Strange Duality Between Hypersurface and Complete Intersection Singularities

During his classification of hypersurface singularities, [Arnold1975] observed a strange duality between the 14 exceptional unimodal singularities. [Ebeling and Wall1985] discovered an extension of this duality embracing on one hand series of bimodal singularities and on the other, isolated complete intersection singularities (ICIS) in ${\mathbb{C}}^{4}$. The duals of the complete intersection singularities are not themselves singularities, but are virtual ($k=-1$) cases of series (e.g. $W_{1,k}:k\geq 0$) of bimodal singularities. They associated to these well-defined Coxeter–Dynkin diagrams and Milnor lattices and showed that all numerical features of Arnold’s strange duality continue to hold. The $k=-1$ cases of the series were called virtual singularities in [Ebeling and Wall1985], because setting $k=-1$ in Arnold’s equations of the series one gets exceptional unimodal singularities with a smaller Milnor number as germs at the origin.

The objective of this paper is to derive this extended strange duality from the mirror symmetry and the Berglund–Hübsch transposition of invertible polynomials. Moreover, we show that the virtual singularities exist in the sense that the equations have to be considered as global polynomials. The bimodal series start with singularities with $k=0$ (e.g. $W_{1,0}$). They can be given by polynomials with two moduli. Setting one of the moduli equal to zero, one is left with a one-parameter family of weighted homogeneous polynomials. It is natural from the mirror symmetry view point to expect that adding one monomial to an invertible polynomial is dual to having another ${\mathbb{C}}^{\ast}$-action on the dual polynomial, which leads to our duality between virtual singularities and complete intersection singularities.

We shall proceed as follows. We first classify the non-degenerate invertible polynomials with a ${\mathbb{Z}}/2{\mathbb{Z}}$-action. They are defined by certain $3\times 3$-matrices. Then we shall classify the possibilities to extend such a $3\times 3$-matrix to a certain $4\times 3$-matrix satisfying certain conditions. Such a matrix defines a polynomial ${\bf f}(x,y,z)$ with four monomials with a non-isolated singularity. We shall consider the Berglund–Hübsch transpose of this polynomial. The kernel of the transpose $3\times 4$-matrix defines a ${\mathbb{C}}^{\ast}$-action on the space ${\mathbb{C}}^{4}$ and this matrix and the degree 0 polynomials define a complete intersection singularity in ${\mathbb{C}}^{4}$ as the zero set of two polynomials.

Following [Ebeling and Takahashi2011], we consider the polynomial ${\bf f}(x,y,z)-xyz$. Under certain conditions, there is a coordinate transformation which transforms this polynomial to a polynomial ${\bf h}(x,y,z)-xyz$, where ${\bf h}$ is again a polynomial with four monomials, but now has an isolated singularity at the origin. We call this a virtual singularity. The polynomial ${\bf h}$ is no longer weighted homogeneous but its Newton polygon at infinity has two two-dimensional faces. We thus obtain a duality between the virtual hypersurface singularities and complete intersection singularities.

We show that this duality has the features of Arnold’s strange duality. Namely, we associate Dolgachev and Gabrielov numbers to the polynomials ${\bf h}$ and the equations defining the complete intersection singularities generalizing the approach of [Ebeling and Takahashi2011]. It turns out that the Dolgachev numbers of the polynomial ${\bf h}$ are the Gabrielov numbers of the pair of polynomials defining the complete intersection singularity and vice versa, the Gabrielov numbers of the polynomial ${\bf h}$ are the Dolgachev numbers of the pair of polynomials defining the complete intersection singularity. Moreover, we show that the reduced zeta function of the monodromy at infinity of a virtual singularity coincides with the product of the Poincaré series of the coordinate ring of the dual complete intersection singularity and a polynomial encoding its Dolgachev numbers.

As an example we consider those singularities with Gorenstein parameter being equal to 1. In this way, we recover precisely the virtual singularities of the bimodal series and the extension of Arnold’s strange duality found in [Ebeling and Wall1985]. Therefore we have shown that these virtual singularities exist as global polynomials. Moreover, the Dolgachev and Gabrielov numbers which we have associated to them agree with the ones predicted in [Ebeling and Wall1985]. Finally, we construct Coxeter–Dynkin diagrams for the virtual bimodal singularities and show that they can be transformed to graphs which have the same shape as in the exceptional unimodal case used in Gabrielov’s original definition of the numbers now named after him.

We recall some general definitions about invertible polynomials.

Let $f(x_{1},\ldots,x_{n})$ be a weighted homogeneous polynomial, namely, a polynomial with the property that there are positive integers $w_{1},\ldots,w_{n}$ and $d$ such that $f(\lambda^{w_{1}}x_{1},\ldots,\lambda^{w_{n}}x_{n})=\lambda^{d}f(x_{1},\ldots,% x_{n})$ for $\lambda\in{\mathbb{C}}^{\ast}$. We call $(w_{1},\ldots,w_{n};d)$ a system of weights.

Without loss of generality one may assume that $a_{i}=1$ for $i=1,\ldots,n$. This can be achieved by rescaling the variables. We may and shall also assume that $\det E>0$.

An invertible polynomial has a canonical system of weights $W_{f}=(w_{1},\ldots,w_{n};d)$ given by the unique solution of the equation

This system of weights is in general non-reduced, i.e. in general $c_{f}:={\rm gcd}(w_{1},\ldots,{}w_{n},d)>1$.

Let $f(x_{1},\ldots,x_{n})$ be an invertible polynomial and $W_{f}=(w_{1},\ldots,w_{n};d)$ be the canonical system of weights associated to $f$. Set

Note that $G_{f}$ always contains the exponential grading operator

Let $G_{0}$ be the subgroup of $G_{f}$ generated by $g_{0}$. One has (cf. [Ebeling and Takahashi2013])

Let $f(x_{1},\ldots,x_{n})=\sum_{i=1}^{n}a_{i}\prod_{j=1}^{n}x_{j}^{E_{ij}}$ be an invertible polynomial. Following [Berglund and Hübsch1993], the Berglund–Hübsch transpose $\widetilde{f}(x_{1},\ldots,x_{n})$ of $f$ is defined by

By [Berglund and Henningson1995], for a subgroup $G\subset G_{f}$ its dual group $\widetilde{G}$ is defined by

Note that ${\rm Hom}(G_{f},{\mathbb{C}}^{\ast})$ is isomorphic to $G_{\widetilde{f}}$, see [Berglund and Henningson1995]. By [Krawitz2009], we have

Moreover, by Ebeling and Takahashi ([Ebeling and Takahashi2013], Proposition 3.1), we have $|\widetilde{G}_{0}|=c_{f}$.

Let $f(x,y,z)$ be a non-degenerate invertible polynomial with $[G_{f}:G_{0}]=2$. We shall now classify the non-degenerate invertible polynomials with such a group action.

Let $\widehat{G}_{0}$ be the subgroup of $\widehat{G}_{f}$ defined by the commutative diagram of short exact sequences

and ${\mathcal{C}}_{(F,G_{0})}:=[(F^{-1}(0){\setminus}\{0\})/\widehat{G}_{0}]$, where $F:=\sum_{i=1}^{4}a_{i}x^{E_{i1}}y^{E_{i2}}z^{E_{i3}}$, is a smooth projective line with 4 isotropic points whose orders are $\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}$, where $A_{(f,G_{0})}=(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4})$ are the Dolgachev numbers of the pair $(f,G_{0})$ defined in [Ebeling and Takahashi2013], for general $a_{1},a_{2},a_{3},a_{4}$.

We associate to these matrices a pair of polynomials as follows. We observe that the kernel of the matrix $E^{T}$ is either generated by the vector $(1,1,0,-2)^{T}$ or by the vector $(1,1,-1,-1)^{T}$. The second case occurs precisely for the matrices of type ${\rm IIB}^{\sharp}$ and ${\rm IV}^{\sharp}$. Let $R:={\mathbb{C}}[x,y,z,w]$. In the first case, there exists a ${\mathbb{Z}}$-graded structure on $R$ given by the ${\mathbb{C}}^{\ast}$-action

In the second case, there exists a ${\mathbb{Z}}$-graded structure on $R$ given by the ${\mathbb{C}}^{\ast}$-action

Let $R=\bigoplus_{i\in{\mathbb{Z}}}R_{i}$ be the decomposition of $R$ according to one of these ${\mathbb{Z}}$-gradings. Let $E^{T}$ be the transposed matrix. We associate to this the polynomial

In the first case, we have $\widetilde{f}\in R_{0}={\mathbb{C}}[x^{2}w,y^{2}w,z,xyw]$. Let

In these new coordinates, we obtain a pair of polynomials

In the second case, we have $\widetilde{f}\in R_{0}={\mathbb{C}}[xw,yz,xz,yw]$. Let

In these new coordinates, we obtain a pair of polynomials

Now we choose for each of the matrices $E$ special values $a_{1},a_{2},a_{3},a_{4}$ such that the corresponding polynomial $F$ has a non-isolated singularity. We denote this polynomial by ${\bf f}$. We summarize the results in Table 1.

We now associate other equations to the polynomials from above. For each type, consider the polynomial ${\bf f}$ from Table 1 and assume that the conditions indicated in Table 2 are satisfied. In all cases except ${\rm IIB}^{\sharp}$ and ${\rm IV}^{\sharp}$, the polynomial ${\bf f}(x,y,z)$ is of the form

or

for some monomials $u(x,y,z)$ and $v(x,y,z)$ and some integer $e\geq 2$. We consider the cusp singularity ${\bf f}(x,y,z)-xyz$ and perform the coordinate change $x\mapsto x+y^{e}$ or $y\mapsto y+x^{e}$ respectively. The corresponding coordinate change is indicated in Table 2. Then ${\bf f}(x,y,z)-xyz$ is transformed to ${\bf h}(x,y,z)-xyz$ where ${\bf h}(x,y,z)$ is indicated in the last column of Table 2.

By inspection of Table 2, we see that some of the polynomials ${\bf h}$ have 4 monomials and others only 3. We restrict our consideration to the cases where the polynomial ${\bf h}$ has 4 monomials. These cases are listed in Table 3. The singularities defined by the polynomials ${\bf h}(x,y,z)$ will be called virtual singularities. We consider the duality between the virtual singularities on one side and the complete intersection singularities on the other side.

Let

be the polynomial defining a virtual singularity and let ${\rm Supp}({\bf h})=\{(A_{i1},A_{i2},A_{i3})\in{\mathbb{Z}}^{3}\,|\,i=1,\ldots% ,4\}$. Let $\Delta_{\infty}({\bf h})$ be the Newton polygon of ${\bf h}$ at infinity ([Kouchnirenko1976]), i.e. $\Delta_{\infty}({\bf h})$ is the convex closure in ${\mathbb{R}}^{n}$ of ${\rm Supp}({\bf h})\cup\{0\}$. The Newton polygon $\Delta_{\infty}({\bf h})$ has two faces which do not contain the origin. Call these faces $\Sigma_{1}$ and $\Sigma_{2}$. Let $I_{k}:=\{i\in\{1,\ldots,4\}\,|\,(A_{i1},A_{i2},A_{i3})\in\Sigma_{k}\}$, $k=1,2$, and let

Then ${\bf h}_{k}$ is an invertible polynomial with a non-isolated singularity at the origin. The polynomials ${\bf h}_{1}$ and ${\bf h}_{2}$ are listed in Table 3. Their canonical systems of weights are reduced. One of the systems of weights of ${\bf h}_{1}$ and ${\bf h}_{2}$ coincides with the reduced system of weights of the non-degenerate invertible polynomial $f$ we started with. Let the numbering be chosen such that this is the system of weights of ${\bf h}_{2}$. The systems of weights are listed in Table 4.

The dual complete intersection singularity defined by $\widetilde{{\bf f}}_{1}=\widetilde{{\bf f}}_{2}=0$ is weighted homogeneous. We list the systems of weights of these complete intersection singularities in Table 5. It turns out that the degrees of the systems of weights of ${\bf h}_{1}$ and ${\bf h}_{2}$ coincide with the degrees of the two polynomials $\widetilde{{\bf f}}_{1}$ and $\widetilde{{\bf f}}_{2}$ respectively.

We shall now define Dolgachev and Gabrielov numbers for the polynomials ${\bf h}$ and the pairs of polynomials $(\widetilde{{\bf f}}_{1},\widetilde{{\bf f}}_{2})$ occurring in our duality.

We first define these numbers for the pairs $(\widetilde{{\bf f}}_{1},\widetilde{{\bf f}}_{2})$. Let $X_{\widetilde{{\bf f}}_{1},\widetilde{{\bf f}}_{2}}\subset{\mathbb{C}}^{4}$ be the weighted homogeneous complete intersection in ${\mathbb{C}}^{4}$ defined by the two equations $\widetilde{{\bf f}}_{1}(W,X,Y,Z)=\widetilde{{\bf f}}_{2}(W,X,Y,Z)=0$, where $\widetilde{{\bf f}}_{1}(W,X,Y,Z)=XY-W^{2}$ or $\widetilde{{\bf f}}_{1}(W,X,Y,Z)=XY-ZW$.

The Gabrielov numbers are defined similarly as in the hypersurface case. We consider the complete intersection singularity $(X^{\prime},0)$ defined by

As in [Ebeling and Takahashi2011] one can show that one can find a holomorphic change of coordinates such that the singularity $(X^{\prime},0)$ is also given by equations of the form

This means that $(X^{\prime},0)$ is a cusp singularity of type $T^{2}_{\gamma_{1},\gamma_{3},\gamma_{2},\gamma_{4}}$ in the notation of Ebeling ([Ebeling1987], 3.1).

The Dolgachev and Gabrielov numbers for the pairs $(\widetilde{{\bf f}}_{1},\widetilde{{\bf f}}_{2})$ of Table 1 are indicated in Table 6.

Now let ${\bf h}$ be the polynomial of Table 3 defining a virtual singularity. The Gabrielov numbers of ${\bf h}$ are defined as in [Ebeling and Takahashi2011]. Namely, we consider the polynomial ${\bf h}(x,y,z)-xyz$. As in [Ebeling and Takahashi2011], one can show that the germ at the origin of this polynomial is right equivalent to a cusp singularity

i.e. it can be transformed to such a polynomial by a holomorphic change of coordinates at the origin. We define the Gabrielov numbers of ${\bf h}$ to be the triple $(\gamma_{1},\gamma_{2},\gamma_{3})$.

The Dolgachev numbers of the polynomial ${\bf h}$ are defined as follows. We associated to ${\bf h}$ two weighted homogeneous polynomials ${\bf h}_{1}$ and ${\bf h}_{2}$. Let $i=1,2$ and let $V_{i}:=\{(x,y,z)\in{\mathbb{C}}^{3}\,|\,{\bf h}_{i}(x,y,z)=0\}$. We consider the ${\mathbb{C}}^{\ast}$-action on $V_{i}$ given by the system of weights of ${\bf h}_{i}$ (see Table 4). We consider the exceptional orbits (i.e. orbits with a non-trivial isotropy group) of this action. We distinguish between two cases:

1. (A)

   $V_{i}$ contains a coordinate hyperplane.

2. (B)

   $V_{i}$ does not contain a coordinate hyperplane.

In case (A) we consider those exceptional orbits which are not contained in the coordinate hyperplane which is contained in $V_{i}$. In case (B) we consider those exceptional orbits which do not coincide with the singular locus of $V_{i}$. We call these the principal orbits. It turns out that in all cases we have exactly two principal orbits.

We list the Dolgachev and Gabrielov numbers of the polynomials ${\bf h}$ corresponding to the virtual singularities in Table 7.

Comparing Table 7 with Table 6, we obtain the following result.

Let $f_{1},\ldots,f_{k}$ be quasihomogeneous functions on ${\mathbb{C}}^{n}$ of degrees $d_{1},\ldots,d_{k}$ with respect to weights $w_{1},\ldots,w_{n}$. Here $w_{1},\ldots,w_{n}$ are positive integers with $\gcd(w_{1},\ldots,w_{n})=1$, $f_{j}(\lambda^{w_{1}}x_{1},\ldots,\lambda^{w_{n}}x_{n})=\lambda^{d_{j}}f_{j}(x% _{1},\ldots,x_{n})$, $\lambda\in{\mathbb{C}}$. We suppose that the equations $f_{1}=f_{2}=\ldots=f_{k}=0$ define a complete intersection $X$ in ${\mathbb{C}}^{n}$. There is a natural ${\mathbb{C}}^{\ast}$-action on the space ${\mathbb{C}}^{n}$ defined by $\lambda\ast(x_{1},\ldots,x_{n})=(\lambda^{w_{1}}x_{1},\ldots,\lambda^{w_{n}}x_% {n})$, $\lambda\in{\mathbb{C}}^{\ast}$.

Let $A={\mathbb{C}}[x]/(f_{1},\ldots,f_{k})$ be the coordinate ring of $X$. There is a natural grading on the ring $A$: $A_{s}$ is the set of functions $g\in A$ such that $g(\lambda\ast x)=\lambda^{s}g(x)$. Let $P_{X}(t)=\sum_{s=0}^{\infty}\dim A_{s}\cdot t^{s}$ be the Poincaré series of the graded algebra $A=\oplus_{s=0}^{\infty}A_{s}$. One has

For $0\leq j\leq k$, let $X^{(j)}$ be the complete intersection given by the equations $f_{1}=\cdots=f_{j}=0$ ($X^{(0)}={\mathbb{C}}^{n}$, $X^{(k)}=X$). The restriction of the function $f_{j}$ ($j=1,\ldots,k$) to the variety $X^{(j-1)}$ defines a locally trivial fibration $X^{(j-1)}{\setminus}X^{(j)}\to{\mathbb{C}}^{\ast}$. Let $V^{(j)}=f_{j}^{-1}(1)\cap X^{(j-1)}$ be the (Milnor) fibre of this fibration (the fibre $V^{(j)}$ is not necessarily smooth) and $\varphi^{(j)}:V^{(j)}\to V^{(j)}$ be the classical monodromy transformation of it. For a map $\varphi:Z\to Z$ of a topological space $Z$, let $\zeta_{\varphi}(t)$ be its zeta function

If, in the definition, we use the actions of the operators ${\varphi}_{\ast}$ on the homology groups $\overline{H}_{p}(Z;{\mathbb{Z}})$ reduced modulo a point, we get the reduced zeta function

Let

If both $X^{(j)}$ and $X^{(j-1)}$ have isolated singularities at the origin then $\overline{H}_{p}(V^{(j)};{\mathbb{Z}})$ is non-trivial only for $p=n-j$ and therefore, if $n-j\geq 1$,

is the characteristic polynomial of the classical monodromy operator ${\varphi}_{\ast}^{(j)}$.

One can show that $({\varphi}^{(j)}_{\ast})^{d_{j}}=\hbox{id}$ and therefore $\overline{\zeta}_{X,j}(t)$ can be written in the form

Following [Saito1998a]; [Saito1998b], we define the Saito dual to $\overline{\zeta}_{X,j}(t)$ to be the rational function

(note that different degrees $d_{j}$ are used for different $j$).

Let $Y^{(k)}=(X^{(k)}{\setminus}\{0\})/{\mathbb{C}}^{\ast}$ be the space of orbits of the ${\mathbb{C}}^{\ast}$-action on $X^{(k)}{\setminus}\{0\}$ and $Y^{(k)}_{m}$ be the set of orbits for which the isotropy group is the cyclic group of order $m$. Let

be the product of cyclotomic polynomials with exponents corresponding to the partition of the complete intersection $X=X^{(k)}$ into parts of different orbit types; here $\chi(Z)$ denotes the Euler characteristic of a topological space $Z$.

Let $(X,0)$ be the virtual hypersurface singularity defined by ${\bf h}=0$ and $(\widetilde{X},0)$ be the dual complete intersection singularity given by the equations $\widetilde{{\bf f}}_{1}=\widetilde{{\bf f}}_{2}=0$ according to Theorem 5. The function $\hbox{Or}_{\widetilde{X}}(t)$ is equal to the polynomial

where $\alpha_{1},\alpha_{2},\alpha_{3}$ are the Dolgachev numbers of the pair $(\widetilde{{\bf f}}_{1},\widetilde{{\bf f}}_{2})$, see Sect. 5.

Finally, let $\zeta_{X}(t)$ be the zeta function of the monodromy at infinity of ${\bf h}$ and

be the reduced zeta function of ${\bf h}$.

From [Ebeling and Gusein-Zade2004] we get the following corollary:

Let $f(x,y,z)$ be a weighted homogeneous polynomial with reduced system of weights $W=(w_{1},w_{2},w_{3};d)$. The Gorenstein parameter $a_{f}$ of $f$ is defined to be

We now consider the classification of virtual singularities according to the Gorenstein parameter $a_{f}$ of the non-degenerate invertible polynomial $f$.

The classification of the non-degenerate invertible polynomials $f$ with $[G_{f}:G_{0}]=2$ and with $a_{f}<0$ can be extracted from Ebeling and Takahashi ([Ebeling and Takahashi2013], Table 3). From this we derive the classification of virtual singularities given in Table 8.

One can also classify the non-degenerate invertible polynomials with $[G_{f}:G_{0}]=2$ with $a_{f}=0,1$. It turns out that there are no such polynomials with $a_{f}=0$. The virtual singularities corresponding to polynomials with $a_{f}=1$ are listed in Table 9.

I turns out that the virtual singularities with $a_{f}=1$ are exactly the virtual singularities corresponding to the bimodal series. According to Arnold’s classification [Arnold1975], there are 8 series of bimodal hypersurface singularities. The virtual bimodal singularities are defined by setting $k=-1$ in the equations of these singularities. The names of Arnold are used in Table 9 and the equations for $k=-1$ are listed in Table 10. We compare them with our polynomials ${\bf h}$. We also indicate the names of the dual isolated complete intersection singularities according to the notation of [Wall1983]. It turns out that these are exactly the singularities in the extension of Arnold’s strange duality of [Ebeling and Wall1985].

We indicate the values of the Dolgachev and Gabrielov numbers of the polynomials ${\bf h}$ associated to the virtual bimodal singularites and the Dolgachev and Gabrielov numbers of the corresponding dual pairs of polynomials defining the isolated complete intersection singularities (ICIS) in Table 11.

Let ${\bf h}(x,y,z)=0$ be the equation for one of the virtual bimodal singularities. By inspection, one sees that the germ at the origin is an exceptional unimodal singularity. The corresponding singularity is indicated in Table 12. Moreover the global polynomial ${\bf h}$ has besides the origin an additional critical point which is of type $A_{1}$.¹¹Note that this is different for some of the original equations of Arnold. There in the cases $W_{1,-1}$, $S_{1,-1}$, and $U_{1,-1}$ we have two additional critical points and the singularities $W_{12}$, $S_{11}$, and $Q_{11}$ respectively at the origin. This also gives an explanation of the deformation ${\bf h}(x,y,z)-xyz$ which we used to define the Gabrielov numbers.

Now we want to consider Coxeter–Dynkin diagrams of these singularities. Let $X:=\{(x,y,z)\in{\mathbb{C}}^{3}\,|\,{\bf h}(x,y,z)=0\}$. The function ${\bf h}$ defines a locally trivial fibration ${\bf h}:{\mathbb{C}}^{3}{\setminus}X\to{\mathbb{C}}^{\ast}$. Let $V={\bf h}^{-1}(1)\cap X$ be the Milnor fibre of this fibration. We shall consider a (strongly) distinguished basis of vanishing cycles of the homology group $H_{2}(V;{\mathbb{Z}})$ (see e.g. [Arnold et al.1988]; [Ebeling2007]). The critical point outside the origin gives an additional vanishing cycle in $H_{2}(V;{\mathbb{Z}})$. We define the Milnor number $\mu$ of $X$ to be the rank of $H_{2}(V;{\mathbb{Z}})$. It is equal to the sum of the Milnor numbers of the singular points of ${\bf h}$. It is indicated in Table 12.

In order to compute a Coxeter–Dynkin diagram for a distinguished basis of vanishing cycles we use the method of [Gabrielov1979]. We have to consider the polar curve corresponding to a choice of a linear function $z:{\mathbb{C}}^{n}\to{\mathbb{C}}$. The choice of the function is indicated in Table 12. The additional critical point lies on the polar curve. One can easily generalize the method of Gabrielov to include this additional critical point. By [Gabrielov1979], one obtains an intersection matrix of a distinguished basis of ${\bf h}$ from the one of a distinguished basis for ${\bf h}|_{z=0}$ by the following formulas. Let $(e_{j})$ ($j=1,2$ in case a, $j=1,2,3$ in case b and $j=1,2,3,4$ in case c) be a distinguished basis of ${\bf h}|_{z=0}$ corresponding to the Coxeter–Dynkin diagram presented in Fig. 1. Let $M_{j}$ be the numbers indicated in Table 12. Then there is a distinguished basis $(e_{j}^{m},1\leq m\leq M_{j})$ with the following intersection numbers

In Table 12, the contribution of the additional critical point to the numbers $M_{j}$ is indicated. By the sequences of elementary basis transformations indicated in [Ebeling1996], the distinguished basis $(e_{j}^{m})$ can be transformed to a distinguished basis

with a Coxeter–Dynkin diagram of the form of Fig. 2 where $\gamma_{1},\gamma_{2},\gamma_{3}$ are the Gabrielov numbers of $X$. We call this graph $S_{\gamma_{1},\gamma_{2},\gamma_{3}}$.

Now let us consider the dual pair $(\widetilde{{\bf f}}_{1},\widetilde{{\bf f}}_{2})$ and the isolated complete intersection singularity defined by it. According to Ebeling ([Ebeling1987], Proposition 3.6.1) one can find a Coxeter–Dynkin diagram with respect to a distinguished basis of thimbles of the form $\Pi_{\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4}}$ of Fig. 3 where $\gamma_{1},\gamma_{2};\gamma_{3},\gamma_{4}$ are the Gabrielov numbers of the pair $(\widetilde{{\bf f}}_{1},\widetilde{{\bf f}}_{2})$.