singularity. Proof. The computation is given by Definition 2.1. Here $\chi(Z)$ can be computed by Riemann–Roch theorem. $\Box$ In above cases, $3-A_{14}$ and $4-A_{11}$ are not rational or elliptic. Cases 2.2, (3), 4.3, (5), (6), and (7) are minimal elliptic. The others are rational. Definition 5.2. Let $(V,p)$ be a germ of weakly elliptic singularity. Let $\pi\colon M\,{\to}\,V$ be the minimal resolution with $\pi^{-1}(p)=A=\bigcup A_i$, $1\leqslant i\leqslant n$, the irreducible decomposition of the exceptional set, and $Z$ be the fundamental cycle. The set of effective cycles $\{A_{*1},\dots, A_{*l}\}$ is the set $\{A_i\colon A_i\cdot Z<0\}$. Remark 5.3. A simple way to compute $\chi(Z)$ is by checking effective cycles. Let $c_1,\dots,c_l$ be the coefficient of effective cycles $A_{*1},\dots,A_{*l}$ of $Z$. Let $d$ be the coefficient of $-3$-cycle, where $d$ may equal to some $c_i$. Let $p_i$ be $-A_{*i}\cdot Z$, then
$$
\begin{equation*}
\chi(Z)=\frac{1}{2}\biggl(\sum_{i=1}^{l}p_ic_i-d\biggr).
\end{equation*}
\notag
$$
For the sake of conciseness, we shall not draw all the dual graphs below, one remembers that a number on fundamental cycle represents a point in the dual graph. An underlined number represents the $-3$-cycle. Remark 5.6. One can see that $1-A_3$ is the limiting case of $D_n$, as their coefficients are the same. (For example, in 1.2.1 and 2.1, in 1.4.1, 2.2 and (6).) Remark 5.9. It is easy to see that when the coefficient of $-3$-cycle grows larger, then the singularity may always be not elliptic. § 6. The arithmetic and geometric genera of star-shaped graphs In this section, we shall illustrate the methods for computing the arithmetic and geometric genera of star-shaped singularities contained in our classifications. With abuse of notations, $E_i$ denotes exceptional curve, and $E=\bigcup E_i$ is the exceptional divisor. Definition 6.1. A weighted dual graph is called star-shaped if it is a connected tree where at most one vertex is connected to more than two other vertices. 6.1. Splice quotient singularity One can refer to [25] for the definition of splice quotient singularity. In this section, we recall the method for computing the geometric genus of splice quotient singularity. Let $\delta_{v}=(E-E_{v}) \cdot E_{v}$ be the number of irreducible components of $E$ intersecting with $E_{v}$. A curve $E_{v}$ is called an end (respectively, a node) if $\delta_{v}=1$ (respectively, $\delta_{v} \geqslant 3$). Let $\mathcal{E}$ (respectively, $\mathcal{N}$) denotes the set of indices of ends (respectively, nodes). A connected component of $E-E_{v}$ is called a branch of $E_{v}$. Let $(a_{v w})=-I^{-1}$, where $I$ denotes the intersection matrix $(E_{v} \cdot E_{w})$. Then every $a_{v w}$ is a positive rational number and $E_{v}^{*}=\sum_{w \in \mathcal{V}} a_{v w} E_{w}$. We define positive integers $e_{v}, \ell_{v w}$ and $m_{v w}$ as follows:
$$
\begin{equation*}
\ell_{v w}=|{\det I}| a_{v w}, \qquad e_{v}=|{\det I}| / \operatorname{gcd}\{\ell_{v w} \mid w \in \mathcal{V}\}, \qquad m_{v w}=e_{v} a_{v w}.
\end{equation*}
\notag
$$
Definition 6.2. An element of a semigroup $\sum_{w \in \mathcal{E}} \mathbb{Z}_{\geqslant 0} E_{w}^{*}$, where $\mathbb{Z}_{\geqslant 0}$ is the set of nonnegative integers, is called a monomial cycle. Let $\mathbb{C}[z]:=\mathbb{C}[z_{w};\, w \in \mathcal{E}]$ be the polynomial ring in $\# \mathcal{E}$ variables. For a monomial cycle $D=\sum_{w \in \mathcal{E}} \alpha_{w} E_{w}^{*}$, we associate a monomial $z(D):=\prod_{w \in \mathcal{E}} z_{w}^{\alpha_{w}} \in \mathbb{C}[z]$. Definition 6.3 (monomial condition). We say that $E$ (or its weighted dual graph) satisfies the monomial condition if for any branch $C$ of any node $E_{v}$, there exists a monomial cycle $D$ such that $D-E_{v}^{*}$ is an effective integral cycle supported on $C$. In this case, $z(D)$ is called an admissible monomial belonging to the branch $C$. Notation 6.4. For a star-shaped graph with $r$ branches, we assume that $n_i$ is the number of points on $i$th branch and $n_1\geqslant n_2\geqslant \dots\geqslant n_r$. We denote the central curve by $E_1$, and $E_2,\dots,E_{n_1+1}$ are the curves on the first branch, with $E_{n_1+1}$ being the end curve:
$$
\begin{equation*}
E_{n_1+1}-E_{n_1}-\dots -E_2-\stackrel{\stackrel{\stackrel{\textstyle{E_{n_1+n_2+n_3+1}}}{\scriptstyle\vdots}}{\textstyle{E_{n_1+n_2+2}}}}{\stackrel{|}{\underset{\underset{\underset{\underset{\textstyle{E_{n_1+n_2+n_3+n_4}}}{\scriptstyle\vdots}}{\textstyle{E_{n_1+n_2+n_3+2}}}}{|}}{E_1}}} -E_{n_1+2}-\dots-E_{n_1+n_2+1}.
\end{equation*}
\notag
$$
Proposition 6.5 ([26; Section 8]). The star-shaped graph satisfies the monomial condition. Theorem 6.6 ([27; Theorem 3.8]). Let $(V, p)$ be a normal two-dimensional singularity and $\psi\colon (\widetilde{V}, A) \to(V, p)$ the minimal good resolution. Assume that the dual graph of $A$ is star-shaped with central curve $E$. Then the arithmetic genus $p_{a}(V, p)$ of $(V, p)$ is
$$
\begin{equation*}
p_{a}(V, p)=\max _{1 \leqq r}\biggl\{r(g-1)-\biggl(\sum_{k=0}^{r-1} \operatorname{deg}(D^{(k)})\biggr)+1\biggr\},
\end{equation*}
\notag
$$
$$
\begin{equation*}
D^{(k)}=k D-\sum_i\biggl\{\frac{k e_i}{d_i}\biggr\} P_i,
\end{equation*}
\notag
$$
Proposition 6.7. Assume the weight dual graph is $\Gamma(n_1,n_2,n_3,n_4)$, then $\operatorname{deg}(D^{(k)})=\sum_{i=1}^{4}\lfloor k/(n_i+1)\rfloor-k$. Proof. By definition, on the one hand, $D^{(k)}=kD-\sum_i \lceil e_i/d_i \rceil P_i$, where $D$ is any divisor such that $O_{E_1}(D)$ is the conormal sheaf of $E_1$. Thus, $\operatorname{deg}(D)=-E_1\cdot E_1=3$. On the other hand, one has
$$
\begin{equation*}
\frac{d_i}{e_i}=2-\frac{1}{2-\cfrac{1}{\ddots\ -\cfrac{1}{2}}}=[2,\dots,2]=\frac{n_i+1}{n_i}.
\end{equation*}
\notag
$$
$$
\begin{equation*}
\operatorname{deg}(D^{(k)})=3k-\sum_{i=1}^{4} \biggl\lceil \frac{n_i}{n_i+1}\biggr\rceil=\sum_{i=1}^{4}\biggl\lfloor \frac{k}{n_i+1}\biggr\rfloor-k.
\end{equation*}
\notag
$$
Corollary 6.8.
$$
\begin{equation*}
p_a=1+\max_{r\geqslant 1}\biggl\{-r-\biggl(\sum_{k=0}^{r-1}\sum_{i=1}^{4}\biggl\lfloor \frac{k}{n_i+1}\biggr\rfloor-k\biggr)\biggr\}.
\end{equation*}
\notag
$$
In a sum, we can list the $p_a$ of maximal graphs of Theorem 4.10 in Tab. 1. Table 1.Arithmetic genera of star-shaped graphs
| $n_1$ | $n_2$ | $n_3$ | $n_4$ | $p_a$ | $n_1$ | $n_2$ | $n_3$ | $n_4$ | $p_a$ |
|---|
| $40$ | $6$ | $2$ | $1$ | $216$ | $22$ | $7$ | $2$ | $1$ | $69$ |
| $16$ | $8$ | $2$ | $1$ | $39$ | $13$ | $9$ | $2$ | $1$ | $26$ |
| $12$ | $10$ | $2$ | $1$ | $108$ | $18$ | $4$ | $3$ | $1$ | $48$ |
| $10$ | $5$ | $3$ | $1$ | $17$ | $8$ | $6$ | $3$ | $1$ | $32$ |
| $8$ | $4$ | $4$ | $1$ | $12$ | $6$ | $5$ | $4$ | $1$ | $13$ |
| $10$ | $3$ | $2$ | $2$ | $17$ | $6$ | $4$ | $2$ | $2$ | $13$ |
| $4$ | $3$ | $3$ | $2$ | $4$ | | | | | |
Next, we consider the formula for the geometric genus of splice-qoutient singularity with star-shaped graph in our special case. Lemma 6.9. Let $X$ be a splice-quotient singularity with $E_1$, the only node in the resolution graph. Let
$$
\begin{equation*}
H(t)=\frac{1}{|\mathbb{H}|}\sum_{\lambda \in \Lambda}\, \prod_{w\in \mathcal{V}} \bigl(1-\exp\bigl(2\pi\sqrt{-1}\, E_{\lambda}^*\cdot E_w^*\bigr)t^{m_{1w}}\bigr)^{\delta_w-2}.
\end{equation*}
\notag
$$
Proof. By Proposition 3.8 in [25]. $\Box$ Example 6.10. Let the weighted dual graph be $\Gamma(2,1,1,1,1)$:
Then $V$ is elliptic singularity but not minimally elliptic. Let $I$ be the intersection matrix of exceptional curves, i.e.,
$$
\begin{equation*}
I=\begin{bmatrix} -3 & 1 & 0 & 1 & 1 & 1 & 1 \\ 1 & -2 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & -2 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & -2 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & -2 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & -2 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & -2 \end{bmatrix},
\end{equation*}
\notag
$$
and $|{\det(I)}|=16=|\mathbb{H}|$. Let us denote the central curve as $E_1$. The basis of $|\mathbb{H}|$ is given by $E_1^*$, $E_4^*$, $E_5^*$, $E_6^*$. By computing $I^{-1}$, we obtain
$$
\begin{equation*}
-I^{-1}= \begin{bmatrix} 3 & 2 & 1 & 3/2 & 3/2 & 3/2 & 3/2 \\ 2 & 2 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1/2 & 1/2 & 1/2 & 1/2 \\ 3/2 & 1 & 1/2 & 5/4 & 3/4 & 3/4 & 3/4 \\ 3/2 & 1 & 1/2 & 3/4 & 5/4 & 3/4 & 3/4 \\ 3/2 & 1 & 1/2 & 3/4 & 3/4 & 5/4 & 3/4 \\ 3/2 & 1 & 1/2 & 3/4 & 3/4 & 3/4 & 5/4 \end{bmatrix},
\end{equation*}
\notag
$$
thus,
$$
\begin{equation*}
m_{1w}=(6,4,2,3,3,3,3).
\end{equation*}
\notag
$$
Notice that $\Lambda=\{\lambda\in (\mathbb{Z}_{\geqslant 0}^4)|\lambda_i=0 \text{ or }1)\}$ and $E_{\lambda}^*=\lambda_1E_1^*+\lambda_2E_4^*+\lambda_3E_5^*+\lambda_4E_6^*$. The Hilbert polynomial with respect to $E_1$ is given by
$$
\begin{equation*}
H(t)=\frac{1}{|\mathbb{H}|}\sum_{\lambda \in \Lambda} \prod_{w=1}^{7}\bigl(1-\exp\bigl(2\pi\sqrt{-1}\, E_{\lambda}^*\cdot E_w^*\bigr)t^{m_{1w}}\bigr)^{\delta_w-2}.
\end{equation*}
\notag
$$
We write $H(t)=p(t)/q(t)+r(t)$, thus $p_g=r(1)=1$. However, $Z_K$ is not an integral divisor, $Z_K\neq Z$, thus $V$ is not minimally elliptic. Remark 6.11. Nagy and Némethi [28] have shown that for a generic analytic structure on a given graph with a RHS (Rational Homology Sphere) link, the geometric genus equals to arithmetic genus. The following theorem gives the computation formula for graphs classified in Theorem 4.10. Theorem 6.12. Let $X$ be a splice-quotient singularity with the resolution graph $\Gamma(n_1,n_2,n_3,n_4)$. All the exceptional curves are assumed to be rational and $E_1^2=-3$, $E_i^2=-2$ $\forall\, i\neq 1$. Let
$$
\begin{equation*}
S=\frac{1}{-1+\sum_{i=1}^4\frac{1}{n_i+1}}.
\end{equation*}
\notag
$$
$$
\begin{equation*}
\begin{aligned} \, &H(t)=\frac{1}{\prod_{i=1}^4m(n_i+1)} \\ &\times\sum_{k_i=0,i=1,2,3,4}^{mn_i} \frac{\bigl(1-\exp\bigl(2\pi i\bigl(\sum_{i=1}^{4}\frac{k_i}{n_i+1}\bigr)\cdot S\bigr)t^{mS}\bigr)^2} {\prod_{j=1}^{4}\bigl(1-\exp\bigl(2\pi i\bigl(\frac{S}{n_j+1}\sum_{i=1}^{4}\frac{k_i}{n_i+1}+\frac{n_jk_j}{n_j+1}\bigr)t^{mS/(n_j+1)}\bigr)\bigr)}. \end{aligned}
\end{equation*}
\notag
$$
Proof. We only detail the proof, when $X$ is numerical Gorenstein. The general case is similar. Notice that $Z_K\cdot E_1=-1$ and $Z_K\cdot E_i=0$, for $ i\neq 1$. Thus, $Z_K=E_1^*$. The first row of $-I^{-1}$ (i.e., $m_{1w}$) is
$$
\begin{equation*}
\biggl(S,\frac{n_1S}{n_1{+}\,1},\dots,\frac{S}{n_1{+}\,1},\frac{n_1S}{n_2{+}\,1},\dots, \frac{S}{n_2{+}\,1}, \frac{n_3S}{n_3{+}\,1},\dots,\frac{S}{n_3{+}\,1}, \frac{n_4S}{n_4{+}\,1},\dots,\frac{S}{n_4{+}\,1}\biggr).
\end{equation*}
\notag
$$
$$
\begin{equation*}
H(t)=\frac{1}{|\mathbb{H}|}\sum_{\lambda \in \Lambda}\prod_{w\in \mathcal{V}} \bigl(1-\exp\bigl(2\pi\sqrt{-1}\, E_{\lambda}^*\cdot E_w^*\bigr)t^{m_{1w}}\bigr)^{\delta_w-2}.
\end{equation*}
\notag
$$
$$
\begin{equation*}
\begin{gathered} \, n_1E_{n_1+1}^*=E_1^*,\qquad n_2E_{n_1+n_2+1}^*=E_1^*, \\ n_3E_{n_1+n_2+n_3+1}^*=E_1^*,\qquad n_4E_{n_1+n_2+n_3+n_4+1}^*=E_1^*. \end{gathered}
\end{equation*}
\notag
$$
$$
\begin{equation*}
\lambda_1E_{n_1+1}^*+\lambda_2E_{n_1+n_2+1}^*+\lambda_3E_{n_1+n_2+n_3+1}^ *+\lambda_4E_{n_1+n_2+n_3+n_4+1}^*,\qquad 0\leqslant \lambda_i\leqslant n_i.
\end{equation*}
\notag
$$
$$
\begin{equation*}
{ {\begin{bmatrix} \begin{matrix} S &\frac{n_1S}{a} &\dots &\frac{S}{a} &\frac{n_1S}{b} &\dots &\frac{S}{b} \\ \frac{n_1S}{a} &\frac {\frac{n_1S}{a}+1}{a} n_1 & \dots &\frac{\frac{n_1S}{a}+1}{a}&\frac{n_1n_2S}{ab}&\dots&\frac{n_1S}{ab} \\ \vdots &\vdots & \ddots & \vdots& \vdots &\ddots&\vdots \\ \frac{S}{a} &\frac{\frac{n_1S}{a}+1}{a} & \dots & \frac{\frac{n_1S}{a}+n_1}{a}&\frac{n_2S}{ab}&\dots&\frac{S}{ab} \\ &&&&&\dots& \end{matrix}\quad \begin{matrix} \frac{n_3S}{c} &\dots &\frac{S}{c} &\frac{n_4S}{d} &\dots &\frac{S}{d} \\ \frac{n_1n_3S}{ac} &\dots&\frac{n_1S}{ac} & \frac{n_1n_4S}{ad}&\dots&\frac{n_1S}{ad} \\ \vdots&\ddots&\vdots & \vdots&\ddots&\vdots \\ \frac{n_3S}{ac}&\dots&\frac{S}{ac} & \frac{n_4S}{ad}&\dots&\frac{S}{ad} \\ &&&&& \end{matrix} \end{bmatrix}}. }
\end{equation*}
\notag
$$
$$
\begin{equation*}
\begin{gathered} \, E_{n_1+1}^*\cdot E_1^*=\frac{S}{a}, \\ E_{n_1+1}^*\cdot E_{n_1+1}^*=\frac{n_1S}{a^2}+\frac{n_1}{a}, \\ E_{n_1+1}^*\cdot E_{n_1+n_2+1}^*=\frac{S}{ab}, \\ E_{n_1+1}^*\cdot E_{n_1+n_2+n_3+1}^*=\frac{S}{ac}, \\ E_{n_1+1}^*\cdot E_{n_1+n_2+n_3+n_4+1}^*=\frac{S}{ad}. \end{gathered}
\end{equation*}
\notag
$$
$$
\begin{equation*}
\begin{gathered} \, E_{n_1+n_2+1}^*\cdot E_1^*=\frac{S}{b}, \\ E_{n_1+n_2+1}^*\cdot E_{n_1+1}^*=\frac{S}{ba}, \\ E_{n_1+n_2+1}^*\cdot E_{n_1+n_2+1}^*=\frac{n_2S}{b^2}+\frac{n_2}{b}, \\ E_{n_1+n_2+1}^*\cdot E_{n_1+n_2+n_3+1}^*=\frac{S}{bc}, \\ E_{n_1+n_2+1}^*\cdot E_{n_1+n_2+n_3+n_4+1}^*=\frac{S}{bd}. \end{gathered}
\end{equation*}
\notag
$$
Example 6.13. Let us consider $X$ to be a splice-quotient singularity with the resolution graph $\Gamma(6,4,2,2)$. Then
$$
\begin{equation*}
S=\frac1{(-1+1/7+1/5+1/3+1/3)=105}
\end{equation*}
\notag
$$
$$
\begin{equation*}
\begin{aligned} \, H(t) &=\frac{1}{3}\biggl(2\cdot \frac{{(1-t^{105})}^2}{(1-t^{15})(1-t^{21})(1-\exp(2\pi i\cdot 1/3)t^{35})(1-\exp(2\pi i\cdot 2/3)t^{35})} \\ &\qquad +\frac{(1-t^{105})^2}{(1-t^{15})(1-t^{21})(1-t^{35})^2}\biggr). \end{aligned}
\end{equation*}
\notag
$$
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Образец цитирования:
S. S.-T. Yau, Qiwei Zhu, Huaiqing Zuo, “Classification of weighted dual graphs consisting of $-2$-curves and exactly one $-3$-curve”, Изв. РАН. Сер. матем., 87:5 (2023), 232–270; Izv. Math., 87:5 (2023), 1078–1116
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/im9337https://doi.org/10.4213/im9337 https://www.mathnet.ru/rus/im/v87/i5/p232
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