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УДК 517.55
Three Dimensional Saito Free Divisors and Singular Curves
Jiro Sekiguchi*
Department of Mathematics, Faculty of Engineering, Tokyo University of Agriculture and Technology, Koganei, Tokyo 184-8588, Japan
Received 10.10.2007, accepted 05.12.2007 The purpose of the present study is to find out examples of Saito free divisors by constructing Lie algebras generated by logarithmic vector fields along them. In the course of the study, the author recognized a deep connection between Saito free divisors and deformations of curve singularities. In this paper, we will explain a method of constructing three dimensional Saito free divisors and show some examples.
Key words: Saito free divisors, Lie algebras.
Introduction
The notion of Saito free divisors was introduced by K. Saito (see [12]) in connection with the study of universal unfoldings of isolated hypersurface singularity. In two dimensional case, any divisor is Saito free (cf. [12], (1.7) Cor.). Since divisors in higher dimensional spaces are not Saito free in general, it is interesting to find Saito free divisors and study their properties. There are many studies on them independent of singularity theory (for examples, see [2], [7], [10], [17] and the references therein).
The primitive purpose of the present study is to find out examples of Saito free divisors by constructing Lie albegras generated by logarithmic vector fields along them. In the course of the study, the author recognized a deep connection between Saito free divisors and deformations of curve singularities. In this paper, we will explain a method of constructing three dimensional Saito free divisors and show some examples.
The contents of this paper is as follows. We will first review the results in [13], [14]. The main purpose of [14] (which is a new version of [13]) is a classification of weighted homogeneous polynomials of three variables which have the same properties as the discriminant sets of quotient spaces of irreducible Coxeter groups of rank three. In particular, they define Saito free divisors. There is a connection between such divisors and deformations of plane curves with simple singularities of exceptional types ([14]). It should be noted here that the fundamental group of the complement of each divisor classified in [14] is determined by T. Ishibe and K. Saito (cf. [11]). The second purpose of this paper is to explain a basic idea how to obtain Saito free divisors from the eight of the so-called 14 exceptional families of isolated singularities due to V. Arnol'd [5], generalizing the formulation given in [14]. The author obtained several Saito free divisors by this method. Partial results are given in [15]. In this paper, we shall show complete lists of such divisors related with singularities of Types E12,Z11, W12 and some interesting examples of divisors related with singularities of Types E13, E14, Z12, Z13, W13.
There are many studies still to be done. The first one is to classify Saito free divisors associated with the eight singularities. The author almost succeeded to answer this question with the assistance of M. Noro (Kobe Univ.). The second one is to determine the fundamental group of the complement of each of the divisors obtained in this paper. The third one is to determine the b-functions of the
* e-mail: sekiguti@cc.tuat.ac.jp
© Siberian Federal University. All rights reserved
polynomials obtained by our method. The answer to this question is quite hopefull because a software Risa/Asir is effecitive for such a purpose. In fact, the b-functions of the 17 polynomials in [14] are already determined by H. Nakayama (Kobe Univ.). The fourth is to determine the types of singularities of the divisors and give a relationship between the root systems of the singularities and those of exceptional families determined by Gabrielov [9] (see also Ebeling [8]). The work of Urabe [18] is suggestive to develop further in this direction. The fifth is to generalize the study in this paper to the case of more complicated singularities.
Acknowledgments. The author would like to thank Alexander G. Alexandrov and Kyoji Saito for fruitful discussions and valuable comments.
1. Definition of Saito Free Divisors
Let X be the n-dimensional affine space Cn and let x = (xi,x2,... , xn) be its coordinate system. A polynomial f (x) of x is weighted homogeneous of type r = (r i,... , rn) if f (trixi,... ,trnxn) = thf(x) for any t € C, where h is a certain number. In this note, we always take ri,...,rn as positive integers. Then h is the degree of f (x).
Let f (x) be a reduced polynomial and let D be the hypersurface of X defined by f (x) = 0. A vector field V on X with polynomial coefficients is logarithmic along D if Vf/f is a polynomial. Let DerX(log D) be the totality of logarithmic vector fields along D. It is clear that DerX(log D) is a Lie algebra defined over R = C[x1,..., xn]. Then D is Saito free if DerX (log D) is a free R-module. Since the theory of logarithmic vector fields goes back to the paper of K. Saito [12], it is better to start with reviewing a criterion by Saito himself for a divisor to be Saito free.
n
Lemma 1 (cf. [12], Lemma (1.9)). Let V = aj(x)dXj (i = 1,.. ., n) be vector fields on X, where
j= i
aj(x) are polynomials. We assume that V (i = 1, ... ,n) satisfy the conditions (i), (ii):
n
(i) [V, Vj] = (x)Vk for some polynomials bj(x).
k= 1
(ii) F(x) = det(aj (x)) has no multiple factor.
Then each V is logarithmic along the divisor D defined by F(x) and they form a system of generators of the Lie algebra consisting of vevtor fields logarithmic along D.
K. Saito showed that the discriminant set of the parameter space of a versal family of a hyper-surface with an isolated singularity is a Saito free divisor.
A typical example arises from the study of rational double points. More generally, we here treat the case of real reflection groups to show examples of Saito free divisors ([12], [19], [20]). Let W be an irreducible reflection group acting on a real vetor space E of dimensions n. Let £ = (£i,..., £n) be its coordinate system and let R^ be the ring of polynomials of £. Then by the theorem of Chevalley, there are algebraically independent homogeneous polynomials Pj(£) (i = 1,..., n) such that RW = C[Pi,..., Pn]. We put dj = deg(Pj) (j = 1, 2,..., n) and assume that d1 < d2 < • • • < dn. If a.j (j = 1,..., N) forms the set of linear functions defining reflection hyperplanes, then the product D = nN=i "2 is W-invariant. In particular there is a polynomial F(Xi,..., Xn) such that D = F(Pi,..., Pn). Then D = 0 defines a Saito free divisor on the affine space with coordinate (Pi,..., Pn) and F(Pi,..., Pn) is a Saito free polynomial of type (d1,..., dn).
In the subsequent sections, we restrict our attention to the case n = 3 and formulate a problem of finding Lie algebras generated by three vector fields containing an Euler vector with weight. In virtue of Lemma 1, it is plausible that the polynomial defined as the determinant of the 3 x 3 matrix consisting of coefficients of the vector fields defines a Saito free divisor. This approach leads us to very successful results in the problem of constructing Saito free divisors.
2. Singular Curves
In this section, we review some of the singular curves which are used in the subsequent considerations.
2.1. Simple Singularities
We recall the defining equations of curves with simple singularities at the origin in C2.
An : un+1 + v2 + w2 = 0 (n > 1)
Dn : u(un-2 + v2) + w2 = 0 (n > 4)
E6 : u4 + v3 + w2 =0 (1)
E7 : u(u2 + v3) + w2 =0
E8 : u5 + v3 + w2 =0
2.2. Unimodular singularities
We use the notation in Arnol'd [5]. Our interest is related with singular curves. Among the 14 exceptional families in [5], p.93, there are 8 which are realized by curve singularities. First we give these 8 families in Table I. Putting w = 0, we obtain curves in uv-space. In Table I, the triplet of numbers (a, b, c) and h mean the type and the degrees of the polynomials in question.
Table I
Type Equation (a, 6, c) h (p ^ r)
E12 u7 + v3 + w2 =0 (6, 14, 21) 42 (2, 3, 7)
E13 w(w2 + v5) + w2 = 0 (4, 10, 15) 30 (1, 2, 5)
E14 u8 + v3 + w2 =0 (3, 8,12) 24 (1, 3, 8)
Z11 w(w4 + v3) + w2 = 0 (6, 8,15) 30 (2, 3,4)
Z12 uv(u3 + v2) + w2 =0 (4, 6,11) 22 (1, 2, 3)
Z13 u(u5 + v3) + w2 = 0 (3, 5, 9) 18 (1, 3, 5)
W12 u5 + v4 + w2 =0 (4, 5,10) 20 (2,4, 5)
W13 v(u4 + v3) + w2 = 0 (3, 4, 8) 16 (1, 3,4)
We now explain the triplet of numbers (p, q, r). For the triplet of numbers (a, b, c) and h in Table I, we define
= 2 (th-a - 1)(th-b - 1)(th-c - 1) X( ) (t° - 1)(tb - 1)(tc - 1)
Then
x(t) = t-2 + td2 + td3 +-----+ td^
for positive integers ¿2, ¿3,..., where ^ is the Milnor number corresponding to the singularity. So we may take that 0 < ¿2 < ¿3 < • • • < Then (p, q, r) are triplet of numbers proportional to
(¿2, a, b).
3. Saito Free Divisors and Simple Singularities
In this section, we review the results in [14].
Let x, y, z be variables and let p, q, r be natural numbers such that p < q < r. In case 1 < p, we may assume that p, q, r have no common factor. We consider three vector fields on (x, y, z)-space
including the Euler vector field with weight (p, q, r): ' Vo < V V2
where hj (x,y, z) are polynomials of x,y, z. In addition, we define a 3 x 3 matrix M associated with the vector fields V0, V1, V2 by
(px qy rz \
qy h22(x, y, z) h23(x,y, z) I . (3)
rz h32(x, y, z) h33(x,y, z) /
Now we consider the conditions on Vo, Vi, V2:
Condition 1. (i) [Vo, Vi] = (q - p)Vi, [Vo, V2] = (r - p)Vi.
(ii) There exist polynomials fj (x, y, z) (j = 0, 1, 2) such that
[Vi, V2] = fo(x, y, z)Vo + fi(x, y, z)Vi + f2(x, y, z)V2.
dh22
(iii) —— is a non-zero constant.
dz
(iv) The polynomial det(M) is not trivial. (We say that det(M) is trivial if it turns out to be z3 by a weight preserving coordinate transformation.)
Condition 1 (i), (ii) claim that the C[x, y, z]-module L(det(M)) spanned by Vo, Vi, V2 becomes a Lie algebra over C[x, y, z] and Condition 1 (iii) does that degz det(M) = 3. Condition 1 (iv) is supplementary. If Vo, Vi, V2 satisfy Condition 1, it follows that Vj det(M) is divisible by det(M) (j = 0,1, 2). Namely, Vo, V, V2 and therefore all the vector fields of L(det(M)) are logarithmic along the hypersurface {(x, y, z); det(M) = 0} in the sense of [12]. Conversely, it is possible to reconstruct the vector fields Vo, Vl, V2 from the polynomial det(M).
Remark 1. It is not trivial whether the polynomial det(M) has a multiple factor or not. But as will be seen Theorem 1 below, it has no multiple factor. As a result, it follows that the hypresurface defined by det(M) =0 is a Saito free divisor.
Let W be a finite reflection group acting on the vector space E of Dimensions 3. Let x, y, z be basic W-invariant polynomials and let F be the discriminant of W. Then the Lie algebra of logarithmic vector fields along F-1(0) satisfies Condition 1. The primitive interest is to find polynomials which have properties as discriminants. Problem 1 below gives an idea to find such polynomials.
Problem 1. Find all the triples {Vo, Vl, V2} of vector fields satisfying Condition 1. Or equivalently, find all polynomials F(x, y, z) of the form F = det(M).
We now state the first main theorem of [14] which answers to this problem.
Theorem 1. Let x,y, z be variables and let p, q, r be natural numbers such that p < q < r. In case 1 < p, we may assume that p, q, r have no common factor. Then the following assertions hold.
(i) If (p, q, r) = (2, 3, 4), (1, 2, 3), (1, 3, 5), there is no triple {Vo, Vl, V2} of vector fields satisfying Condition 1.
(ii) If (p, q, r) is one of (2, 3, 4), (1, 2, 3), (1, 3, 5), the polynomial F(x,y, z) of the form F = det(M) is reduced to one of the following polynomials by a weight preserving coordinate transformation.
d d d px— + qy— + rz —, dx dy dz
d d d qydX + h22(x,y,z)dy + h23(x,y, z) —, (2)
d d d
rz— + h32(x,y, z)dy + h33(x,y, z) —,
(11.1) The case (p, q, r) = (2, 3, 4). (This case corresponds to the reflection group of Type A3.) FAj1 = 16x4z - 4x3y2 - 128x2z2 + 144xy2z - 27y4 + 256z3.
FA,2 = 2x6 - 3x4z + 18x3y2 - 18xy2z + 27y4 + z3.
(11.2) The case (p, q, r) = (1, 2, 3). (This case corresponds to the reflection group of Type B3.) FBjl = z(x2y2 - 4y3 - 4x3z + 18xyz - 27z2).
Fb,2 = z(-2y3 + 4x3z + 18xyz + 27z2). Fb,3 = z(-2y3 + 9xyz + 45z2). Fb,4 = z(9x2y2 - 4y3 + 18xyz + 9z2). Fb,5 = xy4 + y3z + z3.
FB,6 = 9xy4 + 6x2y2z - 4y3z + x3z2 - 12xyz2 + 4z3. Fbj7 = 2xy4 - 2x2y2z - y3z + 2x3z2 + 2xyz2 + z3.
(11.3) The case (p, q, r) = (1, 3, 5). (This case corresponds to the reflection group of Type H3.) FHj1 = -50z3 + (4x5 -50x2y)z2 + (4x7y + 60x4y2 + 225xy3)z- y5 -115x3y4 - 10x6y3 -4x9y2. Fh,2 = 100x3y4 + y5 + 40x4y2z - 10xy3z + 4x5z2 - 15x2yz2 + z3.
FH,3 = 8x3y4 + 108y5 - 36xy3z - x2yz2 + 4z3. Fh,4 = y5 - 2xy3z + x2yz2 + z3. Fh,5 = x3y4 - y5 + 3xy3z + z3.
FH,6 = x3y4 + y5 - 2x4y2z - 4xy3z + x5z2 + 3x2yz2 + z3. Fh,7 = xy3z + y5 + z3.
Fhj8 = x3y4 + y5 - 8x4y2z - 7xy3z + 16x5z2 + 12x2yz2 + z3.
Remark 2. Let F(x,y, z) be one of the polynomials in Theorem 1. Then the curve C : {(y, z); F(0, y, z) = 0} is regarded as the simple singularity of type E6 E7, E8 if F(x,y, z) is one of the polynomials Faj (j = 1, 2), Fbj (j = 1, ..., 7), FH,j (j = 1, .. ., 8), respectively (cf. [16]). Therefore if we regard x as a parameter, the family of curves Cx : F(x, y, z) = 0 on yz-space is a deformation of the curve Co = C.
We state the second main theorem of [14].
Theorem 2. (i) There is a natural bijection between the set of polynomials of (I) and that of corank one subdiagrams of the Dynkin subdiagram of Type E6 left fixed by its non-trivial involution.
(ii) There is a natural bijection between the set of polynomials of (II) (resp. (III)) and that of corank one subdiagrams of the Dynkin diagram of Type E7 (resp. E8).
Remark 3. It is an interesting problem to determine the fundamental groups of the complements of the hypersurfaces defined by the seventeeen polynomials in Theorem 1. This is done by Ishibe and Saito (cf. [11]).
4. A Method of Constructing Saito Free Divisors from Singular Curves
In this section, we explain an idea how to construct Saito free divisors from some of the 14 exceptional families of isolated singularities classified by Arnol'd (cf. [4]).
As before, let x, y, z be variables and let p, q, r be natural numbers such that p < q < r. In case 1 < p, we may assume that p, q, r have no common factor. We consider three vector fields on (x, y, z)-space including the Euler vector field of Type (p, q, r):
' Vo
< Vi V2
d d d px— + qy— + rz—, dx dy dz
d d d
h-2i(x, y, z) dx + h22(x,y,z) dy + h23(x,y,z) —,
d d d h3i(x,y,z)dx + h-32(x, y, z)dy + h33(x, y, z) —,
where hj (x,y, z) are weighted homogeneous polynomials of x, y, z. Corresponding to the vector fields Vo, Vi, V2, we define a 3 x 3 matrix M by
Now we consider the conditions on Vo, Vi, V2
Condition 2. (i) [Vo, V1] = wL V1, [Vo, V2] = w2V2 for some integers wi, W2.
(ii) There exist polynomials fj (x,y,z)(j = 0,1, 2) such that [V^V^] = fo(x,y,z)Vo + fi(x, y, z)V1 +
f2(x,y,z)V2.
We now give conditions on the matrix M for the eight families of isolated singularities in Table I. First we take the triplet of integers (p, q, r) as in Table I.
(1) The cases E12, E13, E14
h-21 = y2, h.31 = z, h22(0, y, z) = az (a is a non-zero constant) and det(M)x=o coincide with z3 + y7, z3 + y5z, z3 + y8 in the cases E12, E13, E14, respectively up to weight preserving coordinate transformations.
(2) The cases zl1, Z12, Z13
h2i = y2, h3i = z, h22(0,y, z) = ayz (a is a non-zero constant) and det(M)x=o coincide with y(z3 + y4), yz(z2 + y3), y(z3 + y5) in the cases Z11, Z12, Z13, respectively up to weight preserving coordinate transformations.
(3) The cases W12, W13
h2i = y2, h.31 = z, h22(0,y, z) = az2 (a is a non-zero constant) and det(M)x=o coincide with z4 + y5, z4 + y4z in the cases W12, W13, respectively up to weight preserving coordinate transformations.
5. Examples of Saito Free Divisors Related
with the Eight Members of the 14 Exceptional Families
In this section, we show the totality of Saito free divisors related with the singularities of types E12, Zii, W12 obtained by our method explained in the previous section. The classification of these three cases is not difficult compared with the remaining five cases. The author also succeeded to classify the remaining five cases with the assistance of M.Noro (Kobe Univ.). Since we need to spend a lot of pages to show the result, we abandoned to do. Instead we only show Saito free polynomials related with these five singularities having parameters.
The following nine polynomials fi,..., fg are Saito free polynomials each of which defines deformations of a curve on (y, z)-space with a singular point of type E12 at the origin regarding x as a parameter. Note that f8 and fg have complex coefficients and they are conjugate to each other. It is possible to prove the following statement. (For a proof, see [3]).
Let F(x, y, z) be a weighted homogeneous Saito free polynomial with weight (2, 3, 7) and degree 21. Assume that F(0, y, z) = z3 + y7. Then F(x,y, z) coincides with one of the 9 polynomials fi,..., fg up to weight preserving coordinate transformation.
f1 = x~6*y~3/32 + 3*x~3*y~5/28 + 3*y~7/49 - 3/16*x~4*y~2*z - 3/7*x*y~4*z + z"3; f2 = -1/864*x~6*y~3 + 5*x~3*y~5/84 + 3*y~7/49 - 1/48*x~4*y~2*z - 3/7*x*y~4*z + z"3; f3 = x~3*y~5/21 + 3*y~7/49 - 3/7*x*y~4*z + z"3; f4 = 3*y~7/49 - 3/7*x*y~4*z + z"3;
f5 = 78125*x~9*y/200120949 + 44375*x~6*y~3/4840416 + 107*x~3*y~5/1372 + 3*y~7/49 -6250*x~7*z/3176523 - 1375*x~4*y~2*z/16464 - 3/7*x*y~4*z + z"3;
(5)
5.1. The Case E
12
f6 = 64*x~9*y/823543 + 208*x~6*y~3/453789 + 68*x~3*y~5/1029 + 3*y~7/49 +
48*x~7*z/117649 - 40*x~4*y~2*z/1029 - 3/7*x*y~4*z + z"3; f7 = -448*x~9*y/243 + 16*x~6*y~3/9 - 4*x~3*y~5/7 + 3*y~7/49 - 112*x~7*z/27 +
8/3*x"4*y"2*z - 3/7*x*y~4*z + z"3; f8 = -752*x~9*y/823543 - 2017*I*x~9*y/(823543*Sqrt[3]) - 397*x~6*y~3/33614 + 323*I*Sqrt[3]*x~6*y~3/33614 + 39*x~3*y~5/686 + 9/686*I*Sqrt[3]*x~3*y~5 + 3*y~7/49 + 1763*x~7*z/235298 - 249*I*Sqrt[3]*x~7*z/235298 + 3/686*x~4*y~2*z - 37/686*I*Sqrt[3]*x~4*y~2*z - 3/7*x*y~4*z + z"3; f9 = -752*x~9*y/823543 + 2017*I*x~9*y/(823543*Sqrt[3]) - 397*x~6*y~3/33614 -323*I*Sqrt[3]*x~6*y~3/33614 + 39*x~3*y~5/686 - 9/686*I*Sqrt[3]*x~3*y~5 + 3*y~7/49 + 1763*x~7*z/235298 + 249*I*Sqrt[3]*x~7*z/235298 + 3/686*x~4*y~2*z + 37/686*I*Sqrt[3]*x~4*y~2*z - 3/7*x*y~4*z + z"3;
5.2. The Case Zn
The following polynomials /i,/2,/3,/4 are Saito free polynomials each of which defines deformations of a curve on (y, z)-space with a singular point of Type Zii at the origin regarding x as a parameter.
Let F(x, y, z) be a weighted homogeneous Saito free polynomial with the weight (2, 3, 4) and degree 30. Assume that F(0, y, z) = y(z3 + y4). Then F(x, y, z) coincides with one of the 4 polynomials /i, /2, /3, /4 below up to weight preserving coordinate transformation.
f1 = y*(3*y"4 - 3*x*y"2*z + z"3)
f2 = y*(x"3*y"2 + 9*y"4 - 9*x*y"2*z + 3*z"3)
f3 = y*(-4*x~3*y~2 - 27*y~4 + 16*x~4*z + 144*x*y~2*z - 128*x~2*z~2 + 256*z~3) f4 = y*(2*x"6 + 18*x"3*y"2 + 27*y"4 - 3*x"4*z - 18*x*y"2*z + z"3)
Note that /3 = y • Fa^x2, y, z), /4 = y • Fa,2(x2, y, z), where Fa,i, Fa,2 are the polynomials introduced in Theorem 1.
5.3. The Case W12
The following polynomials /i, /2, /3, /4 are Saito free polynomials each of which defines deformations of a curve on (y, z)-space with a singular point of Type Wi2 at the origin regarding x as a parameter.
Let F(x, y, z) be a weighted homogeneous Saito free polynomial with weight (2,4, 5) and degree 20. Assume that F(0, y, z) = y5 + z4. Then F(x,y, z) coincides with one of the 4 polynomials /i, /2, /3, /4 below up to weight preserving coordinate transformation.
f1=2560*x~4*y~3 - 95*x~2*y~4 + y~5 - 65536*x~5*z~2 + 2560*x~3*y*z~2 - 30*x*y~2*z~2 + z~4 f2=16*x"6*y"2 + 24*x"4*y"3 + 9*x"2*y"4 + y"5 - 8*x"3*y*z"2 - 6*x*y"2*z"2 + z"4 f3=25*x~8*y + 100*x~6*y~2 + 110*x~4*y~3 + 20*x~2*y~4 + y~5 - 2*x~5*z~2 - 20*x~3*y*z~2 -
10*x*y"2*z"2 + z"4 f4=x"2*y"4 + y"5 - 2*x*y"2*z"2 + z"4
5.4. The Cases E13, £14, Z12, Z13, W13
The following six polynomials are Saito free as polynomials of x,y, z. Each of them contains a parameter t.
g1 = z*(t*x"2*y"4 + 2*y"5 - 10*x*y"2*z + 5*z"2)
g2 = -1728*x"6*y"6 + t"2*x"6*y"6 - 2*t*x"3*y"7 + y"8 + 432*x"4*y"4*z +
12*t*x"4*y"4*z - 12*x*y"5*z + z"3 g31= z*(y"4 + 2*x*y"2*z + t*x"2*z"2 - y*z"2) g32= y*z*(t*x"2*y"2 + y"3 + 2*x*y*z - z"2)
g4 = y*(243*t"2*x"6*y~3 - 54*t*x"3*y"4 + 3*y"5 + 324*t*x"4*y"2*z - 36*x*y"3*z + 108*x~2*y*z~2 + z"3)
g5 = z*(-12*y"4 + 288*x"2*y"2*z + t"2*x"2*y"2*z - 1728*x"4*z"2 + 2*t*x*y*z"2 + z"3)
Remark 4. With the assistance of M. Noro, the author succeeded to construct 33, 33, 34, 35, 23 numbers of Saito free polynomials with real coefficients and without parameters related with the singularities Ei3, Ei4, Zi2, Zi3, Wi3, respectively. There are many Saito free polynomials with complex coefficients related with the singularities E13, E14, Z12, Z13, W13 as in the case of the polynomials /81/9 in subsection 6.1.
The author was partially supported by Grand-in-Aid for Scientific Research (No. 17540013), Ministry of Education, Culture, Sports, Science and Technology, Japanese Government.
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