Normal surface singularities with an integral homology sphere link related to space monomial curves with a plane semigroup

In this article, we consider an infinite family of normal surface singularities with an integral homology sphere link which is related to the family of space monomial curves with a plane semigroup. These monomial curves appear as the special fibers of equisingular families of curves whose generic fibers are a complex plane branch, and the related surface singularities appear in a proof of the monodromy conjecture for these curves. To investigate whether the link of a normal surface singularity is an integral homology sphere, one can use a characterization that depends on the determinant of the intersection matrix of a (partial) resolution. To study our family, we apply this characterization with a partial toric resolution of our singularities constructed as a sequence of weighted blow-ups.


Introduction
For (S, 0) ⊂ (C n , 0) a germ of a normal surface singularity, the link L(S, 0) is defined as the intersection of S with a small closed ball centered at the origin in C n . The topology of L(S, 0) can provide interesting information about the singularity (S, 0). For example, Mumford [15] showed that L(S, 0) is simply connected if and only if S is smooth at 0, and Neumann [18] showed that L(S, 0) determines and is determined by the dual graph of a good resolution of (S, 0) (it is a graph manifold whose plumbing decorated graph is a dual resolution graph of (S, 0)). In this article, we are interested in normal surface singularities whose link is a integral homology sphere or ZHS, that is, whose link has the same integral homology as a three-dimensional sphere. More generally, we can consider normal surface singularities having a rational homology sphere or QHS link.
More precisely, we will study the link of an infinite family of normal surface singularities (S, 0) ⊂ (C g+1 , 0) with g ≥ 2 related to the family of space monomial curves with a plane semigroup. These monomial curves arise as the special fibers of certain equisingular deformations of irreducible plane curve singularities. Such deformations are classical objects in singularity theory, and have been studied and generalized by, among others, Teissier, Goldin, González-Pérez, and Tevelev. Recently, they have played an important role in the solution of Yano's conjecture [5], and the solution of Dimca-Greuel's conjecture for branches by Alberich-Carramiñana et al. [1]. The motivation of the present work is the question by Némethi whether weighted blow-ups can be used to find examples of normal surface singularities having a ZHS link. Némethi has intensely studied surfaces singularities and, more specifically, surface singularities with a QHS link, see for instance [19] or, for a more modern approach [7,12], and the references listed there. We will also compare our family with two important families of surface singularities. First, we will take a look at the Brieskorn-Pham surface singularities {x a 1 1 + x a 2 2 + x a 3 3 = 0} ⊂ (C 3 , 0), whose link is a ZHS if the exponents a i ≥ 2 for i = 1, 2, 3 are pairwise coprime. Second, we will consider the singularities of splice type, having a ZHS link, introduced by Neumann and Wahl [21]. We will see that our surface singularities (S, 0) ⊂ (C g+1 , 0) with a ZHS link are always of splice type, but they are never Brieskorn-Pham if g ≥ 3.
The monodromy conjecture for these space monomial curves Y ⊂ C g+1 with g ≥ 2 is proven in [16] together with [17]; an overview of these two articles can be found in the short note [14]. Roughly speaking, the monodromy conjecture for Y ⊂ C g+1 states that the poles of the motivic, or related, Igusa zeta function of Y induce monodromy eigenvalues of Y . The computation of the motivic Igusa zeta function and its poles is the main subject of [17]; the study of the monodromy eigenvalues and proof of the monodromy conjecture can be found in [16]. A key ingredient in this proof is the consideration of the curve Y as the Cartier divisor { f i = 0} for some i ∈ {1, . . . , g} on a generic embedding surface S ⊂ C g+1 defined by (1) For generic coefficients (λ 2 , . . . , λ g ) ∈ (C\{0}) g−1 , the scheme S := S(λ 2 , . . . , λ g ) is a normal complete intersection surface which is smooth outside the origin. In this article, we are interested in the link of these normal surface singularities (S, 0) ⊂ (C g+1 , 0). For g = 2, one can easily see that (S, 0) ⊂ (C 3 , 0) is a Brieskorn-Pham surface singularity with exponents n 0 , n 1 and n 2 . For g ≥ 3, we will show that (S, 0) ⊂ (C g+1 , 0) is never Brieskorn-Pham by considering the rupture exceptional curves in the minimal good resolution of (S, 0), that is, the exceptional curves that are either non-rational or have valency at least 3 (i.e., they have at least 3 intersections with other exceptional curves). While a Brieskorn-Pham surface singularity has at most one rupture exceptional curve in its minimal good resolution, our surface singularities have at least g − 1 rupture exceptional curves. To show this, we will make use of a good Q-resolution of (S, 0), see Proposition 3.1. This is a resolution in which the final ambient space can have abelian quotient singularities, and the exceptional locus is a simple Q-normal crossing divisor on such a space. A good Q-resolution can be obtained as a sequence of weighted blow-ups and induces a good resolution by resolving the singularities of the final ambient space, which are Hirzebruch-Jung singularities. The good Q-resolutionφ :Ŝ → S that we will consider consists of the first g − 1 steps of the embedded Q-resolution of Y ⊂ S constructed in [16,Section 5]. In particular, the dual graph ofφ is a tree as in Fig. 1.
Since we already know that (S, 0) for g = 2 has a ZHS link if and only if its exponents n i are pairwise coprime, it remains to investigate when (S, 0) ⊂ (C g+1 , 0) has a ZHS link for g ≥ 3. For this purpose, we will make use of a characterization for a general normal surface singularity (S, 0) ⊂ (C n , 0) to have a ZHS link depending on the determinant of (S, 0). The determinant of a partial or good resolution of a normal surface singularity (S, 0) is defined as the determinant of the intersection matrix of the resolution, that is, it is the determinant of the matrix (E i · E j ) 1≤i, j≤r , where E 1 , …, E r are the exceptional curves of the resolution. Geometrically, the cokernel of the intersection matrix of a good resolution of (S, 0) is equal to the torsion part of H 1 (L(S, 0), Z). The determinant det(S) of a normal surface singularity (S, 0) is the absolute value of the determinant of some good resolution of (S, 0). Since the torsion part of H 1 (L(S, 0), Z) is a finite group of order det(S), this is independent of the chosen good resolution. In practice, det(S) can be computed as the product of the absolute value of the determinant of a partial resolution of (S, 0) and the determinants of the remaining singularities of the new ambient space, which are the orders of the corresponding small groups in the abelian quotient singular case, see (3).
In terms of the determinant det(S) and a good resolution π :S → S of a normal surface singularity (S, 0), the characterization can now be formulated as follows: the link L(S, 0) is a ZHS if and only if det(S) = 1 and π has only rational exceptional curves and a tree as dual graph. Because the good Q-resolutionφ :Ŝ → S of our singularities has a tree as dual graph, and the singularities ofŜ can be resolved with rational exceptional curves and a bamboo-shaped dual graph, we only need to check when both det(S) = 1 and the exceptional curves ofφ are rational. Furthermore, we can express det(S) in terms of the orders of the singularities ofŜ and the determinant ofφ. To compute the latter determinant, we will first prove in Proposition 4.3 a formula for the determinant of a general good Q-resolution with the same dual graph as in Fig. 1 by rewriting it in terms of a specific kind of tridiagonal matrices. For our good Q-resolutionφ :Ŝ → S, this immediately implies the expression for the determinant ofφ and of the singularity (S, 0) in Corollaries 4.4 and 4.6, respectively. Together with the properties ofφ, this yields the following theorem. As the same approach also gives conditions for (S, 0) to have a QHS link, which is true if and only if the dual graph of a good resolution is a tree with only rational exceptional curves, we can state our main result as the following generalization of the characterization for Brieskorn-Pham surface singularities.
Once we have shown this result, it is easy to check that our surface singularities (S, 0) ⊂ (C g+1 , 0) with g ≥ 2 having a ZHS link are of splice type. To this end, we will determine their splice diagram. This is a finite tree in which every vertex has either valency 1, called a leaf, or valency at least 3, called a node, and in which a weight is assigned to each edge starting at a node. Every dual graph of a normal surface singularity with a ZHS link corresponds to a unique splice diagram of special type. Hence, such a splice diagram also determines and is determined by the link. Furthermore, if a splice diagram of a ZHS link satisfies the socalled semigroup condition, then Neumann and Wahl constructed in [21] an isolated complete intersection surface singularity having this link, called a singularity of splice type. In addition, they conjectured that every normal complete intersection surface singularity with a ZHS link is of splice type. To compute the splice diagram of our surface singularities having a ZHS link, we will once more consider the good Q-resolutionφ :Ŝ → S. We will see that the semigroup condition is fulfilled and that our surface singularities with a ZHS link are always of splice type. In particular, they support the conjecture of Neumann and Wahl.
This article is organized as follows. We start in Sect. 2 by briefly discussing the necessary background. In Sect. 3, we will introduce our surface singularities (S, 0) ⊂ (C g+1 , 0) in more detail, list the main properties of the considered good Q-resolution of (S, 0), and use this resolution to show that (S, 0) is not Brieskorn-Pham for g ≥ 3 and to show the conditions for its link to be a QHS. In Sect. 4, we will prove the characterization for (S, 0) to have a ZHS link by computing its determinant, give some concrete examples in Example 4.7, and show that our surface singularities with a ZHS link are always of splice type.

Preliminaries
In this preliminary section, we give a short overview of the background needed in this article. We start by fixing some notation and conventions. First, by a (complex) variety, we mean a reduced separated scheme of finite type over C, which is not necessarily irreducible. A one-dimensional ( resp. two-dimensional) variety is called a curve (resp. surface). Sec-ond, for a rational number a b , we denote by [ a b ] its integer part. Third, for a set of integers m 1 , . . . , m r ∈ Z, we denote by gcd(m 1 , . . . , m r ) and lcm(m 1 , . . . , m r ) their greatest common divisor and lowest common multiple, respectively. To shorten the notation, we will sometimes use (m 1 , . . . , m r ) for the greatest common divisor.

Space monomial curves with a plane semigroup
( f , h) be its associated valuation. The semigroup (C) is the image of this valuation and can be generated by a unique minimal system of generators (β 0 , . . . ,β g ) withβ 0 < · · · <β g and gcd(β 0 , . . . ,β g ) = 1. Furthermore, the sequence (β 0 , . . . ,β g ) determines and is determined by the topological type of C, see for instance [28]. Therefore, it is a natural question how one can recover the equation of a plane curve singularity from a given topological type.
Using a minimal generating sequence of the valuation ν C , one can construct a family η : (χ, 0) ⊂ (C g+1 × C, 0) → (C, 0) of germs of curves in (C g+1 × C, 0), which is equisingular, for instance, in the sense that (C) is the semigroup of all curves in the family. The generic fiber η −1 (v) for v = 0 is isomorphic to C, and the special fiber The coefficients c i are needed to see that any irreducible plane curve singularity with semigroup (C) is an equisingular deformation of such a monomial curve. However, for simplicity, we can assume that every c i = 1, which is always possible after a suitable change of coordinates. This yields the monomial curve Y .
Clearly, we can also consider the global curve in C g+1 defined by the above binomial equations; from now on, we define a (space) monomial curve Y ⊂ C g+1 as the complete intersection curve given by . . .
This is still an irreducible curve which is smooth outside the origin. In [16], the monodromy eigenvalues for such a space monomial curve Y ⊂ C g+1 with g ≥ 2 are investigated by considering Y as a Cartier divisor a generic embedding surface S ⊂ C g+1 . Together with the results from [17], this yields a proof of the monodromy conjecture for Y ⊂ C g+1 . In this article, we are interested in the topology of these generic embedding surface singularities (S, 0) ⊂ (C g+1 , 0). We will introduce them in detail in Sect. 3.

Link of a normal surface singularity
Let (S, 0) ⊂ (C n , 0) be a germ of a normal surface singularity. Its link L(S, 0) is an oriented three-dimensional manifold which is defined as the intersection of S with a small enough closed ball centered at the origin in C n . In this article, we are interested in normal surface singularities whose link is a rational ( resp. integral) homology sphere, that is, whose link has the same rational ( resp. integral) homology as a three-dimensional sphere. In this case, we will say that the link is a QHS ( resp. a ZHS). To study when the link L(S, 0) is a QHS or a ZHS, we can make use of a practical criterion in terms of the determinant and a good resolution of (S, 0). By a good resolution of (S, 0), we mean a proper birational morphism π :S → S from a smooth surfaceS to S which is an isomorphism over S\{0} and whose exceptional locus π −1 (0) is a simple normal crossing divisor (i.e., its irreducible components, called the exceptional curves, are smooth and intersect normally). It is well known that such a resolution always exists as a sequence of blow-ups at well-chosen points. A good resolution π :S → S is called minimal if every other good resolution of (S, 0) factors through π. Equivalently, π is minimal if there is no exceptional curve that can be contracted (by blowing down) so that the resulting morphism is still a good resolution of (S, 0). It is worth mentioning that, by Castelnuovo's Contractibility Theorem, the only possible exceptional curves that can be contracted in such a way are rational and have self-intersection number −1. Furthermore, a minimal good resolution of a normal surface singularity (S, 0) always exists and is unique up to isomorphism. Therefore, we call it the minimal good resolution of (S, 0).
With a good resolution of (S, 0), we can associate a dual graph whose vertices correspond to the exceptional curves E 1 , …, E r , and two vertices E i and E j are connected by an edge if and only if E i ∩ E j = ∅. Often, each vertex E i is labeled with two numbers (g i , −κ i ), where g i is the genus of E i and −κ i its self-intersection number. It is a classical result that the free part of H 1 (L(S, 0), Z) has rank 2 r i=1 g i + b, where b is the first Betti number of the dual graph, and that its torsion part is equal to coker(A), where A = (E i · E j ) 1≤i, j≤r is the intersection matrix of the good resolution, see for example [8,Ch. 2,Prop. 3.4]. In particular, as A is negative definite, which was originally noted by DuVal but also shown by Mumford in [15], the torsion part of H 1 (L(S, 0), Z) is a finite group of order | det(A)| = det(−A). This result has two immediate consequences. First, it implies that det(−A) is independent of the chosen good resolution of (S, 0). Hence, we can define the determinant of (S, 0) as det(S) := det(−A) with A the intersection matrix of any good resolution of (S, 0). Second, we find the following easy conditions for (S, 0) to have a QHS or ZHS link. Theorem 2.1 Let (S, 0) ⊂ (C n , 0) be a normal surface singularity, and consider a good resolution π :S → S. The link of (S, 0) is a QHS if and only if all exceptional curves of π are rational and the dual graph of π is a tree. The link of (S, 0) is a ZHS if and only if it is a QHS and det(S) = 1.
To compute the determinant of (S, 0) in practice, we do not really need a good resolution of (S, 0): if π :S → S is a proper birational morphism from a normal surfaceS to S which is an isomorphism over S\{0}, then see for instance [2,Lemma 4.7].
where E 1 , . . . , E r are the exceptional curves of π, and det(S, p) is the absolute value of the determinant of the intersection matrix of some good resolution at p. Note that if p ∈ π −1 (0) is written as a Hirzebruch-Jung singularity of type 1 d (1, q) with d and q coprime, then det(S, p) = d, see Sect. 2.5.
Following the approach in [2, 4.3] for weighted Lê-Yomdin singularities, we can consider the curve C : (see Sect. 2.5), which has genus Furthermore, the determinant of S(a 1 , a 2 , a 3 ) is given by has a QHS ( resp. ZHS) link if and only if the above genus is equal to 0 ( resp. and the determinant is equal to 1). This yields the following result.

Proposition 2.2 Using the above notations, the link of a Brieskorn-Pham surface singularity
It is a ZHS if and only if the exponents a 1 , a 2 and a 3 are pairwise coprime.

Remark 2.3
In fact, we do not really need the theory of weighted Lê-Yomdin singularities and the results from [2, 4.3] to obtain this result. Alternatively, one could directly consider Theorem 2.1 with the partial resolution of S(a 1 , a 2 , a 3 ) consisting of one weighted blow-up at the origin with weight vector ω. This resolution has one exceptional curve E ⊂ P 2 ω which is isomorphic to the curve C ⊂ P 2 (a 1 ,a 2 ,a 3 ) , and which contains three sets of singular points, corresponding to the coordinate axes in P 2 ω . This gives the same genus and determinant as above. For more details, see [13, Example 3.6].

Singularities of splice type
Since we know how to recover all plane curve singularities of a given topological type, it is natural to ask whether this is also possible for surface singularities with a given link. Unfortunately, this question is still open, even if the link is a QHS or ZHS. Here, we restrict to briefly explaining some results and conjectures in the ZHS case. For more details, see [21,22].
With a ZHS link, we can associate a unique splice diagram, originally introduced by Siebenmann [25]. This is a finite tree in which every vertex has either valency 1, called a leaf, or valency at least 3, called a node, and in which a weight is assigned to each edge starting at a node. In [9], Eisenbud and Neumann showed that the links of normal surface singularities that are a ZHS are in one-one correspondence with splice diagrams satisfying the following conditions: (i) the weights around a node are positive and pairwise coprime; (ii) the weight on an edge connecting a node with a leaf is greater than 1; and (iii) all edge determinants are positive.
Here, the edge determinant for an edge connecting two nodes is the product of the two weights on the edge minus the product of the weights adjacent to the edge (i.e., the other weights around the two nodes).
The dual graph of a normal surface singularity with a ZHS link yields a unique splice diagram as follows. First, we suppress all vertices with valency 2. Then, for each edge e starting at some node v, its weight d ve is the absolute value of the determinant of the intersection matrix of the subgraph ve of obtained from cutting at v in the direction of e. The other way around, one can obtain the dual graph , and, hence, the link, from the splice diagram by splicing or plumbing. For the details of this construction, we refer to [9] or, for an easier method, to [22].
In [21], Neumann and Wahl constructed for a ZHS link whose splice diagram satisfies the so-called semigroup condition an isolated complete intersection surface singularity (S, 0) with this link, called a complete intersection singularity of splice type. Up to date, there are no known examples of normal complete intersection surface singularities with a ZHS link whose splice diagram does not satisfy the semigroup condition, or that are not of splice type. Therefore, Neumann and Wahl conjectured that every normal complete intersection surface singularity with a ZHS link is of splice type (in particular, its splice diagram satisfies the semigroup condition). Earlier, in [20], Neumann and Wahl already conjectured the same for every Gorenstein normal surface singularity with a ZHS link, but Luengo-Velasco, Melle-Hernández and Némethi [11] found counterexamples to this conjecture. Furthermore, even when the splice diagram satisfies the semigroup condition, there always exist plenty of other analytic types (probably not complete intersections) with the same link.
Let us take a brief look at this semigroup condition and how to write the equations of the associated singularity of splice type. Consider a splice diagram . For any vertices v and w of , we define the linking number l vw of v and w as the product of all weights adjacent to, but not on, the shortest path from v to w, and the number l vw as the same product in which we omit the weights around v and w. Using this notation, is said to satisfy the semigroup condition if and only if for every node v and edge e starting at v, the weight d ve is contained in the semigroup l vw | w is a leaf of in ve ⊂ N, where ve is the subgraph of cut off from v by e. If satisfies this semigroup condition, we can associate admissible monomials with each node v and an edge e starting at v as follows. Relate to each leaf w of a variable z w and give it v-weight l vw . Because satisfies the semigroup condition, we can find (possibly non-unique) integers α vw ∈ N such that Then, an admissible monomial associated with v and e is any monomial Note that its v-weight is equal to the product of all weights around v, also denoted by d v . If the node v has valency δ v , then we choose for every edge e at v one admissible monomial M ve and we make a system of δ v − 2 linear equations of the form where the coefficients a ie are chosen such that all maximal minors of the matrix (a ie ) i,e have full rank. Finally, in each equation, we can add higher order terms with respect to the weights l vw . The total number of equations is equal to n − 2, where n is the number of leaves in , and these equations define an isolated complete intersection singularity in C n of splice type. If one does not allow higher order terms, it is said to be of strict splice type. For other examples of this construction, we refer to [21,22]. In Sect. 4.3, we will check the semigroup condition and write the equations of strict splice type for our normal surface singularities having a ZHS link. In particular, we will see that they are of splice type. Hence, they support the conjecture of Neumann and Wahl.

Quotient singularities and Q-resolutions
To determine the conditions under which our surface singularities (S, 0) have a QHS or ZHS link, we will make use of a good Q-resolution of (S, 0). Roughly speaking, this a resolution in which the final ambient space can have abelian quotient singularities, and the exceptional divisor must have normal crossings on such a variety. In this section, we give a short introduction to quotient singularities and Q-resolutions. We also touch briefly on an intersection theory on surfaces with abelian quotient singularities. More details can be found in [3] and [4].
Consider an abelian quotient space C n /G for G ⊂ G L(n, C) a finite abelian group. If we write G = μ d 1 × · · · × μ d r as a product of finite cyclic groups, where μ d i is the group of the d i th roots of unity, then there exists Note that we can always consider the ith row of A modulo d i . The class of an element is a normal irreducible n-dimensional variety whose singular locus is of codimension at least two and is situated on the coordinate hyperplanes {x i = 0} for i = 1, . . . , n, which are the images of the coordinate hyperplanes {x i = 0} in C n under the natural projection C n → X (d; A). If n = 2, then one can show that each quotient space X (d; A) = C 2 /G is cyclic, that is, it is isomorphic to a quotient space of type (d; a, b). A cyclic type (d; a, b) is said to be normalized, and the corresponding quotient space X (d; a, b) is said to be written in a normalized form, if and only if gcd(d, a) = gcd(d, b) = 1. If this is not the case, we can normalize X (d; a, b) as follows. First, we can assume that gcd(d, , and similarly for some k dividing d and a. Hence, ) .
For general n ≥ 1, we call a (not necessarily cyclic) type (d; A) normalized if μ d is a small subgroup of G L(n, C) (i.e., it does not contain rotations around hyperplanes other than the identity) acting freely on (C * ) n or, equivalently, if for all x ∈ C n with exactly n − 1 coordinates different from 0, the stabilizer subgroup is trivial. It is possible to convert any type into a normalized form. For n = 2, we can simplify a normalized type (d; a, b) even further. More precisely, as gcd(d, a) = 1, there exists an integer a ∈ Z with gcd(d, a ) = 1 such that aa ≡ 1 mod d. 1, a b) under the identity morphism. In other words, every two-dimensional quotient space singularity (X (d; A), [0]) is a Hirzebruch-Jung singularity (C 2 /μ d , 0) where the action of μ d on C 2 is given by (ξ, (x 1 , x 2 )) → (ξ x 1 , ξ q x 2 ) for some integer q ∈ {1, . . . , d − 1} with gcd(d, q) = 1. This is called a Hirzebruch-Jung singularity of type 1 d (1, q). Similarly, we could start with an integer b ∈ Z such that gcd(d, b ) = 1 and bb ≡ 1 mod d. In this case, we find that X (d; a, b) is isomorphic to X (d; q , 1), where q q ≡ 1 mod d. In other words, a Hirzebruch-Jung singularity of some type 1 d (1, q) is always equal to the Hirzebruch-Jung singularity of type 1 d (q , 1) for q ∈ {1, . . . , d − 1} the unique solution of qq ≡ 1 mod d. It is well known that the minimal good resolution of a Hirzebruch-Jung singularity has only rational exceptional curves and a bamboo-shaped (i.e., linear) dual graph.
can be computed from the continued fraction expansion and the positive integers d, q and q are the absolute value of the determinant of the intersection matrix of all exceptional curves, of E 1 , …, E r −1 , and of E 2 , …, E r , respectively.
Before we can give the precise definition of a good Q-resolution, we still need to introduce two notions: V -manifolds and Q-normal crossing divisors. In [24], a V -manifold of dimension n was introduced as a complex analytic space admitting an open covering {U i } in which each U i is analytically isomorphic to some quotient B i /G i for B i ⊆ C n an open ball and G i a finite subgroup of G L(n, C). We consider V -manifolds in which every G i is a finite abelian subgroup of G L(n, C), which are normal varieties that can locally be written like X (d; A). An important example of a V -manifold is the weighted projective space P n ω of type ω for some weight vector ω = ( p 0 , . . . , p n ) of positive integers which is defined as the quotient of C n+1 \{0} under the action C * × (C n+1 \{0}) → C n+1 \{0} given by (t, (x 0 , . . . , x n )) → (t p 0 x 0 , . . . , t p n x n ). A two-dimensional V -manifold with abelian quotient singularities is also called a V -surface. A Q-normal crossing divisor on a V -manifold X is a hypersurface D that is locally isomorphic to the quotient of a normal crossing divisor under an action (d; A). More precisely, for every point p ∈ X , there exists an isomorphism of germs (X , p) (X (d; A), [0]) such that (D, p) ⊆ (X , p) is identified with a germ of the form

Remark 2.5
In modern language, one usually calls a V -manifold an orbifold. We keep saying V -manifold in this article to emphasize that we follow Steenbrink's approach.
We can now define a good Q-resolution for a germ (X , 0) of an isolated singularity as a proper birational morphism π :X → X such that the following properties hold: (i)X is a V -manifold with abelian quotient singularities; (ii) π is an isomorphism over X \{0}; and (iii) the exceptional divisor π −1 (0) is a Q-normal crossing divisor onX .
For (Y , 0) ⊂ (X , 0) a subvariety of codimension one, an embedded Q-resolution is a proper birational morphism π :X → X with the above three properties in which X \{0} is replaced by X \Sing(Y ), and π −1 (0) by π −1 (Y ). As for a classical good or embedded resolution, we can use the construction of blowing up to compute a good or embedded Q-resolution, but in this case, we use weighted blow-ups. Although weighted blow-ups can be placed in the realm of toric resolutions, we follow the approach in [3,4].
We end this section by briefly discussing an intersection theory on surfaces with abelian quotient singularities. On normal surfaces, an intersection theory was first defined by Mumford [15] and further developed by Sakai [23]; a general intersection theory can be found in [10]. For V -manifolds of dimension 2, which are normal surfaces, an equivalent definition was given in [4]. Here, we focus on explaining the definitions and properties presented in the latter article that are needed in the present article. First of all, on a V -surface S, the notions of Weil and Cartier divisor coincide after tensoring with Q. More precisely, for every Weil divisor D on S, there exists an integer k ∈ Z such that k D is locally principal. Therefore, we call the class of divisors on S with rational coefficients modulo linear equivalence the Q-divisors on S, and we can develop a rational intersection theory. In this article, we will only need to compute the local intersection number (D 1 · D 2 ) p of two Q-divisors D 1 and D 2 at a point p ∈ S. For this purpose, we assume that p is the origin [0] in a normalized cyclic quotient space X (d; a, b), that D i = { f i = 0} for i = 1, 2 is given by a reduced polynomial in C[x, y], that the support of D 1 is not contained in the support of D 2 , and that D 1 is irreducible. In this case, the local intersection number at p is well-defined and given by Another property of the intersection product that we will use is that for π :X → X (d; a, b) a weighted blow-up at the origin with exceptional divisor E, and for D a Q-divisor on This can be shown in the same way as the analogous statement for the classical blow-up.

Our family of normal surface singularities
In this section, we introduce the family of normal surface singularities of our interest that appear in the proof from [16] of the monodromy conjecture for a space monomial curve introduced in Sect. 2.1. We also introduce a good Q-resolution, which we immediately use to show that these singularities for g ≥ 3 are not Brieskorn-Pham and to show the conditions for their link to be a QHS. In the next section, we will use the same resolution to identify the singularities in this family with a ZHS link, and to show that these are of splice type.

A good Q-resolution of our surface singularities
In [16, Section 5], the computation of g weighted blow-ups ϕ k for k = 1, . . . , g yields an embedded Q-resolution ϕ = ϕ 1 • · · · • ϕ g :Ŝ → S of the space monomial curve Y given by (2) seen as Cartier divisor on S. Because the surface S is already Q-resolved after the first g − 1 blow-ups, and the last step is needed to desingularize the curve Y , we can consider the good Q-resolutionφ := ϕ 1 •· · ·•ϕ g−1 :Ŝ → S of (S, 0). We will now explain the properties of this resolution that are needed to see that (S, 0) is not Brieskorn-Pham for g ≥ 3, and to prove the characterization for (S, 0) to have a QHS or ZHS link from Theorem 1. For more details, we refer to [16,Section 5].
First of all, for each blow-up ϕ k for k = 1, . . . , g − 1, we denote the exceptional divisor by E k . To ease the notation, we also denote their strict transform under later blow-ups by E k . Hence, in the end, the exceptional curves of the good Q-resolutionφ are the irreducible components of these E k . If we define then each E k is the disjoint union of r k isomorphic irreducible components that we denote by E k j for j = 1, . . . , r k . In particular, the last exceptional divisor E g−1 is always irreducible, and the pull-back of the Cartier divisor Y underφ is given bŷ whereŶ is the strict transform of Y underφ, and N k for k = 1, . . . , g − 1 is the multiplicity of E k , which is equal to lcm(β k e k , n k , . . . , n g ). Furthermore, for g ≥ 3, each divisor E k for k = 2, . . . , g − 2 (if g ≥ 4) only intersects E k−1 and E k+1 , and E g−1 only intersects E g−2 .
For every k = 1, . . . , g − 2, the intersections of E k and E k+1 are equally distributed, that is, each of the components E (k+1) j of E k+1 intersects precisely r k r k+1 components of E k , each component E k j of E k is intersected by only one of the components of E k+1 , and each nonempty intersection between two components E k j and E (k+1) j consists of a single point. In other words, the dual graph of the good Q-resolutionφ :Ŝ → S is a tree as in Fig. 1.
It is important to note thatφ is not a good resolution of (S, 0) asŜ still contains a lot of singularities that need to be resolved. To explain these singularities, we put M k := lcm(β k e k , n k+1 , . . . , n g ) for k = 0, . . . , g, and we consider the divisors H i for i = 0, . . . , g on S defined by {x i = 0} ∩ S ⊂ C g+1 . Again, to ease the notation, we denote their strict transforms also by H i throughout the process. We further consider the curve Y whose strict transform is always denoted byŶ . In the resolution of (S, 0), each H k for k = 1, . . . , g − 1 is separated fromŶ at the kth step and intersects the kth exceptional divisor E k transversely at some singular point(s). More precisely, if we denote a point in the intersection E k ∩ H k by Q k , then there areβ k M k such points which are equally distributed along the r k components of · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · Because gcd gcd(e k−1 , n kβk n k+1 , . . . , n kβk n g ),β k = gcd e k , n kβk n k+1 , . . . , n kβk n g , these points are Hirzebruch-Jung singularities of type 1 d k (1, q k ) with and the third equality follows from the definition n k = e k−1 e k and the fact that gcd(β k e k , n k ) = 1. For later purposes, we can rewrite d k as (n k+1 , . . . , n g ) gcd n kβk e k , lcm(n k+1 , . . . , n g ) = n k gcd n k , lcm(n k+1 , . . . , n g ) = n k r k r k−1 , where we extend the sequence r 1 , . . . , r g−1 with r 0 := = X gcd n 0β0 n 1 , n 0β0 n 2 , . . . , n 0β0 n g ;β 0 , −1 Each component of E 1 contains the same number of such points, which are of type 1 d 0 (q 0 , 1) with d 0 := gcd n 0β0 n 1 , n 0β0 n 2 , . . . , n 0β0 n g gcd β 0 , n 0β0 n 1 , . . . , n 0β0 where we again used relation (8) and the fact thatβ 1 e 1 = n 0 . For g ≥ 3, a next set of singular points ofŜ are the points in an intersection E k ∩ E k+1 for k = 1, . . . , g − 2 that we denote by Q k(k+1) . We have already explained that there are r k such points in total, one on each component of E k , and that each component of E k+1 contains r k r k+1 such points. Furthermore, the local situation around Q k(k+1) can be described in the variables One can show that these are cyclic quotient singularities with the order of the underlying small group as follows. First, by multiplying conveniently, we can rewriteŜ into the form X ( d d | a 1 a 2 a 3 a 4 ), where d = n k+1βk+1 − n kβk . Second, the group automorphism (ξ, η) given by the identity. Using such an automorphism repeatedly yields an isomorphism where α, β ∈ Z such that gcd(a 1 , a 3 ) = αa 1 + βa 3 . Third, every quotient space of the form a 4 ) 2 ) .
Finally, we can rewrite the resulting cyclic singularity into a Hirzebruch-Jung singularity as explained in Sect. 2.5. We do not provide more details as we will not need an explicit expression for d k(k+1) in general. It is, however, worth mentioning that this approach can be used to show that any quotient space X (d; A) = C 2 /μ d is isomorphic to a cyclic quotient space, and that we will illustrate this approach when the link of (S, 0) is a ZHS, see Sect. 4.3.
The last singular point ofŜ for g ≥ 2 is the intersection point P g−1 :  Fig. 2 The good Q-resolution of (S, 0) Clearly, this point is a Hirzebruch-Jung singularity of type 1 To recapitulate, we visualize the good Q-resolutionφ as in Fig. 2, which shows the exceptional curves and the singular points. For simplicity, the components of each E k are represented by lines, but we will see in a moment that they are not rational in general.
Using Corollary 6.5 from [16], we can compute the Euler characteristic of the exceptional curves ofφ. More precisely, this result gives an expression for the Euler characteristic of the exceptional divisor E k for k = 1, . . . , g − 1 without its singularities: it states thať Hence, the Euler characteristic of E k can be easily computed by adding the cardinality of all its singularities. This yields: Because the components E k j for j = 1, . . . , r k are disjoint and isomorphic, their Euler characteristic is equal to χ(E k j ) = χ(E k ) r k . Using that N k M k = n k r k r k−1 = n k gcd(n k ,lcm(n k+1 ,...,n g )) for k = 1, . . . , g − 1, see (9), and thatβ k r k M k = gcd β k e k , lcm(n k+1 , . . . , n g ) , we can rewrite these Euler characteristics as χ(E k j ) = 2 − gcd n k , lcm(n k+1 , . . . , n g ) − 1 gcd β k e k , lcm(n k+1 , . . . , n g ) − 1 .
We indeed see that the exceptional curves are not rational in general. Even more, this implies that the genus of E k j is zero if and only if gcd n k , lcm(n k+1 , . . . , n g ) = 1 or gcd β k e k , lcm(n k+1 , . . . , n g ) = 1 for k = 1, . . . , g − 1.
Since the dual graph of the good Q-resolutionφ :Ŝ → S is a tree and the quotient singularities ofŜ can be resolved with bamboo-shaped dual graphs and rational exceptional curves, these are already the conditions under which (S, 0) has a QHS link. In other words, we have already shown the first part of Theorem 1.
If g ≥ 3, then we claim that (S, 0) ⊂ (C g+1 , 0) is never a Brieskorn-Pham singularity. To prove this, we will show that the minimal good resolution of (S, 0) contains at least g − 1 rupture exceptional curves. An irreducible exceptional curve is called rupture if either its genus is positive, or its genus is zero and it has valency at least 3 (i.e., it intersects at least three times other components of the exceptional locus). This implies that (S, 0) is indeed not Brieskorn-Pham for g ≥ 3 as a Brieskorn-Pham surface singularity has at most one rupture exceptional curve in its minimal good resolution. The latter can be seen by considering a good Q-resolution of a Brieskorn-Pham surface singularity consisting of one weighted blow-up at the origin which yields one irreducible exceptional curve E containing three sets of Hirzebruch-Jung singularities. We refer for more details to [13,Example 3.6]; see also Remark 2.3. As each of these singularities can be minimally resolved with a bamboo-shaped dual graph and rational exceptional curves, the only possible rupture exceptional curve in the obtained good resolution is the strict transform of E. This implies that the minimal good resolution of a Brieskorn-Pham singularity indeed contains at most one rupture exceptional curve. Even more, the minimal good resolution of a Brieskorn-Pham surface singularity has no rupture exceptional curve if and only if it has only rational exceptional curves and a bamboo-shaped dual graph or, thus, if and only if the singularity is a cyclic quotient singularity.
To show that the minimal good resolution of (S, 0) has at least g − 1 rupture exceptional curves, we make use of the good Q-resolutionφ :Ŝ → S of (S, 0), from which we can obtain a (not necessarily minimal) good resolution π :S → S of (S, 0) by minimally resolving the singularities ofŜ. Since these singularities are all Hirzebruch-Jung, the only possible rupture exceptional curves of π are the strict transforms of the exceptional curves of the good Qresolution. The next result immediately implies that the good resolution π has at least g − 1 exceptional curves that are rupture. (S, 0) ⊂ (C g+1 , 0) be a normal surface singularity defined by the Eq. (6) with g ≥ 3. Consider the good Q-resolutionφ :Ŝ → S of (S, 0) introduced in Sect. 3.2. Then, (i) each exceptional curve E k j for k = 1, . . . , g − 2 and j = 1, . . . , r k yields a rupture exceptional curve in the good resolution π :S → S of (S, 0) coming fromφ; and (ii) if r g−2 = 1 (i.e., the exceptional divisor E g−2 is irreducible), then E g−1 yields a rupture exceptional curve in the good resolution π :S → S of (S, 0) coming fromφ.
In other words, E 1 j will indeed always yield a rupture exceptional curve.
It remains to show the second part. If r g−2 = 1, then gcd(n g−1 , n g ) = 1, which implies that E g−1 has zero genus. Furthermore, it has one intersection point with E g−2 , one point P g−1 with order d = n g gcd β g−1 n g , n g , andβ g−1 M g−1 = gcd(β g−1 n g , n g ) points Q g−1 with order d g−1 = n g−1 > 1. We can again conclude: if gcd(β g−1 n g , n g ) ≥ 2, then E g−1 contains at least two singular points Q g−1 with order d g−1 > 1; if gcd(β g−1 n g , n g ) = 1, then E g−1 contains exactly two singular points, namely one Q g−1 with order d g−1 > 1, and P g−1 with order d = n g > 1.
We still need to show that the minimal good resolution of (S, 0) contains at least g − 1 rupture exceptional curves. From Proposition 3.1, it follows that each exceptional curve E k j for k = 1, . . . , g − 2 and j = 1, . . . , r k can not be contracted in the good resolution π :S → S; either its genus is positive so that Castelnuovo's Contractibility Theorem does not apply, or it has at least three intersections with other exceptional curves so that the exceptional locus would not be a simple normal crossing divisor after contracting E k j . The same applies to E g−1 if r g−2 = 1 or r g−2 ≥ 3. In other words, in these cases, the good resolution π is minimal. If r g−2 = 2, it is possible that E g−1 is superfluous as the next example shows. However, the obtained minimal good resolution of (S, 0) coming from contracting E g−1 (and possibly executing subsequent contractions) will still have at least g − 1 rupture exceptional curves: all the exceptional curves E k j for k = 1, . . . , g − 2 and j = 1, . . . , r k are rupture, where r k ≥ 1 for k = 1, . . . , g − 3 (if g ≥ 4) and r g−2 = 2.

Example 3.2
If r g−2 = 2, then it is possible that the good resolution π :S → S is not minimal. For example, consider the surface S ⊂ C 4 defined by The semigroup of the corresponding space monomial curve Y ⊂ C 4 is minimally generated by (8,12,26,53). From the properties of the good Q-resolutionφ explained above, one can easily check the following: (i) the first exceptional divisor E 1 has r 1 = 2 components E 11 and E 12 that each contain two singular points Q 0 of type 1 3 (1, 1), while every point Q 1 is smooth; (ii) the genus of E 2 is zero, and the points P 2 and Q 2 are smooth; and (iii) the intersection of E 1 and E 2 consists of two singular points Q 12 , one on each component of E 1 , that are Hirzebruch-Jung of type 1 7 (1, 3).
It follows that the dual graph of π :S → S is as in Fig. 3, where we denote the strict transforms of E 1 j and E 2 still by E 1 j and E 2 , respectively, and where the exceptional curves E 0 j and E 12 j come from resolving the singularities Q 0 and Q 12 , respectively. Furthermore, one can show that the pull-back of Y is given whereŶ is the strict transform of Y . Because π * Y · E 2 = 0 by (5) andŶ · E 2 = 2, which can be seen from the local equation (13), we find that the self-intersection number of E 2 is −1.
Hence, by Castelnuovo's Contractibility Theorem, the exceptional curve E 2 can be contracted in order to find the minimal good resolution of (S, 0). However, this minimal good resolution has still g − 1 = 2 rupture exceptional curves, namely E 11 and E 12 .

Conditions for integral homology sphere link
In this section, we will prove the second part of Theorem 1 for g ≥ 3 using the good Qresolutionφ :Ŝ → S of (S, 0) introduced in Sect. 3.2. To this end, following Theorem 2.1, we will investigate the determinant of (S, 0) with formula (3) in terms ofφ. Remark 4.1 (i) Note that Theorem 1 generalizes the g = 2 case or, thus, the classification for Brieskorn-Pham surface singularities in Proposition 2.2. Even more, for g = 2, one could also obtain this result by using the good Q-resolutionφ := ϕ 0 :Ŝ → S of (S, 0). (ii) When the link of (S, 0) is a ZHS, we see that r k = 1 and N k = n kβk for every k = 1, . . . , g − 1. Hence, all exceptional divisors E k for k = 1, . . . , g − 1 are irreducible with multiplicity n kβk , and the dual graph of the good Q-resolutionφ :Ŝ → S is bamboo-shaped with quotient singularities as described in Sect. 3.2. In particular, by Proposition 3.1, the good resolution of (S, 0) obtained fromφ by resolving the singularities ofŜ is minimal.

The determinant of the intersection matrix of the good Q-resolution'
Because we already know the singularities ofŜ, we will be able to compute the determinant of (S, 0) once we know the determinant of the intersection matrix A ofφ. To compute the latter, we first need to calculate the (self-)intersection numbers of the exceptional curves E k j for k = 1, . . . , g − 1 and j = 1, . . . , r k . Clearly, from the local situation (10) around Q k(k+1) To find the self-intersection numbers −a k := E 2 k j , we can use the fact thatφ * Y · E k j = 0, see (5), whereφ * Y is given by (7). SinceŶ only intersects E g−1 in the single point P g−1 with local situation (13), we know thatŶ · E g−1 = n g d andŶ · E k j = 0 for k = 1, . . . , g − 2 and j = 1, . . . , r k . We obtain We can now write the intersection matrix A as follows: Here, we denote by A k for k = 1, . . . , g − 1 the (r k × r k )-diagonal matrix with −a k on the diagonal, by A k,k+1 for k = 1, . . . , g − 2 the (r k × r k+1 )-matrix where D k,k+1 is the r k r k+1 -column vector ( 1 d k(k+1) , . . . , 1 d k(k+1) ) t , and by A k+1,k = A t k,k+1 for k = 1, . . . , g −2 the transpose of A k,k+1 . Note that A g−1 = −a g and A g−2,g−1 = D g−2,g−1 .
We will now show a formula for the determinant det(A) of a general matrix A defined as in (16). Hence, this formula can be used to compute the determinant of the intersection matrix for any good Q-resolution with a dual graph as in Fig. 1, in which the horizontally aligned exceptional curves are isomorphic, have the same self-intersection number, and have the same intersection behavior with the other exceptional curves.
We need to show that (a) + (b) + (c) = (d) with It is trivial that (b) corresponds to K = {(l, l + 1)} in (d). Using that [ l+1 2 ] = [ l−1 2 ] + 1, one can also see that (c) yields the part in (d) where (l, l + 1) ∈ K and |K | ≥ 2. Hence, it remains to show that (a) corresponds to the part in (d) where (l, l + 1) / ∈ K . Clearly, we only need to check that the boundaries for |K | agree; in (a), the upper bound is [ l 2 ], while in (d), the upper bound is [ l+1 2 ]. However, in (d), we need to take into account that (l, l + 1) / ∈ K . We remark the following two facts: Hence, if l + 1 is even, then |K | for K in (d) with (l, l + 1) / ∈ K varies between 1 and In other words, the boundaries for |K | agree. Likewise, if l + 1 is odd, then K in (d) with (l, l + 1) / ∈ K can still attain the upper bound [ l+1 2 ] = [ l 2 ]. This recurrence relation will be very useful for showing the next formula for det(A). (16) for some g ≥ 3, r k ≥ 1 for k = 1, . . . , g − 1 with r g−1 = 1, and d k(k+1) ≥ 1 for k = 1, . . . , g − 2. We have

Proposition 4.3 Let A be a matrix defined as in
Using the recurrence relation from Lemma 4.2 and the expressions in (15) for a k for k = 1, . . . , g − 1 in which r k−1 r k = p k−1 , it is not hard to see that, in our case, the expression for R l simplifies to This immediately yields the following expression for the determinant of the intersection matrix of the good Q-resolution of our surface singularities.

.2. The determinant of the intersection matrix A ofφ is given by
.
In order to better understand the idea of the proof of Proposition 4.3, we first consider the simple case where r k = 1 for all k = 1, . . . , g − 1, and A is the tridiagonal matrix ⎛ If we denote this matrix for a moment by A(g) for g ≥ 3, then the general three-term recurrence relation for the determinant of tridiagonal matrices tells us that where, by convention, we put A(1) = 1 and A(2) = (−a 1 ). This recurrence relation can be shown by first expanding the determinant of A(g) along the last column (resp. row) and then expanding the minor corresponding to 1 d (g−2)(g−1) along the last row (resp. column). Note the similarity between this relation and the relation from Lemma 4.2. Even more, by induction on g and with exactly the same argument as in the proof of Lemma 4.2, one can show that det(A(g)) = (−1) g−1 R g−1 for g ≥ 3, in which p k = 1 for all k = 1, . . . , g − 2. In other words, the recurrence relation satisfied by the R l for l = 1, . . . , g − 2 in Lemma 4.2 is a generalization of (18) by allowing general p k ≥ 1 for k = 1, . . . , g − 2.
For general s, we can write the determinant of B s as (−1) g−s R g−s in which we start with a s instead of a 1 . We will write det(A) (for g ≥ 4) in terms of these tridiagonal matrices using the formula in the next result.
Assume that 2 ≤ t ≤ g − 2. Then, Proof First, note that such t ∈ {1, . . . , g − 1} always exists as r g−1 = 1. Furthermore, note that r k = 1 for all k ≥ t so that p k = 1 for all k ≥ t. With the expression for det(B t ) ( resp. det(B t+1 )) in terms of R g−t ( resp. R g−t−1 ) in which we start with a t ( resp. a t+1 ) instead of a 1 and all p k = 1, we can show this formula with similar arguments as in the proof of Lemma 4.2. However, we will prove the stronger result that for all s = t, . . . , g − 2 by using backward induction and the statement of Lemma 4.2. For s = g − 2, we need to consider det(−a g−1 ) 1) , and show that this is equal to (−1) g−s R g−1 = R g−1 . This follows from first applying Lemma 4.2 for l = g − 3 and then for l = g − 2 with p g−2 = 1. If t = g − 2, we are done. Otherwise, suppose it is true for s + 1 ≤ g − 2. For s, we first expand det(B s ) along the first column and then expand the second minor along the first row to get This way of rewriting det(B s ) is the same as the one we can use to show the three-term recurrence relation (18) for the tridiagonal matrices A(g), but with expansion along the first column instead of along the last column. Because of the similarity between the relations in (18)  det(B s+2 ).
Since p s = 1 as s ≥ t, we can conclude with the induction hypothesis.
We are now ready to prove Proposition 4.3 by using these matrices B s .

Proof of Proposition 4.3
As in the previous lemma, let t be the smallest k ∈ {1, . . . , g − 1} such that r k = 1. If t = 1, we already know that det(A) = (−1) g−1 R g−1 . For t ≥ 2, we will show that det(A) = (−1) Because r k = 1 for k ≥ t, this yields the formula given in the proposition. Throughout the proof, we will denote by A(r 1 , . . . , r g−1 ) a matrix defined as in (16) corresponding to some r 1 , . . . , r g−1 ≥ 1 with g ≥ 3 in which we also allow r g−1 > 1. To get an idea on how to show the above formula for general t, we first consider t = 2, t = 3 and t = 4. column corresponding to the first entry of A 2 , and in both steps, we simplify the second minor corresponding to 1 d 12 and 1 d 23 , respectively. In other words, we rewrite det(A) as Because n 0 and n 1 are coprime by assumption, we indeed find that (S, 0) has a ZHS link if and only if the exponents n i for i = 0, . . . , g are pairwise coprime and gcd(β k e k , n k+1 · · · n g ) = gcd(β k e k , e k ) = 1 for k = 2, . . . , g − 1. This ends our proof of Theorem 1.

Our surface singularities with ZHS link versus singularities of splice type
We finish this article by showing that if (S, 0) has a ZHS link, then it is of splice type. In other words, they belong to the family of complete intersection singularities of splice type defined by Neumann and Wahl and support their conjecture on the possible normal complete intersection surface singularities with a ZHS link. Since (S, 0) for g = 2 is trivially of splice type, we assume that g ≥ 3. We first determine the splice diagram of (S, 0). We can again use the good Q-resolutionφ :Ŝ → S. In Remark 4.1, we already mentioned that each exceptional divisor E k for k = 1, . . . , g − 1 is irreducible with multiplicity N k = n kβk , and that the dual graph is bamboo-shaped with quotient singularities as described in Sect. 3.2. Taking a closer look at these singularities, one can check that the resolutionφ is as in Fig. 4, where the numbers in brackets represent the orders of the small groups acting on the singular points.
It immediately follows that the splice diagram is of the form as in Fig. 5, in which the nodes from left to right correspond to E k for k = 1, . . . , g − 1, and the edge weights n k for k = 0, . . . , g come from the singular points Q k for k = 0, . . . , g − 1 and P g−1 . It remains to show that the other weights are given as in the figure. We start by showing that the order d k(k+1) corresponding to Q k(k+1) = E k ∩ E k+1 for k = 1, . . . , g − 2 becomes very easy. Following the approach explained in Sect. 3.2, we need to consider the quotient space Because gcd(β k+1 e k+1 , e k+1 ) = 1 by assumption on the link, and gcd(β k+1 e k+1 , n k+1 ) = 1 by the properties of the semigroup, we see that gcd(β k+1 e k+1 , e k ) = gcd(β k+1 e k+1 , n k+1 e k+1 ) = 1. Hence, the isomorphism in (12) says that this quotient space is isomorphic to X n k+1βk+1 − n kβk 1 −αe k + β n kβk e k n k+1βk+1 − n kβk 0 n kβk − n k+1βk+1 = X n k+1βk+1 − n kβk ; 1, −αe k + β n kβk e k , where αe k − ββ k+1 e k+1 = 1. It follows that d k(k+1) = n k+1βk+1 − n kβk = N k+1 − N k . Note that this is consistent with (11). Using this, one can also easily see that the expressions (15) for the self-intersection numbers −a k := E 2 k j become for k = g − 1.
Let us now take a look at the edge to the right of E 1 . To show that its weight is e 1 , we need to compute the determinant of the intersection matrix corresponding to the dual graph coming from removing E 1 in Fig. 4 and resolving the singular points on E 2 , . . . , E g−1 . By (3), this is equal to