Graded matrix factorizations of size two and reduction

Abstract

We associate a complete intersection singularity to a graded matrix factorization of size two of a polynomial in three variables. We show that we get an inverse to the reduction of singularities considered by C. T. C. Wall. We study this for the full strongly exceptional collections in the triangulated category of graded matrix factorizations constructed by H. Kajiura, K. Saito, and the second author.

1 Introduction

In his classification of singularities [16], C. T. C. Wall noticed that there is a relation between series of singularities given by a reduction. More precisely, he considers a linear reduction L of a complete intersection singularity in ${\mathbb {C}}^4$ to a hypersurface singularity in ${\mathbb {C}}^3$ and a reduction $\Delta$ of a hypersurface singularity in ${\mathbb {C}}^3$ of corank 3 to one of corank 2. The aim of this paper is to show that as an inverse of such a reduction one can consider a graded matrix factorization of size two.

We consider weighted homogeneous deformations of an invertible polynomial. We use the approach of [1] to define graded matrix factorizations for such deformations. We classify weighted homogeneous deformations of invertible polynomials of three variables and we define Dolgachev numbers for them. We consider a graded matrix factorization of size two of such a deformation and associate a complete intersection singularity to it. We show that we get the original singularity back by L- or $\Delta$-reduction. Moreover we consider graded matrix factorizations of size two of invertible polynomials in two variables and show that they are related to a reduction between hypersurface singularities in ${\mathbb {C}}^2$ considered in [10].

In [12], H. Kajiura, K. Saito, and the second author constructed a full strongly exceptional collection in the triangulated category of graded matrix factorizations of a polynomial associated to a regular system of weights whose smallest exponents are equal to $-1$. These polynomials define the 14 exceptional unimodal hypersurface singularities, the 6 heads of the bimodal series of hypersurface singularities, and 2 further singularities. They are weighted homogeneous deformations of invertible polynomials of three variables. The exceptional collection is described by a certain quiver. The graded matrix factorizations of size two listed in the paper correspond to the outermost vertices of certain arms of the quiver. We show that they correspond to singularities where the corresponding length of the arm (which is one of the Dolgachev numbers) is increased by one. We thus obtain the picture of Fig. 2.

There is Arnold’s strange duality between the 14 exceptional unimodal hypersurface singularities and the extension [10] of it to the bimodal series and the complete intersection singularities which reduce to the 14 exceptional unimodal singularities and the 6 heads of the bimodal series. This can be considered as a special kind of mirror symmetry. As a corollary of our main result we get that the reduction is mirror dual to a special adjacency between these singularities.

2 Graded matrix factorizations for invertible polynomials

A polynomial $f(x_1, \ldots , x_n) \in {\mathbb {C}}[x_1, \ldots , x_n]$ is called weighted homogeneous, if 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 $\lambda \in {\mathbb {C}}^*$. We call $(w_1, \ldots , w_n;d)$ a system of weights. If $\mathrm{gcd}(w_1, \ldots , w_n,d)=1$, then the system of weights is called reduced.

Definition 1

A weighted homogeneous polynomial $f(x_1, \ldots , x_n)$ is called invertible if the following conditions are satisfied

• (i) it can be written

  $$\begin{aligned} f(x_1, \ldots , x_n)=\sum _{i=1}^n a_i \prod _{j=1}^n x_j^{E_{ij}}, \end{aligned}$$

  where $a_i \in {\mathbb {C}}^*$, $E_{ij}$ are non-negative integers, and the $n \times n$-matrix $E:=(E_{ij})$ is invertible over ${\mathbb {Q}}$,

• (ii) it has an isolated singularity at the origin.

Let $f(x_1,\dots ,x_n)=\sum _{i=1}^na_i\prod _{j=1}^nx_j^{E_{ij}}$ be an invertible polynomial. Without loss of generality, we assume that $\det E>0$. The canonical system of weights is the system of weights $(w_1, \ldots , w_n;d)$ given by the unique solution of the equation

$$\begin{aligned} E \begin{pmatrix} w_1\\ \vdots \\ w_n \end{pmatrix} =\mathrm{det}(E) \begin{pmatrix} 1\\ \vdots \\ 1 \end{pmatrix} ,\quad d:=\mathrm{det}(E). \end{aligned}$$

The canonical system of weights $(w_1, \ldots , w_n;d)$ is in general non-reduced, define

$$\begin{aligned} c_f := \mathrm{gcd}(w_1, \ldots , w_n,d). \end{aligned}$$

Definition 2

A weighted homogeneous deformation of an invertible polynomial is a polynomial of the form

$$\begin{aligned} f(x_1, \ldots , x_n)= \sum _{i=1}^N a_i \prod _{j=1}^n x_j^{E_{ij}}, \end{aligned}$$

where $N \ge n$, $a_i \in {\mathbb {C}}^*$, $E_{ij}$ are non-negative integers, $\sum _{i=1}^n a_i \prod _{j=1}^n x_j^{E_{ij}}$ is an invertible polynomial with canonical weight system $(w_1, \ldots , w_n;d)$, and $\prod _{j=1}^n x_j^{E_{ij}}$ are monomials of the same degree d for $1 \le i \le N$.

Note that the case $N=n$ is permitted, i.e., the weighted homogeneous deformations of an invertible polynomial include the invertible polynomial itself.

Definition 3

Let $f(x_1,\dots ,x_n)=\sum _{i=1}^N a_i\prod _{j=1}^nx_j^{E_{ij}}$ be a weighted homogeneous deformation of an invertible polynomial. Consider the free abelian group $\oplus _{i=1}^n{\mathbb {Z}}\vec {x}_i\oplus {\mathbb {Z}}\vec {f}$ generated by the symbols $\vec {x}_i$ for the variables $x_i$ for $i=1,\dots , n$ and the symbol $\vec {f}$ for the polynomial f. The maximal grading $L_f$ of the invertible polynomial f is the abelian group defined by the quotient

$$\begin{aligned} L_f:=\bigoplus _{i=1}^n{\mathbb {Z}}\vec {x}_i\oplus {\mathbb {Z}}\vec {f}/I_f, \end{aligned}$$

where $I_f$ is the subgroup generated by the elements

$$\begin{aligned} \vec {f}-\sum _{j=1}^nE_{ij}\vec {x}_j,\quad i=1,\dots ,N. \end{aligned}$$

Note that $L_f$ is an abelian group of rank 1 which is not necessarily free. We have the degree map $\mathrm{deg}: L_f \rightarrow {\mathbb {Z}}$ defined by $\vec {x}_i \mapsto w_i/c_f$, $\vec {f} \mapsto d/c_f$, where $(w_1, \ldots , w_n; d)$ is the canonical system of weights of f. This map is an isomorphism if and only if the canonical system of weights is reduced, see [8, Proposition 17].

Let $f(x_1, \ldots , x_n)$ be a weighted homogeneous deformation of an invertible polynomial. We recall the definition of an $L_f$-graded matrix factorization of f, see [1] and the references therein.

Let $S={\mathbb {C}}[x_1, \ldots , x_n]$. This is naturally an $L_f$-graded algebra:

$$\begin{aligned} S= \bigoplus _{\vec {l} \in L_f} S_{\vec {l}}. \end{aligned}$$

For any $L_f$-graded S-module M, let $M(\vec {l})$ denote the $L_f$-graded S-module, where the grading is shifted by $\vec {l} \in L_f$:

$$\begin{aligned} M(\vec {l})= \bigoplus _{\vec {l}' \in L_f} M(\vec {l})_{\vec {l}'}, \quad M(\vec {l})_{\vec {l}'}:=M_{\vec {l}+\vec {l}'}. \end{aligned}$$

The grading shift by $\vec {l} \in L_f$ induces a functor $(\vec {l})$ on the category of finitely generated $L_f$-graded S-modules.

Definition 4

An $L_f$-graded matrix factorization of f is

where $F_0$ and $F_1$ are $L_f$-graded free S-modules of finite rank and $f_0: F_0 \rightarrow F_1$, $f_1: F_1 \rightarrow F_0(\vec {f})$ are morphisms such that $f_1 \circ f_0= f \cdot \mathrm{id}_{F_0}$ and $f_0(\vec {f}) \circ f_1= f \cdot \mathrm{id}_{F_1}$. The rank of $F_1$ is equal to the rank of $F_0$ and it is called the size or the rank of the matrix factorization ${\overline{F}}$.

Let $f(x_1, \ldots , x_n)$ be a weighted homogeneous deformation of an invertible polynomial with reduced weight system $(w_1, \ldots , w_n;d)$. We define

$$\begin{aligned} \varepsilon _f:= w_1 + \cdots + w_n -d. \end{aligned}$$

Let f(x, y, z) be a weighted homogeneous deformation of an invertible polynomial in three variables. We shall associate Dolgachev numbers to f. This is done as follows. Let $R_f:={\mathbb {C}}[x,y,z]/(f)$. Let $r \ge 3$, $A=(\alpha _1, \ldots , \alpha _r)$ and let ${\varvec{\lambda }}=(\lambda _3, \ldots , \lambda _r)$ be an $(r-2)$-tuple of complex numbers with $\lambda _3=1$, $\lambda _i\ne \lambda _j$ for $i \ne j$, $i,j=3, \ldots , r$. Let $R_{A,{\varvec{\lambda }}}$ be the ring associated to the weighted projective line corresponding to the tuple of numbers $A=(\alpha _1, \ldots , \alpha _r)$ [11], namely the factor algebra

$$\begin{aligned}R_{A,{\varvec{\lambda }}}:={\mathbb {C}}[X_1, \ldots , X_r]/I_{{\varvec{\lambda }}} \end{aligned}$$

where $I_{{\varvec{\lambda }}}$ is the ideal generated by the polynomials

$$\begin{aligned} X_i^{\alpha _i} - X_2^{\alpha _2} + \lambda _i X_1^{\alpha _1}, \quad i=3, \ldots , r. \end{aligned}$$

We shall consider a classification of weighted homogeneous deformations of invertible polynomials according to the classification of invertible polynomials in three variables in [8]. In each type except V, two of the three monomials are replaced by a product of m terms involving parameters $(\lambda _3, \ldots , \lambda _{m+2})$. In the case of type V, we set $m=1$.

Definition 5

The Dolgachev numbers of a weighted homogeneous deformation f of an invertible polynomial is the $(m+2)$-tuple ($m \ge 1$) of numbers $A=(\alpha _1,\alpha _2, \underbrace{\alpha _3, \ldots , \alpha _3}_m)$ defined by the embedding $R_f \hookrightarrow R_{A,{\varvec{\lambda }}}$ given by Table 1.

It follows from the embedding $R_f \hookrightarrow R_{A,{\varvec{\lambda }}}$ into the ring of a weighted projective line that $L_f \cong {\mathbb {Z}}$.

Table 1 The image of the generators in $R_{A,{\varvec{\lambda }}}$

3 Graded matrix factorizations of size two

We shall now consider $L_f$-graded matrix factorizations of size two for the following two cases

• (A) f is a weighted homogeneous deformation of an invertible polynomial in three variables,

• (B) f is an invertible polynomial in two variables.

(A) Let $S={\mathbb {C}}[x,y,z]$ and let f(x, y, z) be a weighted homogeneous deformation of an invertible polynomial of the form

$$\begin{aligned} f(x,y,z)=p_1(x,y,z)h_1(x,y,z)+p_2(x,y,z)h_2(x,y,z), \end{aligned}$$

where $p_i \in S_{\vec {p}_i}$ for some $\vec {p}_i \in L_f$, $i=1,2$, and $p_1,p_2$ forms a regular sequence in S. As in [5, Section 2] (cf. [1, Definition 2.11]), we consider the Koszul resolution of the $L_f$-graded S-module $M=S/(p_1,p_2)$

$$\begin{aligned} 0 \longrightarrow S(-\vec {p}_1-\vec {p}_2) \longrightarrow S(-\vec {p}_1) \oplus S(-\vec {p}_2) \longrightarrow S \longrightarrow S/(p_1,p_2) \longrightarrow 0. \end{aligned}$$

This yields the $L_f$-graded matrix factorization of f such that

$$\begin{aligned} F_0=S(\vec {0}) \oplus S(\vec {f}-\vec {p}_1-\vec {p}_2), \quad F_1:=S(\vec {f}-\vec {p}_1) \oplus S(\vec {f}-\vec {p}_2), \end{aligned}$$

and $f_0$ and $f_1$ are given by the matrices

$$\begin{aligned} q_0 = \begin{pmatrix} h_1(x,y,z) &{} -p_2(x,y,z)\\ h_2(x,y,z) &{} p_1(x,y,z) \end{pmatrix} \text{ and } q_1= \begin{pmatrix} p_1(x,y,z) &{} p_2(x,y,z) \\ -h_2(x,y,z) &{} h_1(x,y,z) \end{pmatrix} \end{aligned}$$

respectively.

We associate to the matrix factorization a complete intersection singularity in ${\mathbb {C}}^4$. Denote the coordinates of ${\mathbb {C}}^4$ by w, x, y, z.

Definition 6

The complete intersection singularity $(X_Q,0)$ is the singularity defined by

$$\begin{aligned} \mathbf {F}_Q(w,x,y,z)= & {} (F_{Q,1}(w,x,y,z),F_{Q,2}(w,x,y,z))\\:= & {} (-h_1(x,y,z)+wp_2(x,y,z), h_2(x,y,z)+wp_1(x,y,z)). \end{aligned}$$

We can get the function f back from $\mathbf {F}_Q$ by the linear reduction considered, in the case $p_1(x,y,z)=y$ and $p_2(x,y,z)=z$, by Wall [16, 7.9]. Namely, let $L_w\mathbf {F}_Q$ be the elimination of the variable w from the functions $F_{Q,1},F_{Q,2}$. This is given by the resultant $R_w(F_{Q,1},F_{Q,2})$ of the functions with respect to the variable w. We have

$$\begin{aligned} L_w\mathbf {F}_Q=R_w(F_{Q,1},F_{Q,2}) = \det q_1 = f . \end{aligned}$$

Now suppose that $p_1(x,y,z)=y$, $p_2(x,y,z)=z$, $h_2(x,y,z)=z$ and $h_1(x,y,z) = 2a(x,y)z +b(x,y)$. Then

$$\begin{aligned} (F_{Q,1}(w,x,y,z),F_{Q,2}(w,x,y,z))=(-h_1(x,y,z)+wz, z+wy) \end{aligned}$$

and the substitution $z=-wy$ in $F_{Q,1}$ gives

$$\begin{aligned} F_Q(w,x,y):&=F_{Q,1}(w,x,y,-wy)=-w^2y-h_1(x,y,-wy)\\&=-w^2y+2wya(x,y)- b(x,y) . \end{aligned}$$

Thus $(X_Q,0)$ is in this case a hypersurface singularity defined by $F_Q=0$. The discriminant of the polynomial $F_Q(w,x,y)$ in the variable w is

$$\begin{aligned} \Delta _wF_Q(w,x,y)= y^2a(x,y)^2 -yb(x,y). \end{aligned}$$

This is the reduction $\Delta _wF_Q$ of the polynomial $F_Q$ considered by Wall in [16, 7.3]. The polynomial f is equal to

$$\begin{aligned} f(x,y,z)=z^2 + 2ya(x,y)z +yb(x,y). \end{aligned}$$

Under the substitution $\widetilde{z}:=z+ya(x,y)$ this becomes

$$\begin{aligned} f(x,y,\widetilde{z})=\widetilde{z}^2-y^2a(x,y)^2+yb(x,y) = \widetilde{z}^2 - \Delta _wF_Q(w,x,y). \end{aligned}$$

(B) Now let $S={\mathbb {C}}[x,y]$ and let f(x, y) be an invertible polynomial of the form $f(x,y)=x^{\alpha }y^{\beta }+y^3$, where $\alpha \ge 5$. We consider the Koszul resolution of the $L_f$-graded S-module $M=S/(x^2,y^2)$

$$\begin{aligned} 0 \longrightarrow S(-2\vec {x}-2\vec {y}) \longrightarrow S(-2\vec {x}) \oplus S(-2\vec {y}) \longrightarrow S \longrightarrow S/(x^2,y^2) \longrightarrow 0. \end{aligned}$$

This yields the $L_f$-graded matrix factorization of f such that

$$\begin{aligned} F_0=S(\vec {0}) \oplus S(\vec {f}-2\vec {x}-2\vec {y}), \quad F_1:=S(\vec {f}-2\vec {x}) \oplus S(\vec {f}-2\vec {y}), \end{aligned}$$

and $f_0$ and $f_1$ are given by the matrices

$$\begin{aligned} q_0 = \begin{pmatrix} x^{\alpha -2}y^{\beta } &{} -y^2\\ y &{} x^2 \end{pmatrix} \text{ and } q_1= \begin{pmatrix} x^2 &{} y^2 \\ -y&{} x^{\alpha -2}y^{\beta } \end{pmatrix} \end{aligned}$$

respectively.

Definition 7

We define a singularity $(X_Q,0)$ by

$$\begin{aligned} \mathbf {F}_Q(w,x,y):= (F_{Q,1}(w,x,y),F_{Q,2}(w,x,y))= \left( x^{\alpha -2}y^{\beta } +\frac{w}{x}y^2 , y-\frac{w}{x} x^2 \right) . \end{aligned}$$

If we substitute $y=\frac{w}{x} x^2$ in $F_{Q,1}$, we obtain the function

$$\begin{aligned} F_Q(w,x): = F_{Q,1}\left( w,x,\frac{w}{x} x^2 \right) = w^3x+x^{\alpha +\beta -2}w^{\beta }. \end{aligned}$$

Thus $(X_Q,0)$ is in fact a hypersurface singularity defined by $F_Q=0$.

We get the function f back from $F_Q$ by the reduction considered in [10, p. 19], namely $F_Q$ is of the form $F_Q(w,x)=w^3x+x^3A\left( w,x\right)$, where $A(w,x)=x^{\alpha +\beta -5}w^{\beta }$, and $RF_Q(w,x)= w^3+x^5A\left( \frac{w}{x},x\right) =f(x,w)$.

4 Graded matrix factorizations of size two for regular systems of weights with $\varepsilon =-1$

We shall now consider the graded matrix factorizations of size two for the regular systems of weights with $\varepsilon =-1$. They were determined in [12]. There are 22 such systems of weights. They define the Fuchsian hypersurface singularities of genus 0 (see, e.g., [7]). Let $W=(a,b,c;h)$ be such a weight system. By definition, it is reduced. The number $\varepsilon _W$ is defined by $\varepsilon _W := a+b+c-h$. Let $A_W=(\alpha _1, \ldots , \alpha _r)$ be the signature of the system of weights W. Then $(\alpha _1, \ldots , \alpha _r)$ corresponds to the signature of the corresponding Fuchsian singularity. For $r=3$, there are 14 such weight systems which correspond to the 14 exceptional unimodal singularities. Here the polynomial $f_W$ is an invertible polynomial and its canonical system of weights coincides with (a, b, c; h). For $r=4$, there are 6 corresponding to the heads of the bimodal series. There are 2 more with $r=5$. The corresponding equations are weighted homogeneous deformations $f_W$ of an invertible polynomial, see Table 2.

Table 2 Type of $f_W$

This again implies that $L_{f_W} \cong {\mathbb {Z}}$. Thus all the regular systems of weights with $\varepsilon _W=-1$ can be given by weighted homogeneous deformations f of invertible polynomials with $\varepsilon _f=-1$ and $L_f \cong {\mathbb {Z}}$ and the $L_f$-graded matrix factorizations of f correspond to the graded matrix factorizations considered in [12]. The Dolgachev numbers are equal to the signature of the system of weights.

It will turn out that the graded matrix factorizations of size 2 correspond to some of these singularities, but also to the Fuchsian complete intersection singularities of genus 0 (see [7]). There are 8+5+2 of them for certain signatures $(\alpha _1, \ldots , \alpha _r)$.

Let W be a regular system of weights with $\varepsilon _W=-1$ and signature $A_W=(\alpha _1, \ldots , \alpha _r)$. In [12], a full strongly exceptional collection in the triangulated category of graded matrix factorizations associated to W was constructed. The full strongly exceptional collection is described by the graph shown in Fig. 1.

Fig. 1

The graph $T^+_{\alpha _1,\ldots ,\alpha _r}$

It turns out that a graded matrix factorization of size 2 corresponds to the end point $V_{i,\alpha _i}$ of some arm.

Theorem 1

Let f be a weighted homogeneous deformation of an invertible polynomial with $\varepsilon _f=-1$ and $L_f \cong {\mathbb {Z}}$. Let $A=(\alpha _1, \ldots , \alpha _r)$ be the Dolgachev numbers of f. If the $L_f$-graded matrix factorization $V_{i,\alpha _i}$ is of rank 2, then the corresponding singularity $\mathbf {F}_{V_{i,\alpha _i}}$ has the signature $({\widetilde{\alpha }}_1, \ldots , {\widetilde{\alpha }}_r)$ with ${\widetilde{\alpha }}_i=\alpha _i+1$ and ${\widetilde{\alpha }}_j=\alpha _j$ for $j \ne i$.

Proof

We consider the matrix factorizations for the cases of Table 1 and examine conditions under which there exist maps of the local ring of the corresponding complete intersection singularity to the ring of a weighted projective line of type $({\widetilde{\alpha }}_1, \ldots , {\widetilde{\alpha }}_r)$. We indicate the matrix factorization, the map**, the index i where we have an increase of $\alpha _i$ by one, and the condition under which the map** gives an embedding.

(I) $m \ge 2$, $m|p_1$, $m|p_2$. We consider the matrix factorization

$$\begin{aligned} q_0= \begin{pmatrix} \prod _{i=4}^{m+2}(y^{\frac{p_2}{m}}-\lambda _i x^{\frac{p_1}{m}}) &{} -z \\ z^{p_3-1} &{} -(y^{\frac{p_2}{m}}-x^{\frac{p_1}{m}}) \end{pmatrix}. \end{aligned}$$

Map**:

$$\begin{aligned}&x=X_1X_3^{\frac{p_2}{m}}, \ y=X_2X_3^{\frac{p_1}{m}}, \ z=X_3^{(m-1)\frac{p_1p_2}{m^2}}X_4 \cdots X_{m+2}, \\&w=X_4^{p_3-1} \cdots X_{m+2}^{p_3-1} \end{aligned}$$

Increase of $\alpha _3$ under the condition:

$$\begin{aligned} (m-1)\frac{p_1p_2}{m^2}(p_3-1)=\frac{p_1p_2}{m^2}+p_3+1. \end{aligned}$$

(1)

($\mathrm{II}_1$) $m \ge 1$, $m|(p_1-1)$, $m|p_2$. We have two essentially different matrix factorizations:

(a)

$$\begin{aligned} q_0= \begin{pmatrix} \prod _{i=3}^{m+2}(y^{\frac{p_2}{m}}-\lambda _i x^{\frac{p_1-1}{m}}) &{} -z \\ z^{p_3-1} &{} -x \end{pmatrix}. \end{aligned}$$

Map**:

$$\begin{aligned}&x=X_1^{p_3+ \frac{m+p_2}{p_1-1}}, \ y=X_1X_2, \ z=X_1^{p_2}X_3 \cdots X_{m+2}, \\&w=X_3^{p_3-1} \cdots X_{m+2}^{p_3-1} \end{aligned}$$

Increase of $\alpha _1$ under the condition:

$$\begin{aligned} p_2(p_3-1)(p_1-1)=p_3(p_1-1)+m+p_2. \end{aligned}$$

(2)

(b)

$$\begin{aligned} q_0= \begin{pmatrix} x\prod _{i=4}^{m+2}(y^{\frac{p_2}{m}}-\lambda _i x^{\frac{p_1-1}{m}}) &{} -z \\ z^{p_3-1} &{} -(y^{\frac{p_2}{m}}-x^{\frac{p_1-1}{m}}) \end{pmatrix}. \end{aligned}$$

Map**:

$$\begin{aligned} x=X_1^{p_3}X_3^{\frac{p_2}{m}},&y=X_2X_3^{\frac{p_1-1}{m}}, \quad z=X_1X_3^{\frac{p_2}{m}+(m-1)\frac{(p_1-1)p_2}{m^2}}X_4 \cdots X_{m+2}, \\&w=X_1^{p_3-1}X_4^{p_3-1} \cdots X_{m+2}^{p_3-1} \end{aligned}$$

Increase of $\alpha _3$ under the condition:

$$\begin{aligned}&\left( \frac{p_2}{m} + (m-1)\frac{(p_1-1)p_2}{m^2} \right) (p_3-1)\nonumber \\&\quad =\frac{(p_1-1)p_2}{m^2}+p_3+1. \end{aligned}$$

(3)

($\mathrm{II}_2$) $m \ge 2$, $m|p_1$, $m|p_3$. We consider the matrix factorization

$$\begin{aligned} q_0= \begin{pmatrix} \prod _{i=4}^{m+2}(z^{\frac{p_3}{m}}-\lambda _i x^{\frac{p_1}{m}}) &{} -y \\ xy^{p_2-1} &{} -(z^{\frac{p_3}{m}}-x^{\frac{p_1}{m}}) \end{pmatrix}. \end{aligned}$$

Map**:

$$\begin{aligned} x=X_1^{p_3}X_3^{\frac{p_3}{m}},&y=X_3^{(m-1)\frac{p_1p_3}{m^2}}X_4 \cdots X_{m+2}, \quad z=X_1X_2X_3^{\frac{p_1}{m}}, \\&w=X_1^{(m-1)\frac{p_3}{m}}X_4^{p_2-1} \cdots X_{m+2}^{p_2-1} \end{aligned}$$

Increase of $\alpha _3$ under the condition:

$$\begin{aligned} \frac{p_3}{m}+(m-1)\frac{p_1p_3}{m^2}(p_2-1)=(m-1)\frac{p_1p_3}{m^2}+p_2+1. \end{aligned}$$

(4)

One can show that the other possible matrix factorization

$$\begin{aligned} q_0= \begin{pmatrix} \prod _{i=4}^{m+2}(z^{\frac{p_3}{m}}-\lambda _i x^{\frac{p_1}{m}}) &{} -x \\ y^{p_2} &{} -(z^{\frac{p_3}{m}}-x^{\frac{p_1}{m}}) \end{pmatrix} \end{aligned}$$

has no solution.

(III) $m \ge 1$, $m|(p_1-1)$, $m|(p_2-1)$. We have the essentially different matrix factorizations:

(a)

$$\begin{aligned} q_0= \begin{pmatrix} y\prod _{i=3}^{m+2}(y^{\frac{p_2-1}{m}}-\lambda _i x^{\frac{p_1-1}{m}}) &{} -z \\ z^{p_3-1} &{} -x \end{pmatrix}. \end{aligned}$$

Map**:

$$\begin{aligned}&x=X_1^{p_3+ \frac{(m-1)+p_2}{p_1-1}}, \ y=X_1X_2^{p_3}, \ z=X_1^{p_2}X_2X_3\cdots X_{m+2}, \\&w=X_2^{p_3-1}X_3^{p_3-1} \cdots X_{m+2}^{p_3-1} \end{aligned}$$

Increase of $\alpha _1$ under the condition:

$$\begin{aligned} p_2(p_3-1)(p_1-1)=p_3(p_1-1)+(m-1)+p_2. \end{aligned}$$

(5)

(b)

$$\begin{aligned} q_0= \begin{pmatrix} xy\prod _{i=4}^{m+2}(y^{\frac{p_2-1}{m}}-\lambda _i x^{\frac{p_1-1}{m}}) &{} -z \\ z^{p_3-1} &{} -(y^{\frac{p_2-1}{m}}-x^{\frac{p_1-1}{m}}) \end{pmatrix}. \end{aligned}$$

Map**:

$$\begin{aligned}&x=X_1^{p_3}X_3^{\frac{p_2-1}{m}}, y=X_2^{p_3}X_3^{\frac{p_1-1}{m}}, \quad z=X_1X_2X_3^{\frac{p_1-1}{m}+\frac{p_2-1}{m}+(m-1)\frac{(p_1-1)(p_2-1)}{m^2}}\\&X_4 \cdots X_{m+2}, w=X_1^{p_3-1}X_2^{p_3-1}X_4^{p_3-1} \cdots X_{m+2}^{p_3-1} \end{aligned}$$

Increase of $\alpha _3$ under the condition:

$$\begin{aligned}&\left( \frac{p_1-1}{m}+\frac{p_2-1}{m} + (m-1)\frac{(p_1-1)(p_2-1)}{m^2} \right) (p_3-1)\nonumber \\&\quad =\frac{(p_1-1)(p_2-1)}{m^2}+p_3+1. \end{aligned}$$

(6)

(IV) $m \ge 1$, $m|(p_1-1)$, $m|p_2$. We have three essentially different factorizations.

(a)

$$\begin{aligned} q_0= \begin{pmatrix} \prod _{i=3}^{m+2}(y^{\frac{p_2}{m}}-\lambda _i x^{\frac{p_1-1}{m}}) &{} -z \\ yz^{p_3-1} &{} -x \end{pmatrix}. \end{aligned}$$

Map**:

$$\begin{aligned}&x=X_1^{p_3+ \frac{m+p_2}{p_1-1}}X_2, \ y=X_1X_2^{p_1}, \ z=X_1^{p_2}X_3 \cdots X_{m+2}, \\&w=X_2^{p_1-1}X_3^{p_3-1} \cdots X_{m+2}^{p_3-1} \end{aligned}$$

Increase of $\alpha _1$ under the condition:

$$\begin{aligned} p_2(p_3-1)(p_1-1)=p_3(p_1-1)+(m+1-p_1)+p_2. \end{aligned}$$

(7)

(b)

$$\begin{aligned} q_0= \begin{pmatrix} \prod _{i=3}^{m+2}(y^{\frac{p_2}{m}}-\lambda _i x^{\frac{p_1-1}{m}}) &{} -y \\ z^{p_3} &{} -x \end{pmatrix}. \end{aligned}$$

Map**:

$$\begin{aligned}&x=X_1^{p_3}X_2^{p_3}, \ y=X_2^{p_1+\frac{(p_1-1)(p_3-1)+m}{p_2}}, \ z=X_1X_2X_3 \cdots X_{m+2}, \\&w=X_3^{p_3} \cdots X_{m+2}^{p_3} \end{aligned}$$

Increase of $\alpha _2$ under the condition:

$$\begin{aligned} p_2(p_1(p_3-1)-p_3)=(p_1-1)(p_3-1)+m. \end{aligned}$$

(8)

This equation is equivalent to Eq. (7).

(c)

$$\begin{aligned} q_0= \begin{pmatrix} x\prod _{i=4}^{m+2}(y^{\frac{p_2}{m}}-\lambda _i x^{\frac{p_1-1}{m}}) &{} -z \\ yz^{p_3-1} &{} -(y^{\frac{p_2}{m}}-x^{\frac{p_1-1}{m}}) \end{pmatrix}. \end{aligned}$$

Map**:

$$\begin{aligned} x=X_1^{p_3}X_2X_3^{\frac{p_2}{m}},&y=X_2^{p_1}X_3^{\frac{p_1-1}{m}}, \quad z=X_1X_3^{\frac{p_2}{m}+(m-1)\frac{(p_1-1)p_2}{m^2}}X_4 \cdots X_{m+2}, \\&w=X_1^{p_3-1}X_2^{1+(m-1)\frac{p_1-1}{m}}X_4^{p_3-1} \cdots X_{m+2}^{p_3-1} \end{aligned}$$

Increase of $\alpha _3$ under the condition:

$$\begin{aligned} \left( \frac{p_2}{m} + (m-1)\frac{(p_1-1)p_2}{m^2} \right) (p_3-1)+\frac{p_1-1}{m}=\frac{(p_1-1)p_2}{m^2}+p_3+1.\nonumber \\ \end{aligned}$$

(9)

Similarly as in $(\mathrm{II}_2)$, one can show that the other matrix factorization

$$\begin{aligned} q_0= \begin{pmatrix} x\prod _{i=4}^{m+2}(y^{\frac{p_2}{m}}-\lambda _i x^{\frac{p_1-1}{m}}) &{} -y \\ z^{p_3} &{} -(y^{\frac{p_2}{m}}-x^{\frac{p_1-1}{m}}) \end{pmatrix}. \end{aligned}$$

has no solution.

(d) We also have to consider a special matrix factorization in the case $m=1$. We can write the equation as

$$\begin{aligned} f(x,y,z)=-y(xy^{p_2-1}-z^{p_3})+x^{p_1}. \end{aligned}$$

Then we have the matrix factorization

$$\begin{aligned} q_0= \begin{pmatrix} xy^{p_2-1}-z^{p_3} &{} -x \\ x^{p_1-1} &{} -y \end{pmatrix} \end{aligned}$$

Map**:

$$\begin{aligned} x=X_2^{p_3}X_3^{p_3}, \quad y=x_2^{p_1+\frac{2}{p_2-1}}, \quad z=X_1X_2X_3, \quad w=X_3^{p_3(p_1-1)} \end{aligned}$$

Increase of $\alpha _2$ under the condition:

$$\begin{aligned} p_3(p_1-1)(p_2-1)=p_1(p_2-1)+2. \end{aligned}$$

(10)

This equation is equivalent to Eq. (8) with $m=1$.

(V) The equation is

$$\begin{aligned} f(x,y,z)=-z(xz^{p_3-1}-y^{p_2})+yx^{p_1}. \end{aligned}$$

Then we have two matrix factorizations:

(a)

$$\begin{aligned} q_0= \begin{pmatrix} xz^{p_3-1}-y^{p_2} &{} -x \\ yx^{p_1-1} &{} -z \end{pmatrix} \end{aligned}$$

Map**:

$$\begin{aligned} x=X_2^{p_2}X_3^{p_3}, \quad y=X_1^{p_3}X_2X_3, \quad z=X_1X_2^{p_1p_2-p_2+1}, \quad w=X_1^{p_3-1}X_3^{p_1p_2-p_2+1} \end{aligned}$$

Increase of $\alpha _2$ under the condition:

$$\begin{aligned} p_2(p_1-1)(p_3-1)=(p_1-1)(p_3-1)+2. \end{aligned}$$

(11)

(b)

$$\begin{aligned} q_0= \begin{pmatrix} xz^{p_3-1}-y^{p_2} &{} -y\\ x^{p_1} &{} -z \end{pmatrix} \end{aligned}$$

Map**:

$$\begin{aligned} x=X_1X_2X_3^{p_2}, \quad y=X_1^{p_1p_3-p_1+1}X_3, \quad z=X_1^{p_1}X_2^{p_1}, \quad w=X_2^{p_1}X_3^{p_1p_2} \end{aligned}$$

Increase of $\alpha _1$ under the condition (11).

In the cases I, $\mathrm{II}_1$(b), $\mathrm{II}_2$, III(b), and IV(c), one can exchange the factor with $\lambda _3(=1)$ in the lower right corner of the matrix factorization by any of the factors corresponding to $\lambda _i$, $i=4, \ldots , m+2$, and interchange $X_3$ and $X_i$. This leads to an increase of $\alpha _i$ instead of $\alpha _3$. In Case III(a), one can interchange the variables x and y to get an increase of $\alpha _2$.

The solutions to the Eqs. (1)–(11) are precisely the entries of Table 2 and two extra ones. Equation (1) has the solution I, $m=2$, $(p_1,p_2,p_3)=(4,4,3)$, $A=(2,2,3,3)$. This corresponds to the minimally elliptic singularity $V'_{(1)}$ of [6] with weight system $W=(3,3,4;12)$ and $\varepsilon _W=-2$. Equation (9) has the additional solution IV, $m=1$, $(p_1,p_2,p_3)=(3,3,4)$, $A=(8,7,4)$. This corresponds to a singularity with the weight system $W=(12,8,7;36)$ with $\varepsilon _W=-9$. $\square$

Remark 1

The proof shows that the statement of Theorem 1 only holds in this very special situation. Namely, it shows that, except for two exceptional cases, the weighted homogeneous deformations f of an invertible polynomial with $\varepsilon _f=-1$ are the only weighted homogeneous deformations of an invertible polynomial with the property that there exists an $L_f$-graded matrix factorization Q of size 2 such that the corresponding singularity $\mathbf {F}_Q$ has the signature $({\widetilde{\alpha }}_1, \ldots , {\widetilde{\alpha }}_r)$ with ${\widetilde{\alpha }}_i=\alpha _i+1$ and ${\widetilde{\alpha }}_j=\alpha _j$ for $j \ne i$ where $A=(\alpha _1, \ldots , \alpha _r)$ are the Dolgachev numbers of f.

Remark 2

The reduction is a relation between the singularities. Not all possible extensions of the signatures are realised. For example, the following pairs are missing: $(2,3,7) \not \leftarrow (2,3,8)$, $(2,4,5) \not \leftarrow (2,4,6)$, $(3,3,4) \not \leftarrow (3,3,5)$, $(2,2,2,3) \not \leftarrow (2,2,2,4)$. In these cases, the graded matrix factorization $V_{r,\alpha _r}$ for the singularities with signature (2, 3, 7), (2, 4, 5), (3, 3, 4), and (2, 2, 2, 3) is of rank 4. In these cases, the function $f_W$ can be given by $f_W(x,y,z)=f'_W(x,y)+z^2$, where $f'_W(x,y)$ is the invertible polynomial in two variables given by

$$\begin{aligned} (2,3,7)\!:\! x^7+y^3 \!\quad (2,4,5)\!:\!x^5y+y^3 \!\quad (3,3,4):x^8+y^3 \quad (2,2,2,3):x^6y+y^3. \end{aligned}$$

We shall consider the $L_f$-graded matrix factorization of size 2 of $f'_W$ and the corresponding reduction R defined in Sect. 2(B). We enhance the reduction by these cases.

Example 1

We consider the case $W=(4,10,15;30)$, $A_W=(2,4,5)$, $f_W=x^5y+y^3+z^2$, $f'_W=x^5y+y^3$. This is the singularity $E_{13}$ in Arnold’s notation. We consider the graded matrix factorization of $f'_W$ given by

$$\begin{aligned} q_0=\begin{pmatrix} x^3y &{} -y^2 \\ y &{}x^2 \end{pmatrix}. \end{aligned}$$

Then $\mathbf {F}_Q(w,x)=w^3x+wx^4$. This is the singularity $Z_{12}$ with signature (2, 4, 6).

The isolated hypersurface singularities defined by regular systems of weights with $\varepsilon =-1$ and $r \le 4$ and the isolated complete intersection singularities which reduce to them are Kodaira singularities [10] or minimally elliptic singularities [13]. The minimal resolution is of Kodaira type II, III, IV or $\mathrm{I}_0^*$ (see [10]) and it carries a fundamental cycle Z (see [3, p. 132]). The self-intersection number of the fundamental cycle $D:=-Z^2$ is called the grade of the singularity [10]. Let (X, 0) be any of the singularities of Table 2 with Dolgachev numbers $\alpha _1, \ldots , \alpha _r$ and $r \le 4$ or of the isolated complete intersection singularities which reduce to them. According to [10] or [13], there is the correspondence between certain types of Dolgachev numbers, Kodaira types, the number $\ell$ of parameters $k_i$ of the Kodaira type, and the grades indicated in Table 3 (see also [15, p. 515]).

Table 3 Dolgachev numbers, Kodaira types, and grades

By [16, Corollary 7.9.2], the Kodaira type and hence the number $\ell$ of a singularity and its reduction are the same.

The possible reductions between these singularities are visualized in Fig. 2. Here a usual arrow refers to a reduction corresponding to Theorem 1, a dashed arrow to the special reduction R described in Sect. 2(B). The Dolgachev numbers of the regular systems of weights for $\varepsilon =-1$ are denoted by usual letters, those of the remaining singularities are denoted by scriptstyle letters. We also indicate the names of Arnold for the hypersurface singularities [2] and of Wall [16] (see also [7]) for the complete intersection singularities.

Fig. 2

The pyramid of reductions

On the other hand, each of these singularities has another triple or quadruple of numbers, the Gabrielov numbers, see [8] for the definition in the case of the hypersurface singularities and [9] in the case of the complete intersection singularities. They also have types like the Dolgachev numbers, but for them the columns corresponding to $\ell$ and D are interchanged, see Table 4. Note that the grade D corresponds to the type of Gabrielov numbers, see [13, Theorem 3.13]. More precisely, according to [13, Theorem 3.13], for $1 \le D \le 3$, the singularity is a hypersurface singularity and, for $D=4$, it is a complete intersection singularity. For $D=1$, the singularity is given by an equation of the form $z^2+y^3+g(x,y,z)=0$, where the polynomial g(x, y, z) has only terms of degree $\ge 4$, see also [13, Table 1]. For $D=2$, the singularity is given by an equation of the form $z^2+h(x,y)=0$, where the polynomial h(x, y) has only terms of degree $\ge 4$, see also [13, Table 2]. Finally, for $D=3$, the equation is $f(x,y,z)=0$, where the polynomial f(x, y, z) has only terms of degree $\ge 3$.

Table 4 Gabrielov numbers, grades, and numbers $\ell$

The hypersurface singularities with $r=3$ are the 14 exceptional unimodal singularities. There is Arnold’s strange duality [2] between these singularities which interchanges Dolgachev and Gabrielov numbers.

There is another partial order between singularities, namely the adjacency. We consider a special type of adjacency, namely the s-adjacency of singularities as defined by H. Laufer in [14, Definition 4.12]. The s-adjacency corresponds to a deformation with constant grade. By [14, Theorem 4.13], the s-adjacencies between the 14 exceptional unimodal singularities are given by Fig. 3, where the numbers correspond to the Gabrielov numbers. This is the pyramid of adjacencies given in [2, p. 255].

Fig. 3

The pyramid of s-adjacencies between the 14 exceptional unimodal singularities

Remark 3

Note that the s-adjacency is a special case of adjacency. For example, the general adjacency between the 14 exceptional unimodal singularities corresponds to the natural partial ordering between the Gabrielov numbers, see [4].

A comparison of Fig. 2 restricted to the 14 exceptional unimodal singularities, Fig. 3, and Arnold’s strange duality yields the following corollary of Theorem 1.

Corollary 1

Denote Arnold’s strange duality by $X \leftrightarrow X^*$. If a singularity Y is a reduction of X, then the dual singularity $X^*$ is s-adjacent to the dual singularity $Y^*$.