Flat semigroups and weighted homogeneous surface singularities

TL;DR

This paper investigates the conditions under which flat semigroups can be realized as numerical semigroups associated with weighted homogeneous surface singularities, establishing a characterization involving quotients of flat semigroups.

Contribution

It provides a complete characterization of representable flat semigroups as quotients of flat semigroups, linking algebraic properties to geometric realizations.

Findings

A semigroup is representable if and only if it is a quotient of a flat semigroup.

Characterization of representability in terms of flat semigroup quotients.

Connection between algebraic semigroup properties and geometric surface singularities.

Abstract

We consider numerical semigroups associated with normal weighted homogeneous surface singularities with rational homology sphere links. We say that a semigroup is representable if it can be realized in this way. In this article, we study the representability of flat semigroups and prove that a numerical semigroup is representable if and only if it can be written as a quotient of a flat semigroup.

Equations

$${\mathcal{S}}_{(X,o)}:=\{\ell \in {\mathbb{Z}}_{\ge 0}|R_{X,\ell }\ne 0\}$$

$$0\to L\to H_2(\tilde{X},M,\mathbb{Z})\to H_1(M,\mathbb{Z})\to 0.$$

$$L_h^{\prime }=\{{\ell }^{\prime }\in L^{\prime }\mid [{\ell }^{\prime }]=h\}\subset L^{\prime }.$$

$${\mathcal{S}}_{\mathbb{Q}}:=\{{\ell }^{\prime }\in L_{\mathbb{Q}}|({\ell }^{\prime },E_v)\le 0\text{mbox}forallv\in \mathcal{V}\},$$

$$Z_K-E=\sum _{v\in \mathcal{V}}({\delta }_v-2)E_v^{\ast },$$

$${\alpha }_i/{\omega }_i=[b_{i1},\dots ,b_{i{\nu }_i}]=b_{i1}-1/(b_{i2}-1/(\cdots -1/b_{i{\nu }_i})\dots )(b_{ij}\ge 2),$$

$${\omega }_i{\omega }_i^{\prime }\equiv 1(\mathrm{mod}{\alpha }_i),0<{\omega }_i^{\prime }<{\alpha }_i.$$

$$Sf=(-b_0,g;({\alpha }_i,{\omega }_i{)}_{i=1}^d).$$

$$\mathfrak{h}={\alpha }_1\cdots {\alpha }_d|e|\text{mbox}and\mathfrak{o}=\alpha |e|.$$

$$\gamma:=\frac{1}{|e|}\cdot\Big{(}d-2-\sum_{i=1}^{d}\frac{1}{\alpha_{i}}\Big{)}\in\mathbb{Q},$$

$$-(Z_K,E_i^{\ast })=1+(\gamma -{\omega }_i^{\prime })/{\alpha }_i.$$

$$E_{v_{ij}}^{\ast }=m_{ij}{E}_i^{\ast }-\sum _{j<r\le {\nu }_i}{m}_{ijr}{E}_{v_{ir}},$$

$$f_{G(a_{1},\dots,a_{n})}\leq T(a_{1},\dots,a_{n}):=\sum_{i=1}^{n}\Big{(}\frac{d_{i-1}}{d_{i}}-1\Big{)}a_{i},$$

$$f=T⟺a_{i+1}/d_{i+1}\in G(a_1/d_i,\dots ,a_i/d_i)\text{mbox}forevery1\le i\le n-1.$$

$$B(a_1,a_2,\dots ,a_n):=(n-1)\mathrm{lcm}(a_1,\dots ,a_n)-\sum _i{a}_i,$$

$$A=\{a_1,\dots ,a_n\}=\{{\hat{s}}_1{\hat{q}}_1,\dots ,{\hat{s}}_n{\hat{q}}_n\}.$$

$$f_{{\mathcal{S}}}=a\Big{(}n-1-\sum_{i}\frac{1}{q_{i}}\Big{)}.$$

$$f_{\mathcal{S}}=\sum _i\left(q_i-1\right)a_i-\prod _i{q}_i,$$

$$R_X={\oplus }_{\ell \ge 0}{R}_{X,\ell }={\oplus }_{\ell \ge 0}{H}^0(E_0,{\mathcal{O}}_{E_0}(D^{(\ell )}))$$

$$N(\ell):=\deg D^{(\ell)}=b_{0}\ell-\sum_{i=1}^{d}\Big{\lceil}\frac{\ell\omega_{i}}{\alpha_{i}}\Big{\rceil}.$$

$${\mathcal{S}}_{(X,o)}=\{\ell \in \mathbb{Z}|N(\ell )\ge 0\}.$$

$$N(\ell )+N(\gamma -\ell )=-2\text{mbox}forany\ell \in \mathbb{Z}.$$

$$f_{{\mathcal{S}}_M}=\gamma +\frac{1}{|e|}-\check{s},$$

$$f_{{\mathcal{S}}_{\Gamma }}=\alpha +\gamma .$$

$$N(\ell )\ge 0\text{mbox}ifandonlyifN(\alpha +\gamma -\ell )<0\text{mbox}forevery\ell \in \mathbb{Z}.$$

$$S{f}_1=\left(-b_0,({\alpha }_i,{\omega }_i{)}_{i=1}^m\right)\text{and}S{f}_2=\left(-c_0,({\beta }_j,w_j{)}_{j=1}^n\right).$$

$$Sf=\left(-b_0-c_0,({\alpha }_i,{\omega }_i{)}_{i=1}^m,({\beta }_j,w_j{)}_{j=1}^n\right).$$

$${\mathcal{S}}_{{\Gamma }_1}\cap {\mathcal{S}}_{{\Gamma }_2}\subset {\mathcal{S}}_{{\Gamma }_1+{\Gamma }_2}\subset {\mathcal{S}}_{{\Gamma }_1}+{\mathcal{S}}_{{\Gamma }_2}.$$

$${\mathcal{S}}_{{\Gamma }_1+{\Gamma }_2}=\{\ell \in {\mathbb{Z}}_{\ge 0}:N_1(\ell )+N_2(\ell )\ge 0\}\supset \{\ell \in {\mathbb{Z}}_{\ge 0}:N_1(\ell )\ge 0\text{mbox}and{N}_2(\ell )\ge 0\}={\mathcal{S}}_{{\Gamma }_1}\cap {\mathcal{S}}_{{\Gamma }_2}.$$

$${\mathcal{S}}_{{\Gamma }_1}+{\mathcal{S}}_{{\Gamma }_2}=⟨{\mathcal{S}}_{{\Gamma }_1}\cup {\mathcal{S}}_{{\Gamma }_2}⟩=⟨\left\{\ell \in {\mathbb{Z}}_{\ge 0}:N_1(\ell )\ge 0\text{ or }{N}_2(\ell )\ge 0\right\}⟩,$$

Full text

In this article, we study the representability of flat semigroups and prove that a numerical semigroup is representable if and only if it can be written as a quotient of a flat semigroup.

TL is partially supported by the NKFIH Grant “Élvonal (Frontier)” KKP 144148. He is very grateful to the ‘Erdős Center’ for the warm hospitality and for providing with him an excellent research environment during his stay in Budapest.

1. Introduction

1.1.

One of the most classical problems in the theory of numerical semigroups is the Diophantine Frobenius problem, asking the following: given the numerical semigroup $G(a_{1},\dots,a_{n})$ generated by relatively prime integers $a_{1},\dots,a_{n}$, find the largest integer - called the Frobenius number - that is contained in the semigroup, ie. it is not representable as non-negative integer combination of $a_{1},\dots,a_{n}$.

Several ideas from different areas of mathematics have been studied to find ‘closed’ formulae and algorithms to calculate the Frobenius number. From the point of view of the formulae, although several results for peculiar cases and general bound estimates exist in the literature, the problem is still open in full generality. In our forthcomming discussions we will only touch a small part of the ‘classical approaches’, therefore the interested reader might consult for more details with the excelent monograph of Ramírez Alfonsín [RA05].

1.2.

The present article focuses on a relatively new method developed by A. Némethi and the second author in [LN20]. This method is based on a subtle connection with the theory of normal surface singularities. Using geometrical and topological techniques of singularity theory, one can solve the Frobenius problem for numerical semigroups which can be related with certain singularities.

In the theory of surface singularities numerical semigroups appear naturally in many diferent contexts. In this article we consider the case of weighted homogeneous surface singularities.

Let $(X,o)$ be a normal weighted homogeneous surface singularity. Being the germ of an affine variety $X$ with good $\mathbb{C}^{*}$-action, its affine coordinate ring is $\mathbb{Z}_{\geq 0}$-graded: $R_{X}=\oplus_{\ell\geq 0}R_{X,\ell}$. Then the set

[TABLE]

is a numerical semigroup associated with $(X,o)$. In fact, by Pinkham [P77] one knows that if the link $M$ of the singularity $(X,o)$ - which is a negative definite Seifert $3$-manifold in this case - is a rational homology sphere, then ${\mathcal{S}}_{(X,o)}$ is a topological invariant associated with the $3$-manifold $M$. Accordingly, it will be also denoted by ${\mathcal{S}}_{M}$.

In this note we will say that a numerical semigroup is representable if it can be realized as ${\mathcal{S}}_{M}$ for some negative definite Seifert rational homology sphere $M$.

One also follows by [P77] that the representable semigroups can be presented as ${\mathcal{S}}_{M}=\{\ell\in\mathbb{Z}\ |\ N(\ell)\neq 0\}$ where $N(\ell)$ is a quasi-linear function defined by the Seifert invariants of $M$, cf. section 3.3.1. Using this presentation and the arithmetics of the star-shaped minimal good dual resolution graph of the singularity, [LN20] succeeded to find a ‘closed’ formula for the Frobenius number. Moreover, in that work the authors proved that a special class of representable semigroups appears naturally in the classical theory of numerical semigroups: the semigroups associated with Seifert integral homology spheres are exactly the ‘strongly flat semigroups’ with at least $3$ generators, considered by the work of Raczunas and Chrza̧stowski-Wachtel [RChW96] as special semigroups realizing a sharp upper bound estimate with respect to the Frobenius problem. Note that the case of two generators will be threated in this article, clarifying completely the class of strongly flat semigroups.

The mentioned connection and results imposed the following natural questions (see [LN20, sect. 8]):

Characterization Problem**.**

How can the representable semigroups be characterized?

Representability Problem**.**

How big is the set of representable semigroups inside the set of all numerical semigroups?

1.3.

The aim of this work is to give an answer to the characterization problem. In the sequel, we will provide an overview of our study and state the main result of the article.

We study the ‘flat semigroups’ appeared in the same classification theme of [RChW96]. It turns out that they serve as a base towards to the characterization problem. First of all, one proves that they are representable. Moreover, for a given presentation of a flat semigroup we construct a canonical representant $M$ whose properties induces many of the properties of the semigroup. In particular, one can show that every flat semigroup is symmetric and its Frobenius number realizes a sharp upper bound considered by Brauer [Ba42]. Furthermore, they are interesting from singularity theoretic point of view as well, since they are semigroups associated with a special family of isolated complete intersection singularities whose equations are calculated explicitely in section 5.3. Then the main theorem is as follows.

Characterization Theorem**.**

A numerical semigroup is representable if and only if it is a quotient of a flat semigroup.

The strategy of the proof is the following: first of all one shows that the flat semigroups, as well as their quotients are representable. For the reverse statement, we fix a negative definite Seifert rational homology sphere as a representant of a given semigroup. Then, one can perturb its Seifert invariants in such a way that the associated semigroup remains stable. On the other hand, one proves that via this perturbation we achieve a representant whose associated semigroup can be written as a quotient of a flat semigroup.

It is worth to emphasize that the characterization theorem rephrases the representability problem completely via the language of semigroup theory:

‘how big is the set of quotients of flat semigroups in the set of numerical semigroups?’

Further speculations on this question and a connection with the work of [RGS07] and [Sw09] regarding the presentation of semigroups as quotients of symmetric semigroups can be found at the end of the article.

1.4.

The structure of the article is as follows. Section 2 presents the necessary ingredients and preliminaries regarding the topology of normal surface singularities and negative definite Seifert 3-manifolds. Section 3 introduces the Frobenius problem with a view towards the ‘flat’ classification of [RChW96] and discusses the case of semigroups associated with weighted homogeneous surface singularities by presenting the main results of [LN20].

Section 4 contains some general observations regarding representable semigroups. Among others, Lemma 4.1.5 proves that every quotient of a representable semigroup is also representable. As an application, in section 4.2 we show a representation of a semigroup with two generators by ‘multiplying’ the minimal embedded resolution graph of an irreducible plane curve singularity with one Puiseaux pair. Furthermore, in section 4.3 we construct monoids as bounds for a representable semigroup. This serves the base idea for the study of flat semigroups in section 5. In this part we prove that they are representable (Theorem 5.1.3). Moreover, we construct their canonical representants and study some of their properties. In particular, in section 5.3 we give explicit equations for a family of isolated complete intersection singularities whose links are the canonical representants of a flat semigroup.

Finally, section 6 contains the needed results with respect to a perturbation of the Seifert invariants of a reresentant, and proves the main theorem (Theorem 6.1.6) of our work. Section 6.2 ends the paper by giving important examples and discussing a new reformulation of the representability problem.

2. Preliminaries

2.1. On the topology of normal surface singularities with $\mathbb{Q}HS^{3}$ links

2.1.1. Resolution and lattices

We consider a complex normal surface singularity $(X,o)$ and let $\pi:\widetilde{X}\to X$ be a good resolution with dual resolution graph $\Gamma$ whose set of vertices are denoted by ${\mathcal{V}}$. That is, $\pi$ is a birational proper analytic map, which is an isomorphism outside $o$, and $\pi^{-1}(o)$ is a simple normal-crossing divisor. Let $\{E_{v}\}_{v\in{\mathcal{V}}}$ be the irreducible components of the exceptional set $\pi^{-1}(o)$.

The link $M$ of $(X,o)$ is a negative definite plumbed $3$-manifold associated with $\Gamma$. The graph $\Gamma$ encodes the non-degenerate negative definite intersection form together with the collection of genera $\{g(E_{v})\}_{v}$. Furthermore, by a theorem of Grauert [G62], any such connected negative definite plumbing graph can be realized as a dual resolution graph of some normal surface singularity.

In the sequel, in most of the cases we will assume that $M$ is a rational homology sphere ($\mathbb{Q}HS^{3}$), or, equivalently, $\Gamma$ is a tree and all $E_{v}$ are rational (ie. $g(E_{v})=0$).

The smooth complex analytic surface $\widetilde{X}$ is the plumbed $4$-manifold associated with $\Gamma$, with boundary $\partial\widetilde{X}=M$. We define the lattice $L$ as $H_{2}(\tilde{X},\mathbb{Z})$, endowed with the non-degenerate negative definite intersection form $I:=(,)$. It is freely generated by the (classes of the) exceptional divisors $E_{v}$, $v\in{\mathcal{V}}$, that is, $L=\oplus_{v\in{\mathcal{V}}}\mathbb{Z}\langle E_{v}\rangle$. In the homology exact sequence of the pair $(\tilde{X},M)$ one has $H_{2}(M,\mathbb{Z})=0$, $H_{1}(\widetilde{X},\mathbb{Z})=0$, hence the exact sequence has the form:

[TABLE]

There is a perfect pairing $L\otimes H_{2}(\widetilde{X},M,\mathbb{Z})\to\mathbb{Z}$ defined by the Lefschetz-Poincaré duality $H_{2}(\widetilde{X},M,\mathbb{Z})\cong H^{2}(\widetilde{X},\mathbb{Z})$. Hence $L^{\prime}:=\operatorname{{\rm Hom}}(H_{2}(\widetilde{X},\mathbb{Z}),\mathbb{Z})$, the dual lattice of $L$, can be identified with $H_{2}(\widetilde{X},M,\mathbb{Z})$. By (2.1.1) $L^{\prime}/L\cong H_{1}(M,\mathbb{Z})$, which is a finite group and it will be denoted by $H$.

Since the intersection form is non–degenerate, $L^{\prime}$ embeds into $L_{{\mathbb{Q}}}:=L\otimes{\mathbb{Q}}$, and it can be identified with the rational cycles $\{l^{\prime}\in L_{{\mathbb{Q}}}\,:\,(l^{\prime},L)_{{\mathbb{Q}}}\subset\mathbb{Z}\}$, where $(\,,\,)_{{\mathbb{Q}}}$ denotes the extension of the intersection form to $L_{{\mathbb{Q}}}$. Hence, in the sequel we regard $L^{\prime}$ as $\oplus_{v\in{\mathcal{V}}}\mathbb{Z}\langle E^{*}_{v}\rangle$, the lattice generated by the (anti-)dual cycles $E^{*}_{v}\in L_{{\mathbb{Q}}}$, $v\in{\mathcal{V}}$, where $(E_{u}^{*},E_{v})_{{\mathbb{Q}}}=-\delta_{u,v}$ (Kronecker delta) for any $u,v\in{\mathcal{V}}$.

2.1.2. The Lipman cone and minimal representatives of $H$

For $l^{\prime}_{1},l^{\prime}_{2}\in L^{\prime}$ with $l^{\prime}_{i}=\sum_{v}l^{\prime}_{i,v}E_{v}$ ($i=\{1,2\}$) one considers a partial order relation $l^{\prime}_{1}\geq l^{\prime}_{2}$ defined coordinatewise by $l^{\prime}_{1,v}\geq l^{\prime}_{2,v}$ for all $v\in{\mathcal{V}}$. In particular, we say that $l^{\prime}$ is an effective cycle if $l^{\prime}\geq 0$. We set also $\min\{l^{\prime}_{1},l^{\prime}_{2}\}:=\sum_{v}\min\{l^{\prime}_{1,v},l^{\prime}_{2,v}\}E_{v}$ and analogously $\min\{F\}$ for a finite subset $F\subset L^{\prime}$.

For a cycle $l^{\prime}\in L^{\prime}$ we denote by $[l^{\prime}]\in H$ its class in $H:=L^{\prime}/L$. Then the lattice $L^{\prime}$ admits a partition parametrized by the group $H$, where for any $h\in H$ one sets

[TABLE]

Note that $L^{\prime}_{0}=L$. Given an $h\in H$ one can define $r_{h}:=\sum_{v}l^{\prime}_{v}E_{v}\in L^{\prime}_{h}$ as the unique element of $L_{h}^{\prime}$ such that $0\leq l^{\prime}_{v}<1$.

We define the rational Lipman cone by

[TABLE]

which is a cone generated by the duals $E^{*}_{v}$ over $\mathbb{Q}_{\geq 0}$. Since $\{E^{*}_{v}\}_{v}$ have positive entries, ${\mathcal{S}}_{\mathbb{Q}}\setminus\{0\}$ is in the open first quadrant. In particular, this gives us the monoid ${\mathcal{S}}^{\prime}:={\mathcal{S}}_{\mathbb{Q}}\cap L^{\prime}$ of anti-nef cycles in $L^{\prime}$, which is called the Lipman cone. It is generated over $\mathbb{Z}_{\geq 0}$ by the duals $E^{*}_{v}$.

The Lipman cone ${\mathcal{S}}^{\prime}$ also admits a natural equivariant partition ${\mathcal{S}}^{\prime}_{h}={\mathcal{S}}^{\prime}\cap L^{\prime}_{h}$ indexed by $H$. Furthermore, we have the following properties:

1. (a)

if $l^{\prime}_{1},l^{\prime}_{2}\in{\mathcal{S}}^{\prime}_{h}$ then $l^{\prime}_{2}-l^{\prime}_{1}\in L$ and $\min\{l^{\prime}_{1},l^{\prime}_{2}\}\in{\mathcal{S}}^{\prime}_{h}$; 2. (b)

for any $s\in L^{\prime}$ the set $\{l^{\prime}\in{\mathcal{S}}^{\prime}\mid l^{\prime}\not\geq s\}$ is finite; 3. (c)

for any $h\in H$ there exists a unique minimal cycle $s_{h}:=\min\{{\mathcal{S}}^{\prime}_{h}\}\in{\mathcal{S}}^{\prime}_{h}$ (cf. 2.1.3);

2.1.3. Generalized Laufer’s algorithm

[N05, Lemma 7.4] We fix $h\in H$. Then for any $l^{\prime}\in L^{\prime}_{h}$ there exists a unique minimal element $s(l^{\prime})\in{\mathcal{S}}^{\prime}_{h}$ satisfying $s(l^{\prime})\geq l^{\prime}$ which can be obtained by the following algorithm. Set $x_{0}:=l^{\prime}$. Then one constructs a computation sequence $\{x_{i}\}_{i}$ as follows. If $x_{i}$ is already constructed and $x_{i}\not\in{\mathcal{S}}^{\prime}$ then there exists some $E_{v_{i}}$ such that $(x_{i},E_{v_{i}})>0$ and we take $x_{i+1}:=x_{i}+E_{v_{i}}$ (for some choice of $E_{v_{i}}$). Then the procedure after finitely many steps stops, say at $x_{t}$, and necessarily $x_{t}=s(l^{\prime})$.

In particular, if we start with $l^{\prime}=E_{v}$ with an arbitrarily chosen $v\in{\mathcal{V}}$ then $s(l^{\prime})=\min\{{\mathcal{S}}^{\prime}\setminus\{0\}\}$ is the minimal (or Artin’s fundamental) cycle $Z_{min}\in L$ of the singularity [A62, A66, L72]. In this special case, the above algorithm is called the ‘Laufer’s algorithm targeting $Z_{min}$’.

In fact, $s(r_{h})=s_{h}$ and $r_{h}\leq s_{h}$, however, in general $r_{h}\neq s_{h}$. Note that this fact does not contradict the minimality of $s_{h}$ in ${\mathcal{S}}^{\prime}_{h}$ since $r_{h}$ might not sit in ${\mathcal{S}}^{\prime}_{h}$.

2.1.4.

We consider the anti-canonical cycle $Z_{K}\in L^{\prime}$ defined by the adjunction formulae $(Z_{K},E_{v})=e_{v}+2$ for all $v$.

We say that the singularity $(X,o)$, or its topological type, is numerically Gorenstein if $Z_{K}\in L$. Note that the $Z_{K}\in L$ property is independent of the resolution, since $Z_{K}\in L$ if and only if the line bundle $\Omega^{2}_{X\setminus\{o\}}$ of holomorphic 2–forms on $X\setminus\{o\}$ is topologically trivial. $(X,o)$ is called Gorenstein if $\Omega^{2}_{\widetilde{X}}$ (the sheaf of holomorphic 2–forms) is isomorphic to ${\mathcal{O}}_{\widetilde{X}}(-Z_{K})$ (or, equivalently, if the line bundle $\Omega^{2}_{X\setminus\{o\}}$ is holomorphically trivial). Note that the adjunction formulae deduce the following expression

[TABLE]

where we denote $E:=\sum_{v\in\mathcal{V}}E_{v}$ and $\delta_{v}$ is the valency of the vertex $v$.

For more details regarding this section we refer to eg. [N99, N22].

2.2. Negative definite Seifert $3$-manifolds

In the present work the focus will be on star-shaped dual resolution graphs. Their associated plumbed $3$-manifolds are negative definite Seifert $3$-manifolds, and they can be analytically realized by weighted homogeneous surface singularities, see section 3.3.1. Therefore, in the sequel we will provide some details and set some notations regarding the arithmetics of such graphs.

2.2.1.

We consider a star–shaped resolution graph $\Gamma$ with $d$ legs, $d\geq 3$. (A leg is a chain of vertices which is connected to the central vertex). Each leg is determined by the so-called normalized Seifert invariant $(\alpha_{i},\omega_{i})$, where $0<\omega_{i}<\alpha_{i}$ and $\gcd(\alpha_{i},\omega_{i})=1$. If we consider the Hirzebruch–Jung (negative) continued fraction expansion

[TABLE]

the $i^{\mathrm{th}}$ leg has $\nu_{i}$ vertices, say $v_{i1},\ldots,v_{i\nu_{i}}$, with Euler decorations (self-intersection numbers) $-b_{i1},\ldots,-b_{i\nu_{i}}$, and $v_{i1}$ is connected to the central vertex $v_{0}$. All these vertices (except $v_{0}$) have genus–decorations zero. It will be also useful to define $\omega_{i}^{\prime}$ satisfying

[TABLE]

One can show that $\alpha_{i}$ is the determinant of the $i^{th}$–leg $\Gamma_{i}$, $\omega_{i}=\det(\Gamma_{i}\setminus v_{i1})$, and $\omega_{i}^{\prime}=\det(\Gamma_{i}\setminus v_{i\nu_{i}})$.

The central vertex $v_{0}$ corresponds to the central genus $g$ curve $E_{0}$ with self-intersection number $-b_{0}$. The negative definite plumbed 3–manifold $M$ associated with such a star–shaped graph has a Seifert structure and it is determined by the normalized Seifert invariants which will be denoted by

[TABLE]

In the sequel we will assume that $M$ is a (Seifert) rational homology sphere, or, equivalently, the central node has $g=0$. In this case, for simplicity we will omit the genus from the notations and we will simply write $Sf=(-b_{0};(\alpha_{i},\omega_{i})_{i=1}^{d})$.

2.2.2. Key numerical invariants

The orbifold Euler number of $M$ is defined as $e:=-b_{0}+\sum_{i}\omega_{i}/\alpha_{i}$. Then the negative definiteness of the intersection form is equivalent with $e<0$.

Let $\mathfrak{h}:=|H|$ be the order of $H=H_{1}(M,\mathbb{Z})=L^{\prime}/L$, and let $\mathfrak{o}$ be the order of the class $[E^{*}_{0}]$ (or of the generic $S^{1}$ Seifert–orbit) in $H$. Then, using the notation $\alpha:=\mathrm{lcm}(\alpha_{1},\ldots,\alpha_{d})$, one shows that (see eg. [Ne81])

[TABLE]

In particular, if $M$ is an integral homology sphere (called Seifert homology sphere) then necessarily all $\alpha_{i}$’s are pairwise relatively prime and by (2.2.2) $\alpha|e|=1$. This reads as the Diophantine equation $(b_{0}-\sum_{i}\omega_{i}/\alpha_{i})\alpha=1$, which determines all $\omega_{i}$ and $b_{0}$ uniquely by the $\alpha_{i}$’s. The corresponding Seifert homology sphere is denoted by $\Sigma(\alpha_{1},\dots,\alpha_{d})$.

Next, we define the following combinatorial number

[TABLE]

which has a central importance regarding properties of weighted homogeneous surface singularities or Seifert rational homology spheres. It has several interpretations: it is the ‘exponent’ of the weighted homogeneous germ $(X,o)$; $-\gamma$ is also called the ‘log-discrepancy’ of $E_{0}$; $\mathfrak{o}\gamma$ is usually named as the Goto–Watanabe $a$–invariant of the universal abelian cover of $(X,o)$, and $e\gamma$ appears as the orbifold Euler characteristic in [Ne81] (see also [NO10, 3.3.6]). Nevertheless, the most important interpretation for our purpose is the following.

In a star–shaped graph the $E_{0}$-coefficients of all $E^{*}_{v}$ associated with the end–vertices are computed by $-(E_{v}^{*},E^{*}_{0})=1/(|e|\alpha_{v})$ and the $E_{0}$-coefficient of $E^{*}_{0}$ is $-(E_{0}^{*},E^{*}_{0})=1/|e|$ (cf. [N05, (11.1)]). Hence, (2.1.3) gives that the $E_{0}$-coefficient of $Z_{K}$ is exactly $\gamma+1$.

For any $i=1,\dots,d$ we rename the base element of the $i^{th}$ end–vertex $v_{i\nu_{i}}$ by $E_{i}$ and compute the $E_{i}$-coefficient of $Z_{K}$. For this one uses the identities $(E_{i}^{*},E_{j}^{*})=(e\alpha_{i}\alpha_{j})^{-1}$ for $i\not=j$ and $(E_{i}^{*},E_{i}^{*})=(e\alpha_{i}^{2})^{-1}-\omega^{\prime}_{i}/\alpha_{i}$ if $i=j$, cf. [N05, (11.1)]. Therefore, (2.1.3) and a computation shows that

[TABLE]

On the other hand, by [LSz17, Lemma 2.2.1] we have the following expression

[TABLE]

where $m_{ij}$ and $m_{ijr}$ are positive integers. Hence, it yields that $-(Z_{K},E^{*}_{v_{ij}})=M_{ij}(Z_{K},-E^{*}_{i})+M^{\prime}_{ij}$ for some $M_{ij},M^{\prime}_{ij}\in\mathbb{Z}$, which gives us the following.

Lemma 2.2.5**.**

$\Gamma$* is numerically Gorenstein if and only if $\gamma\in\mathbb{Z}$ and $\gamma\equiv\omega_{i}^{\prime}\ ({\rm mod}\ \alpha_{i})$ for all $i=1,\dots,d$.*

Remark 2.2.6**.**

Note that $\gamma|e|=d-2-\sum_{i}1/\alpha_{i}$ is negative if and only if $\pi_{1}(M)$ is finite, cf. [CR74]. This can happen only if $d=3$ and $\sum_{i}1/\alpha_{i}>1$, and in this case $(X,o)$ is a quotient singularity, hence rational. Therefore, if $(X,o)$ is not rational then $\gamma\geq 0$, that is, the $E_{0}$–coefficient of $Z_{K}$ is $\geq 1$. In fact, in these cases all the coefficients of $Z_{K}$ are strict positive, see eg. [LN20, 3.2.5]. Moreover, in the numerically Gorenstein non-rational case — when we already know that $\gamma\geq 0$ — by the congruence from Lemma 2.2.5 we get the stronger $\gamma\geq 1$.

3. Numerical semigroups and weighted homogeneous singularities

3.1. The Frobenius problem

The Diophantine problem of Frobenius aims to find an explicit formula for the greatest integer not representable as a nonnegative linear form of a given system of $d$ relatively prime integers $1\leq a_{1}\leq\ldots\leq a_{d}$. The integer defined in this way is called the Frobenius number of the system, or of the numerical semigroup $G(a_{1},\ldots,a_{d})$, generated by the integers from the system itself. We will use the standard notation $f_{G(a_{1},\ldots,a_{d})}$.

The problem is still open in full generality, however several fomulae for special systems and general bounds exist in the literature. For example, the very first result is the Sylvester formula expressing $f_{G(a_{1},a_{2})}=a_{1}a_{2}-a_{1}-a_{2}$.

For our purpose it will be important to present some general bounds for the Frobenius number. These bounds motivated a classification of numerical semigroups due to Raczunas and Chrza̧stowski-Wachtel [RChW96] which will be presented in section 3.2.

For the classical approach and different aspects of the problem the interested reader might consult with the monograph of Ramírez Alfonsín [RA05]. Further details regarding the theory of numerical semigroups can be found in [RGS09] and [ADGS20].

First of all, in [Ba42] it was considered the following bound

[TABLE]

where $d_{0}=0$ and $d_{i}=\gcd(a_{1},\dots,a_{i})$ for all $i\geq 1$. Moreover, in [BaSh62] the authors provided a characterization for semigroups that satisfy the equality $f=T$ by

[TABLE]

However, it is important to emphasize that the value of $T$, as well as the above condition depends on the order of the generators, and in general only an appropiate permutation of them gives the equality $f=T$.

Raczunas and Chrza̧stowski-Wachtel [RChW96] characterized directly a subclass of semigroups for which $f=T$ holds and $T$ can be expressed in a form which is independent on the permutation of the generators . These are the flat semigroups.

Furthermore, they considered the upper bound

[TABLE]

where $f_{G(a_{1},\dots,a_{n})}\leq T(a_{1},\dots,a_{n})\leq B(a_{1},\dots,a_{n})$ and characterized the class of semigroups for which the equality $f=T=B$ holds. They are called strongly flat semigroups and form a subclass of the flat semigroups.

In the next section we define precisely these classes and discuss the classification given in [RChW96].

3.2. A classification of semigroups [RChW96]

Based on the decomposition of the generators, one considers four ‘shades’ of flatness: strongly flat, flat, almost flat and non-flat semigroups.

Let $A=\{a_{1},\dots,a_{n}\}$ be a system of generators of a numerical semigroup ${\mathcal{S}}$, ie. $\gcd(a_{1},\dots,a_{n})=1$. If we consider the numbers $q_{i}:=\gcd(a_{1},\dots,a_{i-1},a_{i+1},\dots,a_{n})$ and $\widehat{q}_{i}:=\prod_{j\neq i}q_{j}$ for all $i\in\{1,\dots,n\}$, then one follows that $\gcd(q_{i},q_{j})=1$ for every $i\neq j$. Hence $\widehat{q}_{i}\ |\ a_{i}$ and we can define $\widehat{s}_{i}:=a_{i}/\widehat{q}_{i}$ (note also that $\gcd(\widehat{s}_{i},q_{i})=1$). Therefore, the system of generators can be presented in the following form:

[TABLE]

Definition 3.2.2**.**

The set $A$ is

• •

Strongly flat (SF) if one has $\widehat{s}_{i}=1$ for all $i$;

• •

Flat (F) if there exists an $i$ such that $\widehat{s}_{i}=1$;

• •

Almost flat (AF) if there exists an $i$ such that $q_{i}>1$ and

• •

Non-flat (NF) if for all $i$ one has $q_{i}=1$.

We say that the numerical semigroup ${\mathcal{S}}$ is strongly flat, flat, almost flat or non-flat if the corresponding condition is satisfied for the minimal set of generators.

Remark 3.2.3**.**

Note that the full semigroup ${\mathcal{S}}=\mathbb{N}$ and the semigroups minimally generated by two elements are automatically SF. On the other hand, we have $\textbf{SF}\subset\textbf{F}\subset\textbf{AF}$. Moreover, if one of these three conditions is satisfied for a non-minimal set of generators, then it is automatically satisfied for the minimal one too. This property does not hold for NF.

As we have already mentioned, the strongly flat semigroups are characterized by the equality $f_{{\mathcal{S}}}=B(a_{1},\dots,a_{n})$, where $\{a_{1},\dots,a_{n}\}$ is the minimal generating set of ${\mathcal{S}}$. Moreover, using the presentation (3.2.1) of the generators and the notation $a:=\mathrm{lcm}(a_{1},\dots,a_{n})$ one rewrites the Frobenius number in the following form

[TABLE]

If ${\mathcal{S}}$ is flat then $f_{{\mathcal{S}}}=T(b_{1},\dots,b_{n}),$ where $b_{1},\dots,b_{n}$ is an appropriate permutation of the generators $a_{1},\dots,a_{n}$. In this case the Frobenius number is expressed in a direct form as

[TABLE]

see [RChW96, Thm. 2.5].

3.3. The case of semigroups associated with weighted homogeneous singularities

3.3.1. Weighted homogeneous surface singularities

A normal weighted homogeneous surface singularity $(X,o)$ is defined as the germ at the origin of an affine variety $X$ with a good $\mathbb{C}^{*}$-action. This means that its affine coordinate ring is $\mathbb{Z}_{\geq 0}$–graded: $R_{X}=\oplus_{\ell\geq 0}R_{X,\ell}$. (In fact, all finitely generated $\mathbb{Z}_{\geq 0}$–graded $\mathbb{C}$-algebra corresponds to an affine variety with good $\mathbb{C}^{*}$-action.)

One considers the set ${\mathcal{S}}_{(X,o)}:=\{\ell\in\mathbb{Z}_{\geq 0}|R_{X,\ell}\neq 0\}$. It is a numerical semigroup by the grading property and it is called the semigroup associated with $(X,o)$.

The minimal good resolution of $(X,o)$ has a star–shaped dual graph, and the $\mathbb{C}^{*}$-action of the singularity induces an $S^{1}$–Seifert action on the link. In particular, the link $M$ of $(X,o)$ is a negative definite Seifert 3–manifold characterized by its normalized Seifert invariants $Sf=(-b_{0},g;(\alpha_{i},\omega_{i})_{i=1}^{d}).$

By the work of [P77] the complex structure is completely recovered by the Seifert invariants and the configuration of points $\{P_{i}:=E_{0}\cap E_{i1}\}_{i=1}^{d}\subset E_{0}$, where $E_{0}$ is the irreducible exceptional divisor indexed by the central vertex $v_{0}$ and $E_{i1}$ corresponds to $v_{i1}$. Furthermore, the graded ring of the local algebra of the singularity is given by the following formula (called the Dolgachev–Pinkham–Demazure formula)

[TABLE]

with $D^{(\ell)}:=\ell(-E_{0}|_{E_{0}})-\sum_{i=1}^{d}\lceil\ell\omega_{i}/\alpha_{i}\rceil P_{i}$, where $\lceil r\rceil$ denotes the smallest integer greater than or equal to $r$.

In particular, when $M$ is a rational homology sphere, ie. $E_{0}\simeq\mathbb{P}^{1}$, (3.3.1) implies that $\dim(R_{X,\ell})=\max\{0,1+N(\ell)\}$ is topological, where $N(\ell)$ is the quasi-linear function

[TABLE]

Since $-\lceil x\rceil\leq-x$ one obtains $N(\ell)\leq|e|\ell$, hence $N(\ell)<0$ for $\ell<0$. This means that the semigroup ${\mathcal{S}}_{(X,o)}$ can be described with the Seifert invariants as follows

[TABLE]

Since in this case ${\mathcal{S}}_{(X,o)}$ is a topological invariant of $(X,o)$, we will frequently use the notations ${\mathcal{S}}_{M}$, or ${\mathcal{S}}_{\Gamma}$ as well, depending whether we associate it with the Seifert rational homology sphere link $M$, or with one of its star-shaped plumbing graph $\Gamma$.

Definition 3.3.4**.**

In the followings we say that a numerical semigroup $\mathcal{S}$ is representable if it can be realized as a semigroup associated with a weighted homogeneous surface singularity $(X,o)$ with $\mathbb{Q}HS^{3}$ link, or, equivalently, with a star-shaped resolution graph. Accordingly, we will say that $(X,0)$, its link $M$, or $\Gamma$ is a representant of the numerical semigroup $\mathcal{S}$.

Finally, in the next proposition we list some important properties of the quasi-linear function $N(\ell)$ which will be used later.

Proposition 3.3.5**.**

[NO10]**, [LN20, Prop. 3.2.11 & 3.2.13]

(a)  $-(\alpha-1)|e|-d\leq N(\ell)-\lceil\ell/\alpha\rceil\alpha|e|\leq-1$. In particular $\lim_{\ell\to\infty}N(\ell)=\infty$.

(b)  If $\ell>\gamma$ then $h^{1}(E_{0},\mathcal{O}_{E_{0}}(D^{(\ell)}))=0$, ie. $N(\ell)\geq-1$.

(c)  $N(\alpha)=\alpha(b_{0}-\sum_{i}\omega_{i}/\alpha_{i})=\alpha|e|=\mathfrak{o}>0$.

(d)  $N(\ell+\alpha)=N(\ell)+N(\alpha)=N(\ell)+\mathfrak{o}>N(\ell)$ for any $\ell\geq 0$.

(e)  $N(\ell)\geq 0$ for any $\ell>\alpha+\gamma$.

(f)  If the graph is numerically Gorenstein (that is, $Z_{K}\in L$), then

[TABLE]

3.4. The Frobenius number of representable semigroups

If the star-shaped plumbing graph $\Gamma$ of $M$ satisfies $b_{0}\geq d$ then the weighted homogeneous singularity $(X,o)$ supported on this topological type is minimal rational. Moreover, in this case ${\mathcal{S}}_{M}=\mathbb{N}$. For the non-trivial cases the Frobenius number of ${\mathcal{S}}_{M}$ can be expressed as follows:

Theorem 3.4.1**.**

[LN20]** If $b_{0}<d$ then one has

[TABLE]

where $\check{s}$ denotes the $E_{0}$-coefficient of $s_{[Z_{K}+E_{0}^{*}]}$, which is the unique minimal element in $\mathcal{S}^{\prime}_{[Z_{K}+E_{0}^{*}]}$ given by the generalized Laufer’s algorithm 2.1.3.

In a very important special case, when $\Gamma$ is numerically Gorenstein (ie. $Z_{K}\in L$) and $\mathfrak{o}=1$ then the ‘algorithmic term’ $\check{s}$ vanishes, and the corresponding semigroup is symmetric as will be shown by the next proposition.

Proposition 3.4.3**.**

Let $\Gamma$ be a numerically Gorenstein graph which satisfies $\mathfrak{o}=1$. Then ${\mathcal{S}}_{\Gamma}$ is symmetric. Moreover, if $\Gamma$ is determined by the Seifert invariants $Sf=(-b_{0},(\alpha_{1},\omega_{1}),\dots,(\alpha_{n},\omega_{n}))$ then the Frobenius number of ${\mathcal{S}}_{\Gamma}$ simplifies to

[TABLE]

Proof.

Regarding the Frobenius number, the assumptions imply $1/|e|=\alpha$ and $\check{s}=0$, hence (3.4.2) immediately gives the simplified form, cf. [LN20, Corollary 3.2.12 or Example 6.2.4(1)].

The proof of the symmetry goes analogously as in the particular case of strongly flat semigroups presented in [LN20, 4.1.1]. Nevertheless, for the sake of completeness we will clarify the details here as well.

One needs to verify that $\ell\in{\mathcal{S}}_{\Gamma}$ if and only if $f_{{\mathcal{S}}_{\Gamma}}-\ell\not\in{\mathcal{S}}_{\Gamma}$ for every $\ell\in\mathbb{Z}.$ Using the quasi-linear function $N(\ell)$ and the expression (3.4.4) of the Frobenius number, this reads as

[TABLE]

Since $\Gamma$ is numerically Gorenstein, by Proposition 3.3.5(f) we have $N(\ell)+N(\gamma-\ell)=-2$. On the other hand, by part (d) of the same proposition one deduces that $N(\alpha+\gamma-\ell)=N(\gamma-\ell)+\mathfrak{o}=N(\gamma-\ell)+1$, hence $N(\ell)+N(\alpha+\gamma-\ell)=-1$. The fact that $N(\ell)$ and $N(\alpha+\gamma-\ell)$ are integers yields (3.4.5). ∎

3.5. Representability of strongly flat semigroups

Let $M=\Sigma(\alpha_{1},\dots,\alpha_{d})$ be a Seifert integral homology sphere. Thus, $\alpha_{1},\ldots,\alpha_{d}\geq 2$ ($d\geq 3$) are pairwise relatively prime integers and both $b_{0}$ and $(\omega_{1},\ldots,\omega_{d})$ are uniquely determined by the Diophantine equation $\alpha(b_{0}-\sum_{i=1}^{d}\omega_{i}/\alpha_{i})=1$.

If we consider the integers $a_{i}:=\alpha/\alpha_{i}$ then the greatest common divisor of $a_{1},\ldots,a_{i-1},a_{i+1},\ldots,a_{d}$ is $\alpha_{i}$, hence the system $\{a_{i}\}_{i=1}^{d}$ generates a strongly flat semigroup $G(a_{1},\ldots,a_{d})$. In fact, in this case $\mathcal{S}_{M}=G(a_{1},\ldots,a_{d})$ and one has the next theorem.

Theorem 3.5.1**.**

[LN20]** The strongly flat semigroups with at least three generators are representable. They can be realized as numerical semigroups associated with Seifert integral homology spheres $\Sigma(\alpha_{1},\dots,\alpha_{d})$.

In particular, the theorem implies that strongly flat semigroups (with $d\geq 3$) are semigroups associated with numerically Gorenstein graphs with $\mathfrak{o}=1$. Consequently, they are symmetric and their Frobenius number is expressed as $f_{G(a_{1},\ldots,a_{n})}=\alpha+\gamma$. Its identification with the bound $B(a_{1},\dots,a_{d})$ can be seen using (3.1.2), (2.2.2) and by noticing that in this case ${\rm lcm}(a_{1},\ldots,a_{d})=\alpha$.

Since the representation of a semigroup is not unique (see Remark 4.1.3), we say that a Seifert integral homology sphere is the canonical representant of its associated semigroup.

4. On representable semigroups

Let $\Gamma$ be a star-shaped minimal good resolution graph whose plumbed 3-manifold is a rational homology sphere (cf. 2.2.1), and consider the numerical semigroup $S_{\Gamma}$ determined by the graph via the quasi-linear function $N(\ell)$.

In the followings, we will define new operations on such graphs which will help us to develop some properties of their associated numerical semigroups.

4.1. Operations on star-shaped graphs

Let $\Gamma_{1}$ and $\Gamma_{2}$ be two star-shaped graphs with Seifert invariants

[TABLE]

The first operation we consider is the addition of graphs in the following sense: we define $\Gamma:=\Gamma_{1}+\Gamma_{2}$ as the plumbing graph determined by the Seifert invariants

[TABLE]

In other words, we glue all the legs of $\Gamma_{1}$ and $\Gamma_{2}$ to a central node with decoration the sum of the original ones.

Note that if $e_{1},e_{2}$ and $e$ are the orbifold Euler numbers of $\Gamma_{1},\Gamma_{2}$ and $\Gamma$ respectively, then $e=e_{1}+e_{2}<0$. Hence the addition is well defined with respect to the negative definiteness.

Now, we examine this addition from the perspective of the associated numerical semigroups as well. Denote the quasi-linear functions associated with the plumbing graphs $\Gamma_{1},\Gamma_{2}$ and $\Gamma$ by $N_{1},N_{2}$ and $N$. Then the above construction yields $N(l)=N_{1}(l)+N_{2}(l)$.

This allows us to give an upper and a lower bound for the numerical semigroup $\mathcal{S}_{\Gamma}$ in terms of $\mathcal{S}_{\Gamma_{1}}$ and $\mathcal{S}_{\Gamma_{2}}$, which will be presented in the following lemma. Before that we need one additional notation: for two sets $A,B\subset\mathbb{N}$ we denote their sum by $A+B:=\{a+b:a\in A,\,b\in B\}$. In particular, if $A,B$ are numerical semigroups then $A+B$ is so.

Lemma 4.1.1**.**

If $\Gamma_{1}$ and $\Gamma_{2}$ are minimal good resolution graphs of weighted homogeneous surface singularities with $\mathbb{Q}HS^{3}$ link, then

[TABLE]

Proof.

First of all, the quasi-linear function associated with $\Gamma_{1}+\Gamma_{2}$ is $N_{1}+N_{2}$, hence one shows that

[TABLE]

For the upper bound, note that we can write

[TABLE]

which clearly implies the inclusion $\mathcal{S}_{\Gamma_{1}+\Gamma_{2}}\subset\mathcal{S}_{\Gamma_{1}}+\mathcal{S}_{\Gamma_{2}}$. ∎

4.1.1.

In the particular case when $\mathcal{S}_{\Gamma_{1}}=\mathcal{S}_{\Gamma_{2}}$, we get identity in (4.1.2) implying that $\mathcal{S}_{2\Gamma_{1}}:=\mathcal{S}_{\Gamma_{1}+\Gamma_{1}}=\mathcal{S}_{\Gamma_{1}}$. More generally, for an arbitrary graph $\Gamma$ and $k\in\mathbb{N}^{*}$, we get $\mathcal{S}_{k\Gamma}=\mathcal{S}_{\Gamma}$, where $k\Gamma$ is defined as

[TABLE]

Remark 4.1.3**.**

Notice that, this also shows that the realization of a numerical semigroup $\mathcal{S}$ as the semigroup associated with a star-shaped resolution graph is not unique. Moreover, for a given numerical semigroup one can construct infinitely many different representants which do not share immediate topological properties. In particular, the semigroup does not characterize completely the topological type of the singularity.

Definition 4.1.4**.**

Let $\mathcal{S}$ be a numerical semigroup and $k\in\mathbb{N}^{*}$. Then the set

[TABLE]

is a numerical semigroup which is called the quotient of the semigroup $\mathcal{S}$ by $k$.

Next we analyze this quotient construction from the perspective of the quasi-linear function.

We consider a representable semigroup ${\mathcal{S}}$ and let $N(\ell)$ be the associated quasi-linear function and $k\in\mathbb{N}^{*}$. We set $N^{(k)}(l):=N(kl)$. Then the semigroup associated with $N^{(k)}$, given by

[TABLE]

is exactly the quotient of $\mathcal{S}$ by $k$. Moreover, one can prove that this quotient is also representable.

Lemma 4.1.5**.**

The semigroup $\mathcal{S}/k$ is representable for any representable semigroup $\mathcal{S}$ and $k\in\mathbb{N}^{*}$.

Proof.

We represent $\mathcal{S}$ by a normal weighted homogeneous surface singularity $(X,o)$ with minimal good dual resolution graph $\Gamma$ whose central vertex is denoted by $E_{0}$. Then the local algebra is expressed as $R_{X}=\oplus_{\ell\geq 0}H^{0}(E_{0},{\mathcal{O}}_{E_{0}}(D^{(\ell)}))$, see (3.3.1) and the definition of $D^{(\ell)}$ therein. Hence the algebra $R_{X}^{(k)}:=\oplus_{\ell\geq 0}H^{0}(E_{0},{\mathcal{O}}_{E_{0}}(D^{(k\ell)}))$ is also normal and therefore it corresponds to a normal weighted homogeneous surface singularity. Moreover, the associated semigroup is exactly $\mathcal{S}/k$. ∎

Remark 4.1.6**.**

For the previous lemma one can give a combinatorial proof as well which constructs explicitely a graph representant $\Gamma^{(k)}$ of $\mathcal{S}/k$ from $\Gamma$ as follows.

We fix $k\geq 1$ and assume that a representation $\Gamma$ of ${\mathcal{S}}$ has Seifert invariants $Sf=(-b_{0},(\alpha_{i},\omega_{i})_{i=1}^{n})$. Then the induced quasi-linear function of the quotient ${\mathcal{S}}/k$ is expressed as

[TABLE]

For every $i=1,\dots,n$ we consider $0\leq r_{i}<\alpha_{i}$ satisfying $k\omega_{i}\equiv r_{i}$ (mod $\alpha_{i}$). Then one gets $N^{(k)}(\ell)=(k|e|+\sum_{i=1}^{n}r_{i}/\alpha_{i})\ell-\sum_{i=1}^{n}\lceil r_{i}/\alpha_{i}\rceil$. First of all, notice that $k|e|+\sum_{i=1}^{n}r_{i}/\alpha_{i}\in\mathbb{Z}_{>0}$ and the new orbifold Euler number $e^{(k)}=ke$ is negative since $e<0$. Hence, we can associate with $N^{(k)}$ a negative definite star-shaped graph $\Gamma^{(k)}$ with Seifert invariants $Sf^{(k)}=(ke-\sum_{i=1}^{n}r_{i}/\alpha_{i},(\alpha_{i},r_{i})_{i=1}^{n})$, which represents the quotient semigroup ${\mathcal{S}}/k$.

Example 4.1.7**.**

As an illustration of the previous construction we consider the semigroup $G(3,5)$. We claim that it can be represented by the graph on the left hand side of Figure 1 (see section 4.2.2). Therefore, by Remark 4.1.6 the associated quasi-linear function of the quotient semigroup $G(3,5)/2$ is written as $N^{(2)}(\ell)=2\ell-2\lceil\ell/5\rceil-2\lceil 2\ell/3\rceil$ which provides the graph drawn on the right hand side of Figure 1. Moreover, since the set of gaps of this quotient is $\{1,2\}$, we have $G(3,5)/2=G(3,4,5)$.

4.2. A representation of $G(p,q)$

A consequence of the previous section is that the strongly flat semigroups $G(p,q)$ generated by two elements are representable. The idea which will be presented in the sequel is that $G(p,q)$ is a semigroup of an irreducible plane curve singularity with one Puiseaux pair $(p,q)$ and then any $k\geq 2$-multiple of the associated minimal embedded resolution graph gives a representant for this semigroup.

4.2.1. A short detour: plane curve singularities

We will recall some classical facts from the theory of plane curve singularities which will be used in the sequel. More details can be found in general references such as [EN85, W04].

Let $(C,0)\subset(\mathbb{C}^{2},0)$ be an irreducible plane curve singularity defined by the analytic germ $f:(\mathbb{C}^{2},0)\to(\mathbb{C},0)$. According to the general theory, $(C,0)\subset(\mathbb{C}^{2},0)$ admits a minimal good embedded resolution whose associated graph $\Gamma_{f}$ is a connected, negative definite tree with an extra arrow representing the strict transform of $C$, and has a very special shape.

The link of an irreducible plane curve singularity is an algebraic knot $K\subset S^{3}$, whose isotopy type can be completely characterized by any of the following invariants: embedded resolution graph, semigroup of $(C,0)$, Puiseaux pairs, linking pairs or the Alexander polynomial of the knot $K\subset S^{3}$.

For our purpose, it will be important to discuss in little more details the Puiseaux/linking pairs and the semigroup of the singularity.

The Puiseaux (or Newton) pairs are pairs of integers $\{(p_{i},q_{i})\}_{i=1}^{r}$, where $p_{i}\geq 2,q_{i}\geq 1$, $q_{1}>p_{1}$ and $\gcd(p_{i},q_{i})=1$. They are the exponents appearing in the normal form of $f$, cf. [W04].

From topological point of view, sometimes it is more convenient to consider the linking pairs $\{(p_{i},a_{i})\}_{i=1}^{r}$, which can be defined using the Puiseaux pairs by the following recursive identities:

[TABLE]

The decorations of the graph $\Gamma_{f}$ can be determined with the Puiseaux (or linking) pairs, see eg. [N98]. The details of this process will be discussed in the next section only in the needed special case when there is exactly one Puiseaux pair.

Another important invariant of an irreducible plane curve singularity is its numerical semigroup $\mathcal{S}_{f}$. This is defined as the set of intersection multiplicities of $f$ with all possible analytic germs. Although the definition is analytic, one can describe $S_{f}$ combinatorially by expressing its minimal set of generators using eg. the linking pairs $(p_{i},a_{i})_{i=1}^{r}$:

[TABLE]

4.2.2. The case of one Puiseaux pair

Now assume that $r=1$, ie. $(C,0)\subset(\mathbb{C}^{2},0)$ has exactly one Puiseaux pair $(p,q)$. In this case the Puiseaux pair coincides with the linking pair and the normal form of the defining equation is $x^{p}+y^{q}=0$. Figure 2 presents the shape of $\Gamma_{f}$,

where the decorations can be determined from $(p,q)$ as follows. We introduce the numbers $0<\omega_{p}<p$ and $0<\omega_{q}<q$ uniquely determined by the Diophantine equation $pq-\omega_{p}q-\omega_{q}p=1$. Then the negative continued fractions $p/\omega_{p}=[u_{1},\dots,u_{k}]$ and $q/\omega_{q}:=[v_{1},\dots,v_{l}]$ give the corresponding decorations of $\Gamma_{f}$.

By [LSz17], the numerical semigroup associated with $(C,0)$ can be presented as

[TABLE]

where $N(\ell):=\ell-\lceil\omega_{p}\ell/p\rceil-\lceil\omega_{q}\ell/q\rceil$ is the quasi-linear function associated with a Seifert structure $Sf=(-1,(p,\omega_{p}),(q,\omega_{q}))$ of $S^{3}$, represented by the negative definite plumbing graph $\widetilde{\Gamma}_{f}:=\Gamma_{f}\setminus\{\mbox{arrow}\}$. Furthermore, one has $\mathcal{S}_{f}=G(p,q)$ (cf. (4.2.1)) which leads the following result.

Theorem 4.2.3**.**

The numerical semigroup $G(p,q)$ is representable.

Proof.

From the previous discussion one knows that $G(p,q)=\{\ell\in\mathbb{Z}_{\geq 0}:N(\ell)\geq 0\}$ where $N(\ell):=\ell-\lceil\omega_{p}\ell/p\rceil-\lceil\omega_{q}\ell/q\rceil$. Moreover, by 4.1.1 $G(p,q)$ equals to the semigroup $\{\ell\in\mathbb{Z}_{\geq 0}:kN(\ell)\geq 0\}$ for any integer $k\geq 1$. Hence, for an arbitrary $k\geq 2$, $G(p,q)$ is realized as the semigroup associated with the negative definite star-shaped graph $k\widetilde{\Gamma}_{f}$, defined by the Seifert invariants $Sf=(-k,k\times(p,\omega_{p}),k\times(q,\omega_{q}))$. (Here the notation means that $(p,\omega_{p})$, as well as $(q,\omega_{q})$ appears $k$ times.) ∎

Remark 4.2.4**.**

Now, by Theorem 3.5.1 and Theorem 4.2.3, the representability of strongly flat semigroup is clarified completely.

4.3. Bounds and representable semigroups

In this section we will give ‘bounds’ for representable semigroups. It will be crucial in the next section for the characterization of a special subclass of semigroups realizing these bounds.

Let $\Gamma$ be a star-shaped resolution graph with Seifert invariants

[TABLE]

where $(\alpha_{i},\omega_{i})\neq(\alpha_{j},\omega_{j})$ for different indices $i\neq j$. Here $s_{i}$ stands for the ‘multiplicity’ of the $(\alpha_{i},\omega_{i})$-type leg. Then, regarding its associated numerical semigroup ${\mathcal{S}}_{\Gamma}$ we obtain the following result.

Theorem 4.3.2**.**

One has the following inclusions

[TABLE]

where $\alpha^{*}_{i}:=\alpha/\alpha_{i}$, $\alpha_{-i}=\mathrm{lcm}_{j\neq i}(\alpha_{j})$ and the bounds are considered as submonoids of $\mathbb{N}$.

Proof.

We will prove first the upper bound. Using (2.2.2) and the definition of $N(\ell)$ one writes the expression

[TABLE]

where $f(x)=\lceil x\rceil-x$ for any $x\in\mathbb{Q}$. Note that each term appearing on the right-hand side of (4.3.4) is a nonnegative integer, except maybe $N(\ell)$. In particular, by considering the elements $\alpha$ and $\{s_{i}\alpha^{*}_{i}\}_{i=1}^{n}$ as generators, this implies that

[TABLE]

where $G(\alpha,s_{1}\alpha^{*}_{1},\dots,s_{n}\alpha^{*}_{n})$ is a submonoid of $\mathbb{N}$, not necessarily a numerical semigroup.

In order to see the lower bound we can proceed as follows. For any fixed $i\in\{1,\dots,n\}$ the definition of the orbifold Euler number gives the expression

[TABLE]

which is used to deduce the following inequality:

[TABLE]

Furthermore, one also has the properties $N(\ell_{1}+\ell_{2})\geq N(\ell_{1})+N(\ell_{2})$ and $N(\alpha)=\mathfrak{o}>0$ (Proposition 3.3.5(c)) which imply that $G(\alpha,s_{1}\alpha_{-1},\dots,s_{n}\alpha_{-n})\subset S_{\Gamma}$. ∎

5. The topology and geometry of flat semigroups

We consider a special case of (4.3.3) and characterize those representable semigroups which realize the bounds. It turns out that these are exactly the flat semigroups. In this section we will provide their characterization, prove that they are representable and discuss the geometry of their ‘canonical representants’.

5.1.

Consider the inclusions from (4.3.3) and assume that $\mathfrak{o}=1$, $\alpha_{-i}=\alpha^{*}_{i}$ and $\gcd(s_{i},\alpha_{i})=1$ for every $i$. In this case the bounds are in fact numerical semigroups, they coincide and (4.3.3) becomes an identity. On the other hand, the condition $\alpha_{-i}=\alpha^{*}_{i}$ can be achieved exactly when the numbers $\alpha_{i}$ are pairwise relatively primes. Hence we get $\alpha_{-i}=\alpha^{*}_{i}=\prod_{j\neq i}\alpha_{j}$ for which we will use the unified notation $\widehat{\alpha}_{i}$.

In summary, one deduces the following consequence: if the graph $\Gamma$ defined by the Seifert invariants Sf=\big{(}-b_{0},s_{1}\times(\alpha_{1},\omega_{1}),\dots,s_{n}\times(\alpha_{n},\omega_{n})\big{)} satisfies $\mathfrak{o}=1$, the numbers $\{\alpha_{i}\}_{i=1}^{n}$ are pairwise relatively prime integers and $\gcd(s_{i},\alpha_{i})=1$ for every $i$, then

[TABLE]

Moreover, in this case $\mathcal{S}_{\Gamma}$ is a flat semigroup. Indeed, using the notations from section 3.2 one gets $q_{0}=\gcd(\{s_{j}\}_{j})$ and $q_{i}=\alpha_{i}$ for any $i\geq 1$. This implies that $\widehat{q}_{0}=\alpha_{1}\dots\alpha_{n}=\alpha$, $\widehat{q}_{i}=q_{0}\cdot\widehat{\alpha}_{i}$ for $i\geq 1$, $\widehat{s}_{0}=1$ and $\widehat{s}_{i}=s_{i}/q_{0}$ for $i\geq 1$, hence it clearly satisfies the flatness condition (cf. Definition 3.2.2) at $i=0$.

Now, we start with a presentation $G(a_{0},\dots,a_{n})$ of a flat semigroup and assume that $\widehat{s}_{0}=1$. Note that $\{a_{0},\dots,a_{n}\}$ is not necessarily the minimal set of generators. In particular, when $n=1$, for the next construction one has to consider a presentation with at least three generators, eg. $G(a_{0}a_{1},a_{0},a_{1})$.

Then, the chosen set of generators can be read as $\{\widehat{q}_{0},\widehat{s}_{1}\widehat{q}_{1},\dots,\widehat{s}_{n}\widehat{q}_{n}\}$ where we set the numbers $q_{i}:=\gcd(a_{0},\dots,a_{i-1},a_{i+1},\dots,a_{n})$, $\widehat{q}_{i}:=\prod_{j\neq i}q_{j}$ and $\widehat{s}_{i}:=a_{i}/\widehat{q}_{i}$ for all $i\in\{0,\dots,n\}$. Note that $\widehat{s}_{i}$ is an integer since $\gcd(q_{i},q_{j})=1$ for every $i\neq j$, and one also follows that $\gcd(\widehat{s}_{i},q_{i})=1$. By setting $\alpha_{i}:=q_{i}$ for $i\geq 1$ and $s_{i}:=q_{0}\cdot\widehat{s}_{i}$ for every $i\geq 0$ implies that $\{\alpha_{i}\}_{i}$ are pairwise relatively primes and $\gcd(s_{i},\alpha_{i})=1$ if $i\geq 1$. Moreover, one identifies $\widehat{q}_{0}=\alpha$ and $\widehat{s}_{i}\widehat{q}_{i}=s_{i}\widehat{\alpha}_{i}$ for every $i\geq 1$, hence the semigroup is presented in the form of (5.1.1).

The previous argument provides the ‘arithmetical’ characterization of flat semigroups which was also proved in [RChW96].

Theorem 5.1.2** ([RChW96]).**

$\mathcal{S}$* is a flat semigroup if and only if there exist pairwise relatively prime integers $\alpha_{i}\geq 2$ ($i\in\{1,\dots,n\}$) such that $\mathcal{S}$ can be presented as $G(\alpha,s_{1}\widehat{\alpha}_{1},\dots,s_{n}\widehat{\alpha}_{n})$ where $\alpha=\alpha_{1}\dots\alpha_{n}$, $\widehat{\alpha}_{i}=\prod_{j\neq i}\alpha_{j}$ and $\gcd(\alpha_{i},s_{i})=1$ for every $i\neq j$.*

In the following we will show that, once the presentation (5.1.1) is fixed, there exists a canonical way to represent a flat semigroup as a semigroup associated with a star-shaped graph.

Theorem 5.1.3**.**

Every flat semigroup is representable.

Proof.

Let ${\mathcal{S}}=G(\alpha,s_{1}\widehat{\alpha}_{1},\dots,s_{n}\widehat{\alpha}_{n})$ be a fixed presentation of the flat semigroup ${\mathcal{S}}$. We would like to find the appropiate $b_{0}\geq 1$ and $\omega_{1},\dots,\omega_{n}$ such that $0<\omega_{i}<\alpha_{i}$ and the Seifert invariants

[TABLE]

define a negative definite star-shaped plumbing graph $\Gamma$ wiht $\mathfrak{o}=\alpha|e|=1$. Note that this later condition is equivalent with $b_{0}-\sum_{i}s_{i}\omega_{i}/\alpha_{i}=1/\alpha$.

If we ignore the $s_{i}$-multiplicities, then the Diophantine equation $\alpha(\widetilde{b}_{0}-\sum_{i}\widetilde{w}_{i}/\alpha_{i})=1$ has a unique solution $\widetilde{b}_{0},\widetilde{\omega}_{1},\dots\widetilde{\omega}_{n}$, see [LN20]. In addition, if $\widetilde{\omega}_{i}$ is divisible by $s_{i}$ for every $i$, then one writes $\widetilde{b}_{0}-\sum_{i}s_{i}\frac{\widetilde{\omega}_{i}/s_{i}}{\alpha_{i}}=1/\alpha$, hence by setting $\omega_{i}:=\widetilde{\omega}_{i}/s_{i}$ the construction is finished. However, in general, this divisibility does not hold and we have to perturb the initial solution $\widetilde{b}_{0},\widetilde{\omega}_{1},\dots,\widetilde{\omega}_{n}$ as follows.

Note that for arbitrary $k_{1},\dots,k_{n}\in\mathbb{Z}_{\geq 0}$ we can write the following identity

[TABLE]

Since the positive integers $s_{i}$ and $\alpha_{i}$ are relatively prime, we choose $k_{i}$ to be the unique non-negative solution of the equation $k_{i}\alpha_{i}+\widetilde{\omega}_{i}\equiv 0\,(\mathrm{mod}\ s_{i})$ such that $0\leq k_{i}<s_{i}$. In this case $k_{i}\alpha_{i}+\widetilde{\omega}_{i}$ is divisible by $s_{i}$, hence we can define

[TABLE]

This yields that

[TABLE]

hence we get $\mathfrak{o}=1$ which finishes the proof.

∎

Remark 5.1.5**.**

In the case when $s_{i}=1$ for any $i$, the graph constructed in the proof of Theorem 5.1.3 is the canonical representant of a strongly flat semigroup. This motivates the following definition.

Definition 5.1.6**.**

We say that the representant constructed in Theorem 5.1.3 is a canonical representant of the flat semigroup ${\mathcal{S}}$ associated with its presentation $G(\alpha,s_{1}\widehat{\alpha}_{1},\dots,s_{n}\widehat{\alpha}_{n})$.

Remark 5.1.7**.**

(a)  The next example will illustrate how does a canonical representant depend on the presentation $G(\alpha,s_{1}\widehat{\alpha}_{1},\dots,s_{n}\widehat{\alpha}_{n})$ of the flat semigroup. Once the presentation is fixed, it is unique by the previous proof.

Consider the numerical semigroup ${\mathcal{S}}$ generated by $a_{0}=6$, $a_{1}=15$ and $a_{2}=20$. One can check that ${\mathcal{S}}$ is flat at the first two generators, ie. $\widehat{s}_{0}=\widehat{s}_{1}=1$. Hence, if we present it first of all as $G(2\cdot 3,5\cdot 3,10\cdot 2)$, then this provides a canonical representant with Seifert invariants $Sf=(-6,5\times(2,1),10\times(3,1))$. On the other hand, if we present ${\mathcal{S}}$ as $G(5\cdot 3,2\cdot 3,4\cdot 5)$ then one gets a graph with Seifert invariants $Sf=(-3,4\times(3,1),2\times(5,4))$.

One can also choose a non-minimal presentation such as $G(2\cdot 3\cdot 5,1\cdot 15,2\cdot 10,1\cdot 6)$. In this case the associated canonical representant is defined by the Seifert invariants $Sf=(-2,(2,1),2\times(3,1),(5,4))$.

(b)  If we run the algorithm from Theorem 5.1.3 for the strongly flat semigroup $G(p,q)$ presented as $G(pq,p,q)$, then the associated canonical representant will be given by the Seifert invariants $Sf=(-b_{0},(p,\omega_{1}),(q,\omega_{2}))$ where $b_{0},\omega_{1},\omega_{2}$ satisfy $pqb_{0}-q\omega_{1}-p\omega_{2}=1$. The last identity immediately implies that $b_{0}=1$ and the corresponding graph is $\widetilde{\Gamma}_{f}$, see section 4.2.2 and Figure 2. In other words, the canonical representant is $S^{3}$ with the corresponding Seifert structure.

5.2.

In the sequel, we discuss some properties of the canonical representants and of flat semigroups. First, we observe the following.

Proposition 5.2.1**.**

A canonical representant of a flat semigroup is numerically Gorenstein.

Proof.

Let $\Gamma$ be the canonical representant associated with a fixed presentation of a flat semigroup and consider its Seifert invariants as in (5.1.4). By Lemma 2.2.5 one has to check that $\gamma\in\mathbb{Z}$ and $\gamma\equiv\omega_{i}^{\prime}\ ({\rm mod}\ \alpha_{i})$ for every $1\leq i\leq n$.

For the first we recall that for a canonical representant we have $\alpha=\alpha_{1}\alpha_{2}\dots\alpha_{n}$ and $\mathfrak{o}=1$, hence $|e|=1/\alpha$. Then, from (2.2.3) we get

[TABLE]

Note that $\gamma\equiv\omega_{i}^{\prime}\ ({\rm mod}\ \alpha_{i})$ is equivalent with $\omega_{i}\gamma\equiv 1\,(\mathrm{mod}\,\alpha_{i})$, see section 2.2.1. The previous calculation gives the expression:

[TABLE]

which tells us that $\omega_{i}\gamma\equiv-s_{i}\widehat{\alpha}_{i}\omega_{i}\,(\mathrm{mod}\,{\alpha_{i}})$. On the other hand, the identity $\mathfrak{o}=1$ reads as

[TABLE]

which implies $-s_{i}\widehat{\alpha}_{i}\omega_{i}\equiv 1(\mathrm{mod}\,{\alpha_{i}})$. Hence, $\omega_{i}\gamma\equiv 1(\mathrm{mod}\,{\alpha_{i}})$ for any $i\in\{1,\dots,n\}$. ∎

Then, we can apply Proposition 3.4.3 to deduce that a flat semigroup is symmetric. Furthermore, in this case the Frobenius number simplifies to $f_{{\mathcal{S}}}=\alpha+\gamma$. This, expressed by the minimal set of generators reproves the formula (3.2.4) from [RChW96, Theorem 2.5].

Theorem 5.2.2**.**

If $\mathcal{S}$ is a flat semigroup, minimally generated by $a_{0},a_{1},\dots,a_{n}$ ($n\geq 2$), then

[TABLE]

where $\alpha_{i}=\gcd(a_{0},\dots,a_{i-1},a_{i+1}\dots,a_{n})$.

Proof.

Using the previous discussions and the notation $\alpha:=\prod_{i=1}^{n}\alpha_{i}$, after a possible permutation of the generators, we can assume that $(a_{0},a_{1},\dots,a_{n})=(\alpha,s_{1}\widehat{\alpha}_{1},\dots,s_{n}\widehat{\alpha}_{n})$. Then one has

[TABLE]

∎

5.3. The geometry of the canonical representants

In the followings we study the canonical representants from geometric point of view. We can construct explicit equations for weighted homogeneous singularities whose link (or minimal good dual resolution graph) is a cannonical representant of a flat semigroup.

5.3.1. The universal abelian cover and the action of $H$

Lemma 5.3.1**.**

If $\Gamma$ is the the canonical representant of a flat semigroup $G(\alpha,s_{1}\widehat{\alpha}_{1},\dots,s_{n}\widehat{\alpha}_{n})$, then one has

[TABLE]

Proof.

Recall that the canonical representant $\Gamma$ associated with the given presentation is defined by the Seifert invariants $Sf=(-b_{0},s_{1}\times(\alpha_{i},\omega_{i}),\dots,s_{n}\times(\alpha_{n},\omega_{n}))$ where $b_{0}$ and $\omega_{i}$ is contructed by Theorem 5.1.3.

Let $E_{0}$ be the base element associated with the central node of $\Gamma$, and for simplicity, we will denote by $E_{j(i)}$ ($i\in\{1,\dots,n\}$, $j\in\{1,\dots,s_{i}\}$) the base elements associated with the end-vertices. The classes in $H=L^{\prime}/L$ of the corresponding dual base elements will be denoted by $g_{0}:=[E^{*}_{0}]$ and $g_{j(i)}:=[E^{*}_{j(i)}]$. Then the group $H$ can be presented as

[TABLE]

cf. Neumann [Ne81].

Since $\mathfrak{o}=1$ for the canonical representant $\Gamma$, one gets $g_{0}=0$ and the relations simplify to $\sum_{i=1}^{n}\sum_{j=1}^{s_{i}}\omega_{i}\cdot g_{j(i)}=0$ and $\alpha_{i}\cdot g_{j(i)}=0$. From the former relation we deduce that $\widehat{\alpha_{i}}\omega_{i}\cdot\sum_{j=1}^{s_{i}}g_{j(i)}=0$, while the later gives $\alpha_{i}\cdot\sum_{j=1}^{s_{i}}g_{j(i)}=0$. These imply that the order of $\sum_{j=1}^{s_{i}}g_{j(i)}$ divides both $\widehat{\alpha_{i}}\omega_{i}$ and $\alpha_{i}$. Since $\{\alpha_{i}\}_{i}$ are pairwise relatively prime, this is possible if and only if $\sum_{j=1}^{s_{i}}g_{j(i)}=0$ for any $i\in\{1,\dots,n\}$. Therefore, we get

[TABLE]

∎

Let $\Gamma$ be a canonical representant of a flat semigroup as before and $(X,o)$ be a weighted homogeneous surface singularity whose minimal good dual resolution graph is $\Gamma$. Then there exists the universal abelian cover $(X^{ab},0)\to(X,o)$ that induces an unramified Galois covering $X^{ab}\setminus 0\to X\setminus o$ with Galois group $H\simeq H_{1}(M,\mathbb{Z})$ ($M$ is the link of $(X,o)$), ie. $(X,o)=(X^{ab}/H,0)$. Furthermore, by a theorem of Neumann [Ne81] (see also [N22, 5.1.40]) this universal abelian cover $(X^{ab},0)$ is a Brieskorn complete intersection singularity, which can be given by the equations

[TABLE]

where the $(d-2)\times d$ matrix $(c^{k}_{j(i)})$ has full rank. Here $d:=s_{1}+\dots+s_{n}$ and the variable $z_{j(i)}$ is assigned to the end-vertex $E_{j(i)}$ for any $i=1,\dots,n$, $j=1,\dots,s_{i}$. In the followings, we define the $H$-action on $X^{ab}$.

Consider the Pontrjagin dual $\hat{H}:=\operatorname{{\rm Hom}}(H,S^{1})$ of $H$ and let $\theta:H\to\hat{H},[l^{\prime}]\mapsto e^{2\pi i(l^{\prime},\cdot)}$ be the isomorphism of $H$ with $\hat{H}$. The $H$ acts on $\mathbb{C}^{d}$ by the diagonal action $\mathrm{diag}(\chi_{j(i)})_{i,j}:H\to\mathrm{Diag}(d)\subset GL_{d}(\mathbb{C})$, where $\chi_{j(i)}:=e^{2\pi i(E^{*}_{j(i)},\cdot)}\in\hat{H}$ is the character corresponding to $[E^{*}_{j(i)}]$. Since the equations $f_{k}$ are eigenvectors we obtain an induced action on $X^{ab}$ too. By [Ne81, Theorem 2.1] this action is free off the origin and the orbit space $(X^{ab}/H,0)\simeq(X,o)$ (for the right choice of $c^{k}_{j(i)}$).

Remark 5.3.4**.**

Note that, although the complex structure of $(X^{ab},0)$ depends on the choice of the matrix $(c^{k}_{j(i)})$, its link $M^{ab}$ (as a Seifert $3$-manifold) is independent of it. Furthermore, by the result of [JN83, Thm. 7.2] (see also [N22, 5.1.17]) one can show that in our case the resolution graph $\Gamma^{ab}$ of $(X^{ab},0)$ inherits from $\Gamma$ the following structure (in the sequel all invariants of $M^{ab}$ will be marked by $\star^{ab}$):

• •

If $s_{i}=1$ then the leg in $\Gamma$ with $(\alpha_{i},\omega_{i})$ induces a leg in $\Gamma^{ab}$ with $\alpha^{ab}_{i}=\alpha_{i}$ and this will appear with multiplicity $s^{ab}_{i}=\prod_{l=1}^{n}\alpha_{l}^{s_{l}-1}$.

• •

If $s_{i}>1$ then for any $j(i)=1,\dots,s_{i}$ one gets $\alpha^{ab}_{j(i)}=1$ and $s^{ab}_{j(i)}=\alpha_{i}^{s_{i}-2}\prod_{l\neq i}\alpha_{l}^{s_{l}-1}$. Note that these legs completely dissapear since $\alpha^{ab}_{j(i)}=1$, however their multiplicity contribute to the genus of the central fiber in $M^{ab}$.

• •

More precisely, one gets $g^{ab}=1+\frac{1}{2}\prod_{l=1}^{n}\alpha_{l}^{s_{l}-1}(\sum_{i,s_{i}>1}s_{i}(1-\frac{1}{\alpha_{i}})-2)$. Furthermore, $b_{0}^{ab}$ and $\omega^{ab}_{i}$ can be determined by the formulae: $\omega^{ab}_{i}\widehat{\alpha}_{i}\equiv-1\,(\mathrm{mod}\,\alpha_{i})$ and $-e^{ab}=\prod_{l=1}^{n}\alpha_{l}^{s_{l}-2}$.

We emphasize that $M^{ab}$ is a rational homology sphere if and only if $s_{i}=1$ for any $i$. Or, equivalently, $\Gamma$ is the canonical representant of a strongly flat semigroup by Remark 5.1.5.

5.3.2. The equations of $(X,o)$

Now we look at the induced action on the polynomial ring $R=\mathbb{C}[(z_{j(i)})_{i,j}]$ and the invariant subring of $R$ is denoted by $R^{H}$. By considering the generators of the invariant monomials and their relations in $R^{H}$, in the sequel we will study the possible equations for the analytic types of $(X,o)$.

Since the characters $\chi_{j(i)}$ generate $\hat{H}$, they satisfy the relations from (5.3.2), namely one has

[TABLE]

A monomial $z^{a}:=\prod_{i}\prod_{j(i)}z_{j(i)}^{a_{j(i)}}$ is in $R^{H}$ if and only if $\prod_{j(i)}\chi_{j(i)}^{a_{j(i)}}=1$ for any $i$. By (5.3.5) one deduces that the monomials $z^{\alpha_{i}}_{j(i)}$ and $\prod_{j(i)=1}^{s_{i}}z_{j(i)}$ are in $R^{H}$ for any $i$ and $j(i)$. If there are any other generators, divided them with the already listed invariant monomials, they must have the form of $\prod_{j(i)=1}^{s_{i}}z^{a_{j(i)}}_{j(i)}$ for some $i$ with the exponents $(a_{j(i)})=(a_{1},\dots,a_{s_{i}-1},0)$ where $0\leq a_{j(i)}<\alpha_{i}$. We claim that in this case $a_{j(i)}=0$ for $j(i)=1,\dots,s_{i}-1$ as well. Indeed, one has the identity $\prod_{j(i)=1}^{s_{i}-1}\chi^{a_{j(i)}}_{j(i)}=1$ which implies that $\sum_{j(i)=1}^{s_{i}-1}a_{j(i)}E^{*}_{j(i)}\in L$. In particular, $\sum_{j(i)=1}^{s_{i}-1}a_{j(i)}g_{j(i)}=0\in H$. Then by the isomorphism from Lemma 5.3.1 and the assumptions on $a_{j(i)}$ follows that $a_{j(i)}=0$.

In summary, the generators associated with $i$ are as follows: if $s_{i}=1$ then $z_{i}:=z_{j(i)}$ is a generator; in the case $s_{i}>0$ we get $w_{j(i)}:=z^{\alpha_{i}}_{j(i)}$ for $j(i)=1,\dots,s_{i}$ and $w_{i}:=\prod_{j(i)=1}^{s_{i}}z_{j(i)}$. Then $R^{H}$ can be presented as $\mathbb{C}[z_{i},w_{i},w_{j(i)}]/I$ where the ideal $I$ is given by the relations

[TABLE]

providing us the equations for the possible analytic types of $(X,o)$. Note that the first type of relations is coming from the equations (5.3.3) of $X^{ab}$, the second type of equations is given by the relations of the monoid algebra $\mathbb{C}[M_{i}]$ associated with the affine monoid $M_{i}=\langle(t_{1}:=\alpha_{i},0,\dots,0),t_{2}:=(0,\alpha_{i},0,\dots,0),\dots,(0,\dots,0,\alpha_{i}),t_{s_{i}+1}:=(1,1,\dots,1)\rangle\subset\mathbb{Z}^{s_{i}}_{\geq 0}$ for every $i$ with $s_{i}>1$. In other words, for a fixed $i$ the corresponding relations generates the toric ideal $I_{i}$ where $\mathbb{C}[M_{i}]\simeq\mathbb{C}[w_{j(i)},w_{i}]/I_{i}$. In fact, in our case one has $I_{i}=(w_{i}^{\alpha_{i}}-\prod_{j(i)=1}^{s_{i}}w_{j(i)})$. Indeed, the generators of the relations in terms of the monoid generators have the form of $\sum_{l\in I}a_{l}t_{l}=\sum_{m\in J}b_{m}t_{m}+b_{s_{i}+1}t_{s_{i}+1}$ for some $a_{l},b_{m},b_{s_{i}+1}\geq 0$ where $I,J\subset\{1,\dots,s_{i}\}$ and $I\cap J=\emptyset$. Looking at the coordinates, this implies that $b_{m}=0$, $I=\{1,\dots,s_{i}\}$ and $a_{l}\alpha_{i}=b_{s_{i}+1}$, which gives us the only generator $\sum_{l=1}^{s_{i}}t_{l}=\alpha_{i}t_{s_{i}+1}$.

Note that many of the variables $w_{j(i)}$ can be eliminated and the number of equations in (5.3.6) can be reduced. In the following we will distinguish three cases depending on the number $K:=\#\{i\ :\ s_{i}=1\}$ of legs with multiplicity $1$.

I.  $K=0$    In this case we have the linear system $\{\sum_{i,s_{i}\neq 1}\sum_{j(i)}c^{k}_{j(i)}w_{j(i)}=0\}_{k=1,\dots,d-2}$. Associated with two fixed, not necesarilly different indices $i_{1}$ and $i_{2}$ we choose $j_{0}(i_{1})\in\{1,\dots,s_{i_{1}}\}$ and $j_{0}(i_{2})\in\{1,\dots s_{i_{2}}\}$. For simplicity, we set the notations $x:=w_{j_{0}(i_{1})}$ and $y:=w_{j_{0}(i_{2})}$. Then, since the coefficients $\{c^{k}_{j(i)}\}$ are generic (ie. the matrix of the system has rank $d-2$), the other variables can be expressed linearly in terms of $x$ and $y$. Therefore, we get that $(X,o)\subset(\mathbb{C}^{n+2},0)$ is defined by the following equations

[TABLE]

where $a_{j_{0}(i_{1})}=b_{j_{0}(i_{2})}=1$, $a_{j_{0}(i_{2})}=b_{j_{0}(i_{1})}=0$ and the other $a_{j(i)},b_{j(i)}\in\mathbb{C}$ are generic coefficients.

II.  $K=1$   We simply write $z$ for the variable associated with the only $i$ with $s_{i}=1$, set also $\alpha:=\alpha_{i}$. On the other hand, we fix $i_{0}\in\{i\ :s_{i}>1\}$ and one of its associated variables will be denoted by $x:=w_{j_{0}(i_{0})}$. The other variables are linearly expressed with $z^{\alpha}$ and $x$, hence the equations of $(X,o)\subset\mathbb{C}^{n+1}$ are

[TABLE]

where $a_{j_{0}(i_{0})}=0$, $b_{j_{0}(i_{0})}=1$ and the others are generic.

III.  $K>1$   In the last case we choose $i_{1},i_{2}$ with $s_{i_{1}}=s_{i_{2}}=1$ and denote their associated variables by $x:=z_{i_{1}}$ and $y:=z_{i_{2}}$. Then, one can express $z_{i}^{\alpha_{i}}$ for $i\in\{i\ :\ s_{i}=1\}$ and $i\neq i_{1},i_{2}$, as well as the variables $w_{j(i)}$ for every $i\in\{i\ :\ s_{i}>1\}$ and $j(i)$, linearly in terms of $x^{\alpha_{i_{1}}}$ and $y^{\alpha_{i_{2}}}$. Therefore we get that $(X,o)\subset(\mathbb{C}^{n},0)$ is defined by the following equations

[TABLE]

where $p_{i},q_{i},a_{j(i)},b_{j(i)}$ are generic coefficients.

Remark 5.3.10**.**

We emphasize that in all three cases the normal surface singularities $(X,o)$ are complete intersections. In particular, they are Gorenstein and Proposition 5.2.1 follows automatically.

Example 5.3.11**.**

Consider the flat semigroup $G(6,15,20)$ discussed in Remark 5.1.7. We look at its (last) canonical representant, which is defined by the Seifert invariants $Sf=(-2,(2,1),2\times(3,1),(5,4))$. Then, by case III. of the above construction, we get a family of suspension hypersurface singularities defined by $(X_{a_{i},b_{i}}=\{f(x,y,z)=(a_{1}x^{2}+b_{1}y^{5})(a_{2}x^{2}+b_{2}y^{5})+z^{3}=0\},0)\subset(\mathbb{C}^{3},0)$ where $a_{i},b_{i}\in\mathbb{C}$ are generic coefficients. In particular, if we consider the hypersurface singularity defined by the equation $x^{4}+y^{10}+z^{3}=0$ for example, one can check that its associated semigroup is $G(6,15,20)$. The other canonical representants considered in Remark 5.1.7 provide (eg. by case I.) other families of complete intersections with the same associated semigroup.

6. The characterization of representable semigroups

6.1. Perturbation of the Seifert invariants and the characterization

In this section we characterize the representable semigroups by proving that they can be viewed as quotients of flat semigroups.

In order to prove this result two technical steps are needed: first of all we show that every Seifert invariant $(\alpha,\omega)$ can be ‘perturbated’ without affecting the quasi-linear function $N(\ell)$; the second step claims that the Seifert invariants $(-b_{0},s_{1}\times(\alpha_{1},\omega_{1}),\dots,s_{n}\times(\alpha_{n},\omega_{n}))$ can be changed to $(-b_{0},s_{1}\times(\alpha^{\prime}_{1},\omega^{\prime}_{1}),\dots,s_{n}\times(\alpha^{\prime}_{n},\omega^{\prime}_{n}))$ in such a way that the associated semigroup will be stable, $\alpha^{\prime}_{1},\dots\alpha^{\prime}_{n}$ are pairwise relatively prime and $\gcd(\alpha^{\prime}_{i},s_{i})=1$.

We start our discussion with the characterization of the later case.

Theorem 6.1.1**.**

Let $\Gamma$ be a graph defined by the Seifert invariants

[TABLE]

If the numbers $\alpha_{i}\geq 2$ are pairwise relatively prime integers and $\gcd(\alpha_{i},s_{i})=1$ for every $i$, then its associated semigroup is a quotient of a flat semigroup. In fact, $\mathcal{S}_{\Gamma}=G(\alpha,s_{1}\widehat{\alpha}_{1},\dots,s_{n}\widehat{\alpha}_{n})/\mathfrak{o}$.

Proof.

First of all we observe that $\gcd(\alpha_{i},\mathfrak{o})\neq 1$ implies $\gcd(\alpha_{i},s_{i})\neq 1$, hence by the assumptions of the theorem we must have $\gcd(\alpha_{i},\mathfrak{o})=1$ for every $i\in\{1,\dots,n\}$. Indeed, for a fixed $i$ the expression $\mathfrak{o}=\alpha b_{0}-\sum_{j\neq i}s_{j}\omega_{j}\widehat{\alpha}_{j}-s_{i}\omega_{i}\widehat{\alpha}_{i}$ implies the identity $s_{i}\omega_{i}\widehat{\alpha}_{i}\equiv 0\,(\mathrm{mod}\,{\gcd(\alpha_{i},\mathfrak{o})})$, which simplifies to $s_{i}\omega_{i}\equiv 0\,(\mathrm{mod}\,{\gcd(\alpha_{i},\mathfrak{o})})$ since $\gcd(\alpha_{i},\alpha_{j})=1$ for $j\neq i$. Then, by multiplying the last congruence with $\omega_{i}^{\prime}$ and using (2.2.1) yields that $s_{i}\equiv 0\,(\mathrm{mod}\,{\gcd(\alpha_{i},\mathfrak{o})})$, which supports our claim.

Now, using similar ideas as in the proof of the Theorem 5.1.3 we consider the non-negative integers $k_{1},\dots,k_{n}$ as the solutions of the equations

[TABLE]

with $0\leq k_{i}<\mathfrak{o}$ for every $1\leq i\leq n$. These, on one hand, allow us to define the numbers

[TABLE]

which satisfy $\gcd(\tilde{\omega}_{i},\alpha_{i})=1$ and $0<\tilde{\omega}_{i}<\alpha_{i}$. On the other hand, they imply that $b_{0}+\sum_{i=1}^{n}s_{i}k_{i}\equiv 0\,(\mathrm{mod}\,{\mathfrak{o}})$ too, hence one can define the new decoration as $\tilde{b}_{0}=(b_{0}+\sum_{i=1}^{n}s_{i}k_{i})/\mathfrak{o}$. Indeed, from the equations (6.1.2) one gets $\sum_{i=1}^{n}\alpha s_{i}k_{i}+s_{i}\omega_{i}\widehat{\alpha}_{i}\equiv 0\,(\mathrm{mod}\,{\mathfrak{o}})$. Furthermore, the definition of $\mathfrak{o}$ reads as $\mathfrak{o}=\alpha b_{0}-\sum_{i}s_{i}\omega_{i}\widehat{\alpha}_{i}$ which, applied to the previous equation gives $\alpha(b_{0}+\sum_{i}s_{i}k_{i})\equiv 0\,(\mathrm{mod}\,{\mathfrak{o}})$. Since $\gcd(\alpha,\mathfrak{o})=1$, we deduce that $(b_{0}+\sum_{i}s_{i}k_{i})\equiv 0\,(\mathrm{mod}\,{\mathfrak{o}})$.

We consider the graph $\tilde{\Gamma}$ defined by the Seifert invariants

[TABLE]

In the followings, any of the numerical data associated with $\tilde{\Gamma}$ will be distinguished by the ‘tilde’  notation, eg. $\tilde{e}$ will stand for the orbifold Euler number of $\tilde{\Gamma}$. The first observation is that $\tilde{e}=e/o<0$, hence $\tilde{\Gamma}$ is negative definite. Furthermore, $\tilde{\alpha}=\alpha$ and one implies that $\tilde{\mathfrak{o}}=1$. Then, by the assumptions of the theorem we conclude that $\tilde{\Gamma}$ is the canonical representant of the flat semigroup ${\mathcal{S}}_{\tilde{\Gamma}}=G(\alpha,s_{1}\widehat{\alpha}_{1},\dots,s_{n}\widehat{\alpha}_{n})$.

On the other hand, if we look at the quasi-linear function $\tilde{N}(l)=\tilde{b}_{0}\ell-\sum_{i=1}^{n}s_{i}\lceil\tilde{\omega}_{i}\ell/\alpha_{i}\rceil$ associated with ${\mathcal{S}}_{\tilde{\Gamma}}$, one can see that $\tilde{N}^{(\mathfrak{o})}(\ell)=\tilde{N}(\mathfrak{o}\ell)=N(\ell)$. Hence $\mathcal{S}_{\Gamma}=\mathcal{S}_{\tilde{\Gamma}}/\mathfrak{o}$ which concludes our claim.

∎

Lemma 6.1.3**.**

Let $r\in\mathbb{Q}_{>0}$ be arbitrary. For every $M>0$ there exists $r_{M}\in\mathbb{Q}$ such that

[TABLE]

Proof.

Since $r\in\mathbb{Q}_{>0}$ one writes $r=\omega/\alpha$, where $\omega,\alpha\in\mathbb{N}$ and $\gcd(\omega,\alpha)=1$. For a fixed $\ell\in\mathbb{N}$ we introduce the notation $x:=\lceil\omega\ell/\alpha\rceil$ for simplicity. Notice that if $\ell=0$ then (6.1.4) holds for any $r^{\prime}$, so in the sequel we assume that $\ell\neq 0$. By the properties of the function $t\mapsto\lceil t\rceil$ one knows the inequalities

[TABLE]

This implies that for any $r^{\prime}\in(r-\frac{1}{\alpha\ell},r]$ we have $x\geq r\ell\geq r^{\prime}\ell>r\ell-\frac{1}{\alpha}\geq x-1$, hence $\lceil r^{\prime}\ell\rceil=x=\lceil r\ell\rceil$.

Thus, for a fixed $\ell$ we have constructed an interval for $r^{\prime}$ such that (6.1.4) is satisfied. When $\ell$ is varying in $[0,M]$ we consider the intersection of these intervals

[TABLE]

∎

Lemma 6.1.5**.**

Let $N\colon\mathbb{Z}\to\mathbb{Z}$, $N(\ell)=b_{0}\ell-\sum_{i=1}^{n}s_{i}\lceil\omega_{i}\ell/\alpha_{i}\rceil$ be a quasi-linear function of a representable semigroup. Then for every $M\in\mathbb{N}$ there exists a modification $N^{\prime}\colon\mathbb{Z}\to\mathbb{Z},N^{\prime}(\ell)=b_{0}\ell-\sum_{i=1}^{n}s_{i}\lceil\omega^{\prime}_{i}\ell/\alpha^{\prime}_{i}\rceil$ ( $\omega^{\prime}_{i},\alpha^{\prime}_{i}\in\mathbb{Z}_{>0}$, $0<\omega_{i}^{\prime}<\alpha^{\prime}_{i}$ and $\gcd(\omega^{\prime}_{i},\alpha^{\prime}_{i})=1$), such that $\alpha^{\prime}_{1},\dots,\alpha^{\prime}_{n}$ are pairwise relatively prime, $\gcd(\alpha^{\prime}_{i},s_{i})=1$ and it satisfies

[TABLE]

Proof.

Let $\Gamma$ be the graph corresponding to $N(\ell)$. We will say that the $i$-th block of $\Gamma$ consists of the $s_{i}$ number of legs with Seifert invariant $(\alpha_{i},\omega_{i})$. For a fixed $M\in\mathbb{N}^{*}$ we will construct the modification by induction on $i$.

First we describe the inductive step. Assume that the $(i-1)$-th block is already modified. This means that the new Seifert invariants $(\alpha^{\prime}_{1},\omega^{\prime}_{1}),\dots,(\alpha^{\prime}_{i-1},\omega^{\prime}_{i-1})$ are constructed in a way that $\alpha^{\prime}_{t}$ are pairwise relatively primes and $\gcd(\alpha^{\prime}_{t},s_{t})=1$ for any $t\in\{1,\dots,i-1\}$.

Then, for a large enough $k\in\mathbb{N}$, by Lemma 6.1.3 one finds a rational number $r^{\prime}\in(r^{\prime}_{M,i},\omega_{i}/\alpha_{i})$ of the form $r^{\prime}=x/(k\alpha^{\prime}_{1}\dots\alpha^{\prime}_{i-1}s_{i}+1)$ ($x\in\mathbb{N}$) satisfying $\lceil r^{\prime}\ell\rceil=\lceil\omega_{i}\ell/\alpha_{i}\rceil$ for any $\ell\leq M$. This, written as $\omega^{\prime}_{i}/\alpha^{\prime}_{i}:=r^{\prime}$ (where $\gcd(\omega^{\prime}_{i},\alpha^{\prime}_{i})=1$) gives us the perturbation. Since $\alpha^{\prime}_{i}$ is a divisor of $k\alpha^{\prime}_{1}\dots\alpha^{\prime}_{i-1}s_{i}+1$, it is relatively prime to all the $\alpha^{\prime}_{t}$ with $t\leq i-1$ and $s_{i}$. In this way, we get the $i$-th block of the modified graph with $s_{i}$ legs, all having the Seifert invariant $(\alpha^{\prime}_{i},\omega^{\prime}_{i})$.

For $i=1$ we can start by distinguishing two cases:

I.  If $\gcd(\alpha_{1},s_{1})=1$ then we do not modify them and we set $(\alpha^{\prime}_{1},\omega^{\prime}_{1}):=(\alpha_{1},\omega_{1})$.

II.  If $\gcd(\alpha_{1},s_{1})\neq 1$ then we do the same as in the inductive step. Therefore, we get a rational number of the form $r^{\prime}=x/(ks_{1}+1)$ ($x\in\mathbb{N}$) satisfying $\lceil r^{\prime}\ell\rceil=\lceil\omega_{1}\ell/\alpha_{1}\rceil$ for any $\ell\leq M$, which provides $\omega^{\prime}_{1}/\alpha^{\prime}_{1}:=r^{\prime}$.

By the construction one follows that $N(\ell)=N^{\prime}(\ell)$ for any $\ell\leq M$. ∎

Theorem 6.1.6**.**

A numerical semigroup is representable if and only if it is a quotient of a flat semigroup.

Proof.

By Lemma 4.1.5 and Theorem 5.1.3 follow that a quotient of a flat semigroup is representable. For the reverse, let $S$ be a representable semigroup and consider one of its representant $\Gamma$ defined by the Seifert invariants $(-b_{0},s_{1}\times(\alpha_{1},\omega_{1}),\dots,s_{n}\times(\alpha_{n},\omega_{n}))$. Let $M$ be the maximum of the largest generator (in the minimal set of generators) and the Frobenius number $f_{\mathcal{S}}$ and we apply Lemma 6.1.5 for this fixed $M$. Then we get a new graph $\Gamma^{\prime}$ defined by the perturbed Seifert invariants $(-b_{0},s_{1}\times(\alpha^{\prime}_{1},\omega^{\prime}_{1}),\dots,s_{n}\times(\alpha^{\prime}_{n},\omega^{\prime}_{n}))$ satisfying that $\alpha^{\prime}_{1},\dots\alpha^{\prime}_{n}$ are pairwise relatively prime integers and $\gcd(\alpha^{\prime}_{i},s^{\prime}_{i^{\prime}})=1$. Moreover, the associated semigroup $S_{\Gamma^{\prime}}$ equals to $S$. Indeed, the identity $N(\ell)=N(\ell^{\prime})$ valid for all the integers up to the Frobenius number $f_{S}$ implies that ${\mathcal{S}}_{\Gamma^{\prime}}\subset{\mathcal{S}}$. On the other hand, $N(\ell)=N(\ell^{\prime})$ for all the integers up to the largest generator deduces that ${\mathcal{S}}\subset{\mathcal{S}}_{\Gamma^{\prime}}$. Finally, Theorem 6.1.1 applied to $\Gamma^{\prime}$ clarifies the statement. ∎

Example 6.1.7**.**

Consider the non flat semigroup $G(4,6,7,9)$. We claim that it is representable and one of its representation is given by the graph with Seifert invariants $Sf=(-2,2\times(2,1),2\times(4,1),(5,1))$ (the calculations were performed using GAP[GAP]). Following our previous discussions, a perturbation of the Seifert invariants can be as follows.

Let us first consider the two legs with $(2,1)$. Note that $M=9$ and $(2,1)$ can be changed to eg. $(11,5)$ since $\frac{5}{11}\in(\frac{1}{2}-\frac{1}{18}=\frac{4}{9},\frac{1}{2})$. Similarly, $(4,1)$ can be changed to $(13,3)$. These perturbations give us a new graph defined by the Seifert invariants $Sf=(-2,2\times(11,5),2\times(13,3),(5,1))$ which has $\mathfrak{o}=307$ and satisfies the assumptions of Theorem 6.1.1. Finally, we have

[TABLE]

6.2. Further speculations and remarks

As a conclusion, we would like to propose a general question and emphasize some directions opened by the problem studied in this article.

In the theory of numerical semigroups one knows by [RGS07] (see also [RGS09]) that every numerical semigroup can be presented as one half of a symmetric numerical semigroup. More generally, for a fixed $d\geq 2$, every semigroup can be presented as one over $d$ of a symmetric numerical semigroup, cf. [Sw09]. Note that the Example 4.1.7 is also an example for the construction given by Rosales and García-Sánchez in [RGS07].

As a consequence, by applying Lemma 4.1.5, one follows that if we can represent every symmetric semigroup as a semigroup associated with a weighted homogeneous surface singularity, then every numerical semigroup is representable. This approach naturally poses the following question.

Question 6.2.1**.**

Is there a symmetric numerical semigroup which can not be represented as a semigroup associated with a weighted homogeneous surface singularity with $\mathbb{Q}HS^{3}$ link?

By the knowledge of the authors, there is no good understanding how the symmetric property of semigroups incarnates on the level of the representants, or how does it fit in the ‘flat’ classification theme of Raczunas and Chrza̧stowski-Wachtel.

For example, one can construct representable semigroups which are not symmetric, but they have a numerically Gorenstein representant:

Example 6.2.2** (Non-symmetric numerically Gorenstein case [LN20]).**

Let $\Gamma$ be the graph defined by the Seifert invariants

[TABLE]

Then $Z_{K}=(86,43,43,29,29,13,13,2)$, hence $\Gamma$ is numerically Gorenstein. In addition, one can compute that $\gamma=85$, $1/|e|=28$ and $\check{s}=28$ (using 2.1.3), thus the Frobenius number equals $f(\mathcal{S}_{\Gamma})=85$. In addition $N(6)=N(85-6)=-1$ which implies $6,f(\mathcal{S}_{\Gamma})-6\notin\mathcal{S}_{\Gamma}$, hence it is not symmetric.

One the other hand, using the proof of Theorem 3.4.3 one can construct numerical semigroups which are symmetric, but not flat, as shown by the next example. Therefore, the set of symmetric semigroups is rather bigger than that of the flat ones.

Example 6.2.3** (Symmetric but not flat).**

Let $\Gamma$ be the graph defined by the Seifert invariants

[TABLE]

Then one can verify eg. with [GAP] that $\mathcal{S}_{\Gamma}=G(8,21,35)$. Hence $\mathcal{S}_{\Gamma}$ is not a flat but an almost flat semigroup. However, it is symmetric.