A new characterization of exceptional unimodal singularities

TL;DR

This paper introduces a novel way to characterize exceptional unimodal singularities, inspired by mirror symmetry in weighted projective spaces, offering new insights into their structure.

Contribution

It provides a new characterization of exceptional unimodal singularities, advancing understanding in singularity theory and mirror symmetry.

Findings

New characterization of exceptional unimodal singularities

Connections to mirror symmetry in weighted projective spaces

Potential implications for classification of singularities

Abstract

Motivated by mirror symmetry for weighted projective spaces, we give a new characterization of exceptionoal unimodal singularities.

Tables

Table 1. Table 1. 14 exceptional unimodal singularities

Table 2. Table 2. Semi-duality of weights

No.	$a$	$C$	$D$	semi-dual
1	$(1,1,1)$			1
3	$(2,2,1)$			5
				3
4	$(4,4,3)$			4 (dual)
5	$(3,1,1)$			3
6	$(5,2,2)$			6
7	$(4,2,1)$			7
				19
8	$(6,3,2)$			10
				24
9	$(10,5,4)$			9 (dual)
10	$(6,4,1)$			8
12	$(9,6,2)$			12
13	$(12,8,3)$			20 (dual)
14	$(21,14,6)$			14 (dual)
18	$(3,3,2)$			25
19	$(3,2,2)$			7
				19
20	$(9,8,6)$			13 (dual)
21	$(2,1,1)$			21
22	$(6,5,3)$			22 (dual)
24	$(5,4,2)$			8
25	$(4,3,1)$			18
28	$(10,7,3)$	–	–	–
37	$(8,4,3)$			58 (dual)
38	$(15,8,6)$			50 (dual)
39	$(9,5,3)$			60 (dual)
40	$(7,4,2)$			40
42	$(5,3,1)$			63
44	$(8,5,2)$	–	–	–
45	$(14,9,4)$	–	–	–
50	$(15,10,4)$			38 (dual)
51	$(18,12,5)$	–	–	–
58	$(6,5,4)$			37 (dual)
59	$(8,7,5)$	–	–	–
60	$(7,6,4)$			39 (dual)
63	$(4,3,2)$			42
66	$(3,2,1)$			66
				66
71	$(7,4,3)$	–	–	–
72	$(7,5,2)$	–	–	–
77	$(13,7,5)$	–	–	–
78	$(11,6,4)$			78 (dual)
82	$(11,7,3)$	–	–	–
87	$(5,4,3)$			87 (dual)
89	$(5,3,2)$	–	–	–

Equations

$$F_a^{-1}(t)$$

$$=\left\{(u_1,u_2,u_3)\in {\left({\mathbb{C}}^{\times }\right)}^3|u_1^{a_1+1}{u}_2^{a_2}{u}_3^{a_3}+u_1^{a_1}{u}_2^{a_2+1}{u}_3^{a_3}+u_1^{a_1}{u}_2^{a_2}{u}_3^{a_3+1}+1=t{u}_1^{a_1}{u}_2^{a_2}{u}_3^{a_3}\right\}.$$

$$G_a(u_1,u_2,u_3)$$

$$H_a(u_1,u_2,u_3)$$

$$G_a(u_1,u_2,u_3)+1=t{H}_a(u_1,u_2,u_3).$$

$$f_C(x,y,z)=\sum _{i=1}^3{x}^{c_{i1}}{y}^{c_{i2}}{z}^{c_{i3}}.$$

$${\phi }_D(u_1,u_2,u_3)={\left(u_1^{d_{i1}}{u}_2^{d_{i2}}{u}_3^{d_{i3}}\right)}_{i=1}^3,$$

$$h(x,y,z)=xyz.$$

$$f_C(x,y,z)+w^{\check{h}}=txyzw$$

$${\check{a}}_i=\sum _{j=1}^3{d}_{ij}$$

$$B_a=CD$$

$$\left(\begin{array}{ccc}{a}_1& a_2& a_3\end{array}\right)=\left(\begin{array}{ccc}1& 1& 1\end{array}\right)D$$

$$B_a=\left(\begin{array}{ccc}{a}_1+1& a_2& a_3\\ a_1& a_2+1& a_3\\ a_1& a_2& a_3+1\end{array}\right).$$

$$|\det C|=\det B_a=a_1+a_2+a_3+1.$$

$$C\left(\begin{array}{c}{a}_1\\ a_2\\ a_3\end{array}\right)=\left(\begin{array}{c}h\\ h\\ h\end{array}\right)\text{and}\left(\begin{array}{ccc}{\check{a}}_1& {\check{a}}_2& {\check{a}}_3\end{array}\right)C=\left(\begin{array}{ccc}\check{h}& \check{h}& \check{h}\end{array}\right)$$

$$|\det C|=h=\check{h}.$$

Full text

1. introduction

Exceptional unimodal singularities are introduced by Arnold [Arn76] as hypersurface singularities of modality one which do not come in infinite family. There are exactly 14 of them which are defined by weighted homogeneous polynomials as shown in Table 1. Dolgachev [Dol74] found another characterization of exceptional unimodal singularities as triangle singularities which are hypersurfaces. The triple $(\delta_{1},\delta_{2},\delta_{3})$ of integers specifying the triangle singularity is called the Dolgachev number.

The Milnor lattices of exceptional unimodal singularities are computed by Gabrielov [Gab74]. The Coxeter–Dynkin diagram of the Milnor lattice with respect to a distinguished basis of vanishing cycles is specified by a triple $(\gamma_{1},\gamma_{2},\gamma_{3})$ of integers called the Gabrielov number. In Table 1, one can see that exceptional unimodal singularities come in pairs in such a way that the Dolgachev number and the Gabrielov number are interchanged. This fact is discovered by Arnold and given the name strange duality. Pinkham [Pin77] and Dolgachev and Nikulin [Dol83, Nik79] gave an interpretation of strange duality as the exchange of the transcendental lattices and algebraic lattices of K3 surfaces. This interpretation can be considered as a precursor of mirror symmetry for K3 surfaces [AM97, Dol96]. See e.g. [KMU13, Ued, LU] and references therein for the relation between exceptional unimodal singularities and mirror symmetry for K3 surfaces.

A weight system is a quadruple $(a_{1},a_{2},a_{3};h)$ of positive integers. The triple $a=(a_{1},a_{2},a_{3})$ is called the weight, and the integer $h$ is called the degree. A polynomial $\sum_{i=(i_{1},i_{2},i_{3})\in\mathbb{N}^{3}}\alpha_{i}x^{i_{1}}y^{i_{2}}z^{i_{3}}\in\mathbb{C}[x,y,z]$ is weighted homogeneous of weight $a$ and degree $h$ if $\sum_{k=1}^{3}i_{k}a_{k}=h$ for any $i\in\mathbb{N}^{3}$ with $\alpha_{i}\neq 0$. We set $a_{0}=h-a_{1}-a_{2}-a_{3}.$ It is known by Reid (unpublished) and Yonemura [Yon90] that there are exactly 95 weights $a$ such that the minimal model of a general anticanonical hypersurface in the weighted projective space $\mathbb{P}(a_{0},a_{1},a_{2},a_{3})$ is a K3 surface. The weight systems in Table 1 associated with exceptional unimodal singularities are on the Reid–Yonemura list. Since all of them satisfy the condition $a_{0}=1$, we will always assume this condition in this paper, and write $W_{a}=(a_{1},a_{2},a_{3};a_{1}+a_{2}+a_{3}+1).$ There are 41 weights with $a_{0}=1$ on the Reid–Yonemura list.

The mirror of $\mathbb{P}(a_{0},a_{1},a_{2},a_{3})$ is given by the regular function $F_{a}=u_{0}+u_{1}+u_{2}+u_{3}$ on $M_{q}=\{(u_{0},u_{1},u_{2},u_{3})\in\left(\mathbb{C}^{\times}\right)^{4}\mid u_{0}^{a_{0}}u_{1}^{a_{1}}u_{2}^{a_{2}}u_{3}^{a_{3}}=q\},$ whose fiber $F_{a}^{-1}(t)$ is the mirror of an anticanonical hypersurface in $\mathbb{P}(a_{0},a_{1},a_{2},a_{3})$ [Giv95]. We set the Kähler parameter $q$ to 1 for simplicity. Since $a_{0}=1$, the fiber $F_{a}^{-1}(t)$ can be described as

[TABLE]

We define polynomial maps $G_{a},H_{a}\colon\mathbb{C}^{3}\to\mathbb{C}$ by

[TABLE]

so that $F_{a}^{-1}(t)$ is defined by

[TABLE]

An integer matrix $C=\left(c_{ij}\right)_{i,j=1}^{3}$ with non-negative entries defines a polynomial map $f_{C}\colon\mathbb{C}^{3}\to\mathbb{C}$ by

[TABLE]

Similarly, an integer matrix $D=\left(d_{ij}\right)_{i,j=1}^{3}$ with non-negative entries defines a polynomial map $\varphi_{D}\colon\mathbb{C}^{3}\to\mathbb{C}^{3}$ by

[TABLE]

which restricts to an isomorphism of tori $(\mathbb{C}^{\times})^{3}$ if $\lvert\det D\rvert=1$. We also define a polynomial map $h\colon\mathbb{C}^{3}\to\mathbb{C}$ by

[TABLE]

Consider the following condition on the weight $a$:

Condition 1**.**

There exist $3\times 3$ integer matrices $C=(c_{ij})_{i,j=1}^{3}$ and $D=(d_{ij})_{i,j=1}^{3}$ with non-negative entries satisfying

1. (i)

$\lvert\det D\rvert=1,$ 2. (ii)

$G_{a}=f_{C}\circ\varphi_{D},$ 3. (iii)

$H_{a}=h\circ\varphi_{D},$ and 4. (iv)

$f_{C}$ has an isolated critical point at the origin.

If Condition 1 is satisfied, then $F_{a}^{-1}(t)$ can be compactified to a quasi-smooth hypersurface

[TABLE]

of Dwork type in $\mathbb{P}(1,\check{a}_{1},\check{a}_{2},\check{a}_{3})$, where

[TABLE]

and $\check{h}=1+\sum_{i=1}^{3}\check{a}_{i}.$ Note that Conditions (1.ii) and (1.iii) can be written as

[TABLE]

and

[TABLE]

respectively, where

[TABLE]

It follows from (11) and Condition (1.i) that

[TABLE]

The main result in this paper is the following characterization of exceptional unimodal singularities:

Theorem 1**.**

A weight $(a_{0},a_{1},a_{2},a_{3})$ with $a_{0}=1$ on the Reid–Yonemura list comes from an exceptional unimodal singularity if and only if $a=(a_{1},a_{2},a_{3})$ satisfies Condition 1. If this is the case, then $C$ is uniquely determined by $a$ up to permutation of rows and columns, and $f_{C}$ is a defining polynomial of the strange dual singularity.

For each $a$ among 41 on the Yonemura–Reid list, there are finitely many integer matrices $D$ with non-negative entries satisfying (12). Each such $D$ determines $C$ by (11), and the complete list of $C$ and $D$ such that $\lvert\det D\rvert=1$, up to permutation of rows and columns, is shown in Table LABEL:tb:semi-dual. If this is the case, then we say that the weight $\check{a}$ defined by (10) is semi-dual to the weight $a$. It follows from Table LABEL:tb:semi-dual that semi-duality is reflexive; $\check{a}$ is semi-dual to $a$ if and only if $a$ is semi-dual to $\check{a}$. The proof Theorem 1 is given by testing if each $C$ satisfies Condition 1.(iv).

Note that a normal form of a weighted homogeneous exceptional unimodal singularity is not unique in general, and Theorem 1 allows us to fix one uniquely. For example, the defining equation for the $W_{12}$-singularity can be written either as $x^{5}+y^{4}+z^{2}$ or $x^{5}+y^{2}z+z^{2}$, and only the latter comes from Theorem 1.

Recall from [Kob08] that weight systems $W_{a}=(a_{1},a_{2},a_{3};h)$ and $W_{\check{a}}=\left(\check{a}_{1},\check{a}_{2},\check{a}_{3};\check{h}\right)$ are said to be Kobayashi dual if there is a $3\times 3$ integer matrix $C$ with non-negative entries satisfying the weighted magic square condition

[TABLE]

and the primitivity

[TABLE]

Kobayashi duality is a generalization of strange duality to weights which may not come from exceptional unimodal singularities [Kob08, Ebe06]. The matrices $C$ appearing in Table LABEL:tb:semi-dual are primitive weighted magic squares, so that our semi-duality is a special case of Kobayashi duality.

Acknowledgements. The author thanks her advisor Kazushi Ueda for guidance and encouragement.