Strange duality between the quadrangle complete intersection singularities
aa r X i v : . [ m a t h . AG ] F e b STRANGE DUALITY BETWEEN THE QUADRANGLE COMPLETEINTERSECTION SINGULARITIES
WOLFGANG EBELING AND ATSUSHI TAKAHASHI
Abstract.
There is a strange duality between the quadrangle isolated complete inter-section singularities discovered by the first author and C. T. C. Wall. We derive thisduality from the mirror symmetry, the Berglund-H¨ubsch transposition of invertible poly-nomials, and our previous results about the strange duality between hypersurface andcomplete intersection singularities using matrix factorizations of size two.
Introduction
V. I. Arnold [A2] observed a strange duality between the 14 exceptional unimodalsingularities. It is well known that this duality is a special case of the Berglund-H¨ubschduality of invertible polynomials, see e.g. [ET1]. C. T. C. Wall and the first author [EW]discovered an extension of this duality embracing on one hand series of bimodal hypersur-face singularities and on the other hand, isolated complete intersection singularities (ICIS)in C . The duals of the ICIS are not themselves singularities but are virtual ( k = − J ,k , k ≥
0) of bimodal singularities. In [EW], the k = − virtual singularities and Milnor lattices were associated to them,but they do not coincide with the Milnor lattices of the germs at the origin by setting k = − triangle hypersurface singular-ities, i.e., they are weighted homogeneous singularities obtained from triangles in thehyperbolic plane. More precisely, they are determined by triangles with angles πb , πb , πb ,where b , b , b are positive integers called the Dolgachev numbers of the singularity. The k = 0 elements of the bimodal series are quadrangle hypersurface singularities, i.e., theyare related in a similar way to quadrangles in the hyperbolic plane. They are determined Mathematics Subject Classification. by four positive integers b , b , b , b . For 6 quadruples ( b , b , b , b ), the correspond-ing quadrangle singularities are hypersurface singularities. The dual ICIS are trianglecomplete intersection singularities in C . There are 8 of them determined by 8 triples( b , b , b ). For another 5 quadruples ( b , b , b , b ), the quadrangle singularities are ICIS.These singularities will be considered in this paper. They are again the k = 0 elements ofcertain series of singularities. These series are the 8 series of K -unimodal ICIS in Wall’sclassification [W1]. Wall and the first author also observed a duality between the k = − virtual singularites aswell. The objective of this paper is to show that these singularities exist as well and toderive this duality from the Berglund-H¨ubsch duality, too. The quadrangle complete in-tersection singularities together with their Dolgachev numbers are listed in Table 1. Thevirtual singularities obtained by setting k = − × × TRANGE DUALITY BETWEEN COMPLETE INTERSECTION SINGULARITIES 3
Name Equations Restrictions Dol Weights J ′ , ( xy + w , ax + xw + yw + z ) a = 0 , − , , ,
4; 8 , K ′ , ( xy + w , ax + xw + y + z ) a = 0 , , , ,
3; 6 , L , ( xy + zw, ax + xw + yw + z ) a = 0 , − , , ,
3; 7 , M , (2 xy + w − z , ax z + x w + 2 yw ) a = 0 , ± , , ,
3; 6 , I , ( x ( y − z ) + w , aw + y ( z − x )) a = 0 , , , ,
2; 6 , Table 1.
The elements with k = 0 of the seriesName Equation J ′ , − ( xy + w , x + xw + yw + z ) K ′ , − ( xy + w , x + xw + y + z ) K ♭ , − ( xy + w , x + x w + 2 xw + y + z ) L , − ( xy + zw, x + xw + yw + z ) L ♯ , − ( xy + zw, − x + x w + xw + yw + z ) M , − (2 xy + w − z , z + x w + 2 yw ) M ♯ , − (2 xy + w − z , z + x ( w − z ) + 2 yw ) I , − ( x ( y − z ) + w , xw + y ( z − x )) Table 2.
Setting k = − k = 0 element of the dual seriesand a polynomial related to its Dolgachev numbers.1. Invertible polynomials
We recall some general definitions.A complete intersection singularity in C n given by polynomial equations f = · · · = f k = 0 is called weighted homogeneous if there are positive integers w , . . . , w n (called weights ) and d , . . . , d k (called degrees ) such that f j ( λ w x , . . . , λ w n x n ) = λ d j f j ( x , . . . , x n )for j = 1 , . . . , k and for λ ∈ C ∗ . We call ( w , . . . , w n ; d , . . . , d k ) a system of weights .A weighted homogeneous polynomial f ( x , . . . , x n ) is called invertible if it can bewritten f ( x , . . . , x n ) = n X i =1 a i n Y j =1 x E ij j , where a i ∈ C ∗ , E ij are non-negative integers, and the n × n -matrix E := ( E ij ) is invertibleover Q .An invertible polynomial is called non-degenerate if it has an isolated singularity atthe origin. WOLFGANG EBELING AND ATSUSHI TAKAHASHI
Let f be an invertible polynomial given as above. By rescaling of the variables, onecan assume that a i = 1 for i = 1 , . . . , n . Moreover, we can assume that det E >
Berglund-H¨ubsch transpose [BH] e f of f is defined by the transpose matrix E T of E , i.e. e f ( x , . . . , x n ) = n X i =1 a i n Y j =1 x E ij j . Let f ( x , . . . , x n ) be an invertible polynomial. The canonical system of weights W f of f is the system of weights ( w , . . . , w n ; d ) given by the unique solution of the equation E w ... w n = det( E ) , d := det( E ) . We define q := w d , . . . , q n := w n d . The maximal group of diagonal symmetries of f is the group G f = { ( λ , . . . , λ n ) ∈ ( C ∗ ) n : f ( λ x , . . . , λ n x n ) = f ( x , . . . , x n ) } . It always contains the exponential grading operator g := ( e πiq , . . . , e πiq n ) . Denote by G the subgroup of G f generated by g .By [BHe] (see also [EG2, Proposition 2]), Hom( G f , C ∗ ) is isomorphic to G e f . For asubgroup G ⊂ G f , Berglund and Henningson [BHe] defined its dual group e G by e G := Hom( G f /G, C ∗ ) . Note that | e G | = c f , see [ET2, Proposition 3.1].2. Wall’s reduction
Let ( X,
0) be an ICIS in C given by an equation F = 0 where F ( x, y, z, w ) = ( xy − a ( z, w ) , yb ( z, w ) + c ( x, z, w ))where a ( z, w ) and c ( x, z, w ) are polynomials of degree ≥ b ( z, w ) is a polynomial ofdegree ≥
1, and x, b ( z, w ) form a regular sequence in C [ x, z, w ]. Then we can consider thereduction L y F ( x, z, w ) = xc ( x, z, w ) + a ( z, w ) b ( z, w )of [W2] corresponding to the variable y . This means that we eliminate the variable y to get the equation of a hypersurface singularity in C . Geometrically, this elimination TRANGE DUALITY BETWEEN COMPLETE INTERSECTION SINGULARITIES 5 corresponds to the projection along the y -axis on the coordinate space of the remainingvariables x, z, w . It is proved in [W2, Theorem 7.9], for the case b ( z, w ) = z , that theMilnor number increases by one.In [ET4], we considered certain polynomials of the form f ( x, z, w ) = xc ( x, z, w ) + a ( z, w ) b ( z, w )with the conditions on a ( z, w ), b ( z, w ), and c ( x, z, w ) as above and associated a completeintersection singularity to a graded matrix factorization of size two of f . We showed that,in this way, we get an inverse to Wall’s reduction. More precisely, a matrix factorizationof f is given by two matrices q = a ( z, w ) − xc ( x, z, w ) b ( z, w ) ! and q = b ( z, w ) x − c ( x, z, w ) a ( z, w ) ! such that q q = q q = f f ! . We associate to this the complete intersection singularity ( X Q ,
0) given by F Q ( x, y, z, w ) = ( F Q, ( x, y, z, w ) , F Q, ( x, y, z, w )) := ( a ( z, w ) − xy, c ( x, z, w ) + yb ( z, w )) . If f = L y F , then we obtain back ( X Q ,
0) = ( X, Q , , S , , S ♯ , , and U , . They are given bynon-degenerate invertible polynomials with [ G f : G ] = 2. In [ET3, Proposition 1], weclassified such polynomials. The coordinates are chosen so that the action of e G = Z / Z on e f is given by ( x, y, z ) ( − x, − y, z ). In [ET3, Proposition 2], we classified certain 4 × a , a , a , a ∈ C which areused in [ET3] and will be used in Section 4.Name Type p , p ( q ) , p ( q ) f a , a , a , a Q , IV a x + a xy + a yz + a x y , , , − S , IV a x + a xy + a yz + a x y , , , − S ♯ , IV ♯ a x z + a xy + a yz + a x y − , , , − U , IV ♯ a xz + a xy + a yz + a x y − , , , − Table 3.
Functions f of 4 of the quadrangle hypersurface singularitiesWe now consider the matrix factorizations of the functions f of Table 3. They aregiven in Table 4, where we use suitable coordinates ( x, z, w ) instead of ( x, y, z ). In the WOLFGANG EBELING AND ATSUSHI TAKAHASHI
Name Coord. change f Name Q , ( x, y, z ) ( w, x, z ) x ( a wx + a z + a xw ) + ( a w ) · w J ′ , S , ( x, y, z ) ( x, z, w ) x ( a x + a z + a x z ) + ( a z ) · w K ′ , S , ( x, y, z ) ( x, z, w ) x ( a x + a z + a x z ) + ( a w ) · zw L , S ♯ , ( x, y, z ) ( x, z, w ) x ( a xw + a z + a x z ) + ( a z ) · w K ♭ , S ♯ , ( x, y, z ) ( x, z, w ) x ( a xw + a z + a x z ) + ( a w ) · zw L ♯ , U , ( x, y, z ) ( w, x, z ) x ( a xw + a z + a w ) + ( a z ) · z w M , U , ( x, y, z ) ( z, w, x ) x ( a x z + a x w ) + ( a w + a z ) · zw M ♯ , U , ( x, y, z ) ( x, z, w ) x ( a w + a z + a xz ) + ( a z ) · w I , Table 4.
Matrix factorizations of the functions f case Q , , the matrix factorization from [ET4, Table 2] q = y − (1 + λ ) xy + λ x − yz − x ! is equivalent to the matrix factorization from Table 4 with a = λ , a = − a = 1, a = 1 + λ , which is seen by adding the second column multiplied by y − (1 + λ ) xy tothe first column.3. An extension of the Berglund-H¨ubsch duality
We shall now show that the duality between the quadrangle complete intersectionsingularities can be derived from the Berglund-H¨ubsch transposition of invertible polyno-mials in 4 variables. We use the procedure in [ET3] to associate a weighted homogeneousnon-invertible polynomial with 4 terms in 4 variables to each of the quadrangle completeintersection singularities. We consider the complete intersection singularities associatedto the matrix factorizations in Table 4 defined by equations ( F Q, , F Q, , where we set a i = 1, i = 1 , . . . ,
4, and where we take a suitable order of the terms. Moreover, in theequation for I , we replace XZ + Y Z by X + Y . We also substitute temporarily thevariables x, y, z, w by capital letters X, Y, Z, W . We have the following 4 cases:(a) F Q, ( X, Y, Z, W ) = XY − W ,(b) F Q, ( X, Y, Z, W ) = XY − W ,(c) F Q, ( X, Y, Z, W ) = XY − ZW ,(d) F Q, ( X, Y, Z, W ) = XY − Z W . TRANGE DUALITY BETWEEN COMPLETE INTERSECTION SINGULARITIES 7
We make the following coordinate substitutions in F Q, ( X, Y, Z, W ):(a) XY − W : X := x w, Y := y w, Z := z, W := xyw, (b) XY − W : X := x w , Y := y w , Z := z, W := x y w . (c) XY − ZW : X := xw, Y := yz, Z := xz, W := yw, (d) XY − Z W : X := y z , Y := x w , Z := xz, W := y w , Then the polynomial F Q, ( X, Y, Z, W ) is transformed to an invertible polynomial f ( x, y, z, w ) = X i =1 x E i y E i z E i w E i for a 4 × E of exponents. The corresponding polynomials are listed in Table 5.This procedure can be explained as follows. We observe that the kernel of the matrix E is generated by one of the following vectors:(a),(b) (1 , , , − T ,(c),(d) (1 , , − , − T ,Let R := C [ x, y, z, w ]. There exists a Z -graded structure on R given by the respective C ∗ -action (here λ ∈ C ∗ ):(a) , (b) λ ∗ ( x, y, z, w ) = ( λx, λy, z, λ − w )(c) , (d) λ ∗ ( x, y, z, w ) = ( λx, λy, λ − z, λ − w )Let R = L i ∈ Z R i be the decomposition of R according to one of these Z -gradings. Thenew coordinates X, Y, Z, W are some invariant polynomials with respect to these actionsand they satisfy the relation given by the corresponding first equation.Name ( F Q, , F Q, ) f Dual J ′ , ( XY − W , X W + Y W + Z + XW ) x yw + xy w + z + x y w J ′ , K ′ , ( XY − W , X + Z + Y Z + X Z ) x w + z + y zw + x zw K ′ , K ♭ , ( XY − W , XW + Z + Y Z + X Z ) x y w + z + y zw + x zw L , L , ( XY − ZW, X + Z + Y W + X Z ) x w + x z + y zw + x zw K ♭ , L ♯ , ( XY − ZW, XW + Z + Y W + X Z ) xy w + x z + y zw + x zw L ♯ , M , ( XY − Z W, Z + W + Y Z + XW ) x z + y w + x zw + y z w M , M ♯ , ( XY − ZW, X W + Y Z + Y W + X Z ) x yw + xyz + y zw + x zw M ♯ , I , ( XY − W , X + Y + Z + W ) x w + y w + z + x y w I , Table 5.
Strange duality
WOLFGANG EBELING AND ATSUSHI TAKAHASHI
An inspection of Table 5 shows that the Berglund-H¨ubsch transpose of the poly-nomial f is either the polynomial f itself or another polynomial appearing in the table.This leads to the indicated duality.4. Virtual isolated complete intersection singularities
We now derive the equations for the virtual singularities.In [ET3, Section 4], we associated a polynomial h to f , which defines the corre-sponding virtual bimodal hypersurface singularity. This is done as follows. We considerthe polynomial f ( x, z, w ) from Table 3 with the choice of coefficients a , a , a , a given inthe last column. Then the corresponding equation defines a non-isolated singularity. Weconsider the cusp singularity f ( x, z, w ) − xzw and perform the coordinate change indicated in Table 6. Then this polynomial is trans-Name f Coord. change h Q , − w + x w + xz − x w w w + x w + x w + xz − x zS , − x + xz + zw − x z z z + x xz + zw + x w − wx S ♯ , − − x w + xz + zw − x z z z + x xz + x z + zw − x wU , − − z w + x w + xz − xw x x + w x w + xw + xz − zw Table 6.
Functions h of 4 of the quadrangle hypersurface singularitiesformed to h ( x, z, w ) − xzw, where the polynomial h ( x, z, w ) is indicated in Table 6. The polynomial h ( x, z, w ) has anisolated singularity at the origin, but also an additional critical point of type A outsidethe origin. Moreover, if we consider the 1-parameter family h ( x, z, w ) − t · xzw for t ∈ [0 , t = 0 ,
1, the polynomial h ( x, z, w ) − t · xzw has two additional critical points oftype A outside the origin. One of them merges with the singularity of h ( x, z, w ) at theorigin for t = 0 and the other one merges with the singularity of f ( x, z, w ) − xzw at theorigin for t = 1. Example 1.
Consider the case Q , . Then f ( x, z, w + x ) − xz ( w + x )= ( w + x ) + x ( w + x ) + xz − x ( w + x ) − xz ( w + x )= w + x w + xz − x z − xzw = h ( x, z, w ) − xzw. TRANGE DUALITY BETWEEN COMPLETE INTERSECTION SINGULARITIES 9
The polynomial h ( x.z, w ) has a singularity of Arnold type Q at the origin. On the otherhand, for t = 0, h (cid:18) x, z, w − t x (cid:19) − t · xz (cid:18) w − t x (cid:19) = (cid:18) w − t x (cid:19) + x (cid:18) w − t x (cid:19) + xz − x z − t · xz (cid:18) w − t x (cid:19) = w + (cid:18) t − t (cid:19) x w + (cid:18) t − t (cid:19) x + xz + (cid:18) − t (cid:19) x w − t · xzw. Using the proof of [ET1, Theorem 10], one can show that, for t = 1, this is a cuspsingularity of type T , , . For t = 1, it is a cusp singularity of type T , , . Using this, onecan check the above statements.Now we are looking at possible matrix factorizations of the polynomials h of Table 6.They are listed together with the corresponding isolated complete intersection singularitiesin Table 7. The corresponding isolated complete intersection singularity is denoted by H = ( h , h ). The resulting singularities defined by H = ( h , h ) are called the virtualsingularities and they are denoted by replacing the index 0 by −
1. There is anotherName Matrix factorization of h ( h , h ) Virtual Q , − x ( − x z + z + xw ) + w · w ( xy − w , − x z + yw + z + xw ) J ′ , − S , − x ( − x w + z + xw ) + z · w ( xy − w , − x w + z + yz + xw ) K ′ , − S ♯ , − x ( − x w + z + x z ) + z · w ( xy − w , − x w + z + yz + x z ) K ♭ , − S , − x ( − x w + z + xw ) + zw · w ( xy − zw, − x w + z + yw + xw ) L , − S ♯ , − x ( − x w + z + x z ) + zw · w ( xy − zw, − x w + z + yw + x z ) L ♯ , − U , − x ( xw + z + w ) − z · w ( xy − w , − yz + xw + z + w ) M , − U , − x ( − z + x w ) + zw · ( z + w ) ( xy − zw, − z + yw + x w + yz ) M ♯ , − U , − x ( − xw + z + xz ) + z · w ( xy − w , − xw + z + yz + xz ) I , − Table 7.
Virtual singularitiesmatrix factorization in the case U , − , namely x ( xw + z + w ) − zw · w. It is equivalent to the matrix factorization corresponding to I , − .Let H ( x, y, z, w ) = ( h ( x, y, z, w ) , h ( x, y, z, w )) with h ( x, y, z, w ) = X i =1 a i x A i y A i z A i w A i be the equations defining a virtual singularity and let Supp( h ) = { ( A i , A i , A i , A i ) ∈ Z | i = 1 , . . . , } . Let Γ ∞ ( h ) be the Newton polygon of h at infinity [Ko], i.e. Γ ∞ ( h )is the convex closure in R of Supp( h ) ∪ { } . The Newton polygon Γ ∞ ( h ) has twofaces which do not contain the origin. Call these faces Σ and Σ . Let I k := { i ∈{ , . . . , } | ( A i , A i , A i , A i ) ∈ Σ k } , k = 1 ,
2, and let h ,k = X i ∈ I k a i x A i y A i z A i w A i . Then ( h , h ,k ) defines a non-isolated weighted homogeneous complete intersection sin-gularity. The polynomials h and h and their systems of weights are listed in Table 8.Name h , Weights h , Weights J ′ , − − x z + z + xw , , ,
3; 6 , z + xw + yw , , ,
4; 8 , K ′ , − − x w + z + xw , , ,
2; 4 , z + xw + yz , , ,
3; 6 , K ♭ , − − x w + x z + yz , , ,
3; 6 , x z + yz + z , , ,
3; 6 , L , − − x w + z + xw , , ,
2; 5 , z + xw + yw , , ,
3; 7 , L ♯ , − − x w + x z + yw , , ,
3; 6 , x z + yw + z , , ,
3; 7 , M , − xw + z + w , , ,
3; 6 , z + w − yz , , ,
3; 6 , M ♯ , − − z + x w + yz , , ,
2; 5 , x w + yz + yw , , ,
3; 6 , I , − − xw + yz + xz , , ,
2; 6 , yz + xz + z , , ,
2; 6 , Table 8.
Decomposition of equations5.
Dolgachev numbers
We shall now define Dolgachev numbers for our virtual singularities.The Dolgachev numbers of the virtual singularity ( h , h ) are defined in a similarway as [ET3, Section 5]. Let i = 1 , V i := { ( x, y, z, w ) ∈ C | h ( x, y, z, w ) =0 , h ,i ( x, y, z, w ) = 0 } . We consider the C ∗ -action on V i given by the system of weightsof ( h , h ,i ) (see Table 8). We consider exceptional orbits (i.e. orbits with a non-trivialisotropy group) of this action. We distinguish between three cases:(A) V i contains a linear subspace L of C of codimension 2 obtained by setting twocoordinates to be zero. TRANGE DUALITY BETWEEN COMPLETE INTERSECTION SINGULARITIES 11 (B) V i = U ∪ U ′ , where( h ( x, y, z, w ) , h ,i ( x, y, z, w )) = ( g ( x, y, w ) , zg ( x, y, z )) ,U = { ( x, y, z, w ) ∈ C | g ( x, y, w ) = z = 0 } ,U ′ = { ( x, y, z, w ) ∈ C | g ( x, y, w ) = g ( x, y, z ) = 0 } . (C) V i is not of the form of (A) or (B).In case (A) we consider those exceptional orbits which are not contained in L . In case(B) we consider those exceptional orbits which are not contained in U . In case (C) weconsider those exceptional orbits which do not coincide with the singular locus of V i . Wecall these the principal orbits. It turns out that in all cases we have exactly two principalorbits. Definition.
The
Dolgachev numbers of the virtual singularity ( h , h ) are the numbers α , α ; α , α where α , α and α , α are the orders of the isotropy groups of the principalexceptional orbits of ( h , h , ) and ( h , h , ) respectively.The Dolgachev numbers of the virtual singularities are computed as follows. Thetwo pairs of polynomials ( h , h ,i ), i = 1 ,
2, of Table 8 define non-isolated weighted ho-mogeneous complete intersection singularities of certain types. The systems of weightscorrespond to the five quadrangle ICIS and three elliptic complete intersection singulari-ties as considered by Wall [W3]. We indicate the notation of Wall [W3] in Table 9. Thecorresponding orbifold curves have genus zero. We list the orders of the isotropy groupsof the exceptional orbits of these ICIS in this table (see also [E3]). Some of them corre-spond to the orders of the isotropy groups of the exceptional orbits for the non-isolatedsingularities given by the pairs ( h , h ,i ), i = 1 ,
2. Those ones which do not occur arestroken out. The orders of the isotropy groups of the principal orbits are indicated inbold face. We also indicate for each pair which of the corresponding cases (A), (B), or(C) applies. An exceptional orbit which coincides with the singular locus is marked by ∗ .The Dolgachev numbers α , α ; α , α of the virtual singularities are indicated in the lastcolumn. Example 2. (a) We consider the singularity K ♭ , − . We have ( h , h , ) = ( xy − w , − x w + x z + yz ) with the system of weights (2 , , ,
3; 6 , x = y = w = 0 singular line, order of isotropy group: 3 y = z = w = 0 order of isotropy group: 2 x = z = w = 0 order of isotropy group: 4 Name ( h , h , ) orbits ( h , h , ) orbits α , α ; α , α J ′ , − K ′ , (C) ,4 ∗ , J ′ , (C) , ∗ , K ′ , − δ ,2 ∗ , , K ′ , (C) , ∗ , K ♭ , − M , (C) , , ∗ , K ′ , (B) ,2,4, L , − α (2) (A) ,2,3 ∗ , L , (C) , ∗ , L ♯ , − M , (A) ,3 ∗ ,4 L , (A) ,2,3, M , − I , (C) ,3 ∗ , α (1) (C) , , ∗ , M ♯ , − α (2) (A) , , ∗ ,3, M , (A) 2,3, ∗ , I , − α (2) (C) , , ∗ , I , (B) 3,3, Table 9.
Dolgachev numbersThis gives ( α , α ) = (2 , K ♭ , − , but now ( h , h , ) = ( xy − w , x z + yz + z ) with the system of weights (2 , , ,
3; 6 , z = x + y = x + w = 0 with trivial isotropy group. It is contained in U = { xy − w = z = 0 } . The exceptional orbits contained in U are y = z = w = 0 order of isotropy group: 2 x = z = w = 0 order of isotropy group: 4The exceptional orbits not contained in U are y = w = x + z = 0 order of isotropy group: 2 x = w = y + z = 0 order of isotropy group: 4This gives ( α , α ) = (2 , L , − with ( h , h , ) = ( xy − zw, − x w + xw + z ). Thesystem of weights is (2 , , ,
2; 5 , V contains the hyperplane L = { x = z = 0 } , sowe are in case (A). The exceptional orbits contained in L are x = z = w = 0 singular line, order of isotropy group: 3 x = y = z = 0 order of isotropy group: 2The exceptional orbits not contained in L are y = z = w = 0 order of isotropy group: 2 x − w = y = z = 0 order of isotropy group: 2This gives ( α , α ) = (2 , Remark 3.
Using the primary decomposition algorithm of the computer algebra software
Singular [DGPS], one can show that, for each pair ( h , h ,i ) where we have case (A),the subspace L is an irreducible component of V i . If one removes this component L incase (A), the component U in case (B), and the point corresponding to the singular line in TRANGE DUALITY BETWEEN COMPLETE INTERSECTION SINGULARITIES 13 case (C), one gets P α i − ,α i with one point removed. Here P α i − ,α i denotes the complexprojective line with two orbifold points with singularities Z /α j Z , j = 2 i − , i .6. Gabrielov numbers
We now want to define Gabrielov numbers. They will be defined as in [ET3, Sec-tion 5]. For this purpose, we consider the pairs H = ( h , h ) of polynomials of Ta-ble 7. Here the first three cases J ′ , − , K ′ , − , and K ♭ , − are suspensions of the curvesingularities J , − , K , − , and K ♯ , − . In these cases, we consider as the first polynomial h ( x, y, z, w ) := xy − w − z . Then we consider the complete intersection singularity( X ′ ,
0) defined by ( h ( x, y, z, w ) , h ( x, y, z, w ) − zw. As in [ET1], one can show that the singularity ( X ′ ,
0) is K -equivalent to the singularitydefined by ( xy − z γ − w γ ,x γ + y γ − zw. This means that ( X ′ ,
0) is a cusp singularity of type T γ ,γ ,γ ,γ in the notation of [E1, 3.1]. Definition.
The
Gabrielov numbers of the virtual singularity given by the pair ( h , h )are the numbers γ , γ ; γ , γ . Proposition 4.
The Gabrielov numbers of the virtual quadrangle complete intersectionsingularities are given by Table 10.Proof.
We consider Wall’s reduction of the virtual quadrangle singularities according toTable 7. The cusp singularity H ′ := ( h ( x, y, z, w ) , h ( x, y, z, w ) − zw ) corresponds to thehypersurface cusp singularity L y H ′ = h ( x, z, w ) − xzw . In all cases except I , − , by trans-formations similar to the transformations in [ET3], we obtain the indicated Gabrielovnumbers. More precisely, for a suitable permutation σ : { , , , } → { , , , } , theGabrielov numbers satisfy ( γ σ (1) , γ σ (2) , γ σ (3) , γ σ (4) ) = (2 , e γ − , e γ − , e γ − e γ , e γ , e γ ) are the Gabrielov numbers of the corresponding virtual hypersurface singu-larity.In the case I , − , we indicate the claimed K -equivalence. We add the first polynomial h ( x, y, z, w ) to the second one h ( x ; y, z, w ) − zw and obtain( xy − w , xy − w − xw + z + yz + xz − zw ) . (6.1)By the transformation w w + y , this is transformed to( xy − w − p ( y, w ) , − y − w − q ( y, w ) − xw + z + xz − zw ) , (6.2) where p ( y, w ) and q ( y, w ) are certain polynomials of degree 3 in the variables y and w . Using [A1, Lemma 7.3] and the fact that y divides p ( y, w ), one can get rid of thepolynomial p ( y, w ) with the help of the term xy . Similarly, one can get rid of thepolynomial q ( y, w ) in the second equation with the help of the term zw . Now we applythe transformation w w + x . Then the pair (6.2) gets( xy − w − x − p ( x, w ) , − x − y − q ( x, w ) − xw − x + z − zw ) , (6.3)where p ( x, w ) and q ( x, w ) are again polynomials which can be removed. Applying thetransformation x x + z , one gets( xy − w − z − p ( x, z ) , − x − y − q ( x, z ) − xw − x − xz − zw ) , (6.4)again with certain removable polynomials p ( x, z ) and q ( x, z ). Finally, by the transfor-mation z ( z − x ) followed by w w − x and rescaling, we obtain( xy − z − w + p ( x, w ) , x + y − zw ) , (6.5)again with a removable polynomial p ( x, w ). (cid:3) By [E1], one can compute Coxeter-Dynkin diagrams of the (global) singularitiesdefined by H = ( h , h ). Let X (1) := { ( x, y, z, w ) ∈ C | h ( x, y, z, w ) = 0 } and considerthe function h : X (1) → C . It has besides the origin one or two additional critical pointswhich are of type A . The singularity at the origin is indicated in Table 10. We definethe Milnor number of the virtual singularity by the sum of the Milnor numbers of thesingular points. It is equal to 12 in all cases.One can compute that there exists a (strongly) distinguished basis of thimbles( e , . . . , e µ +1 ) = ( e rj | ≤ j ≤ , ≤ r ≤ M j ), where the intersection matrix of ( e , . . . , e ) =( e , . . . , e ) coincides with the intersection matrix of the system ( b δ ′ , . . . , b δ ′ ) of [E1, Sect. 2.3],the numbers M , M , M are equal to one and the other numbers M , M , M , M , M areindicated in Table 10, and the intersection matrix of ( e rj | ≤ j ≤ , ≤ r ≤ M j ) is com-puted according to [E1, Theorem 2.2.3]. By the proof of [E1, Proposition 3.6.1], one cantransform these bases to (strongly) distinguished bases of thimbles with Coxeter-Dynkindiagrams of the form Π γ ,γ ,γ ,γ of Fig. 1, where γ , γ ; γ , γ are the Gabrielov numbersof the virtual singularity. 7. Strange duality
We now consider the duality defined in Section 1. We summarise the results onthe Dolgachev and Gabrielov numbers of the virtual singularities in Table 11. From thistable, we get the following result:
TRANGE DUALITY BETWEEN COMPLETE INTERSECTION SINGULARITIES 15 • ⑧ ⑧ ⑧ ⑧ δ ρ − • ✤✤✤✤✤✤✤✤ ❂❂❂❂❂❂❂❂❂✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ δ ρ − • ✤✤✤✤✤✤✤✤ ❂❂❂❂❂❂❂❂❂ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ δ ρ • δ · · · • ✁✁✁✁✁✁✁✁✁ δ γ − • ✁✁✁✁✁✁✁✁✁ ✁✁✁✁✁✁✁✁✁ δ ρ − • ❂❂❂❂❂❂❂❂❂ δ ρ − • δ γ − · · · • δ • ⑤⑤⑤⑤⑤⑤⑤⑤ δ γ − • ❇❇❇❇❇❇❇❇ δ γ − · · · ⑤⑤⑤⑤⑤⑤⑤⑤ · · · ❇❇❇❇❇❇❇❇ • δ • δ Figure 1.
The graph Π γ ,γ ,γ ,γ Virtual Germ at 0 M M M M M γ , γ ; γ , γ J ′ , − J ′ ,
2; 2 , K ′ , − K ′ ,
2; 3 + 1 , K ♭ , − K ′ ,
2; 3 , L , − L , , L ♯ , − L ,
3; 2 , M , − J ′ , , M ♯ , − M ,
3; 3 , I , − M ,
3; 3 , Table 10.
Coxeter-Dynkin diagrams of virtual singularities
Theorem 5.
The Gabrielov numbers of a virtual quadrangle complete intersection singu-larity coincide with the Dolgachev numbers of the dual one, and vice versa.
For another feature of this duality, we have to introduce some notions.Let f , . . . , f k be quasihomogeneous functions on C n of degrees d , . . . , d k withrespect to weights w , . . . , w n . We suppose that the equations f = f = . . . = f k = 0define a complete intersection X in C n . There is a natural C ∗ -action on the space C n defined by λ ∗ ( x , . . . , x n ) = ( λ w x , . . . , λ w n x n ), λ ∈ C ∗ . Let A = C [ x , . . . , x n ] / ( f , . . . , f k ) be the coordinate ring of X . Then the C ∗ -actionon C n induces a natural grading A = ⊕ ∞ s =0 A s on the ring A , where A s := { g ∈ A | g ( λ ∗ ( x , . . . , x n )) = λ s g ( x , . . . , x n ) for λ ∈ C ∗ } . We shall consider the Poincar´e series P X ( t ) = P ∞ s =0 dim A s · t s of this graded algebra.One has P X ( t ) = Q kj =1 (1 − t d j ) Q ni =1 (1 − t w i ) . For a map ϕ : Z → Z of a topological space Z , the zeta function is defined to be ζ ϕ ( t ) = Y p ≥ (cid:8) det (cid:0) id − t · ϕ ∗ | H p ( Z ; C ) (cid:1)(cid:9) ( − p . If, in the definition, we use the actions of the operators ϕ ∗ on the reduced homologygroups H p ( Z ; Z ), we get the reduced zeta function ζ ϕ ( t ) = ζ ϕ ( t )(1 − t ) . For 0 ≤ j ≤ k , let X ( j ) be the complete intersection given by the equations f = . . . = f j = 0 ( X (0) = C n , X ( k ) = X ). The restriction of the function f j ( j = 1 , . . . , k )to the variety X ( j − defines a locally trivial fibration X ( j − \ X ( j ) → C ∗ . Let V ( j ) = f − j (1) ∩ X ( j − be the typical fibre (Milnor fibre) of this fibration. Note that it is notnecessarily smooth. There is a monodromy transformation ϕ ( j ) : V ( j ) → V ( j ) on it. Let ζ X,j ( t ) := ζ ϕ ( j ) ( t ) . One can show that ( ϕ ( j ) ∗ ) d j = id and therefore ζ X,j ( t ) can be written in the form Y ℓ | d j (1 − t ℓ ) α ℓ , α ℓ ∈ Z . Following K. Saito [S1, S2], we define the Saito dual to ζ X,j ( t ) to be the rational function ζ ∗ X,j ( t ) = Y m | d j (1 − t m ) − α ( dj/m ) (note that different degrees d j are used for different j ).Let Y ( k ) = ( X ( k ) \ { } ) / C ∗ be the space of orbits of the C ∗ -action on X ( k ) \ { } and Y ( k ) m be the set of orbits for which the isotropy group is the cyclic group of order m . Fora topological space Z , denote by χ ( Z ) its Euler characteristic. DefineOr X ( t ) := Y m ≥ (1 − t m ) χ ( Y ( k ) m ) . Now let X be an ICIS in C defined by two polynomial equations f = f = 0 andassume that both X (1) = f − (0) and X (2) = f − (0) have isolated singularities at the TRANGE DUALITY BETWEEN COMPLETE INTERSECTION SINGULARITIES 17 origin. Moreover, assume that X (1) has a singularity of type A . Consider the mapping F := ( f , f ) : C → C . Let C F be the critical locus of F and D F = F ( C F ). Themapping F | C − F − ( D F ) : C − F − ( D F ) → C − D F defines a locally trivial fibration.Assume that (1 , D F . Let V (1) = f − (1) and V = f − (1) ∩ V (1) . (Note that V = V (2) but V and V (2) are homeomorphic to each other.) Then V ⊂ V (1) and themonodromy transformation ϕ (1) : V (1) → V (1) induces a relative monodromy operator b ϕ ∗ : H ( V (1) , V ; Z ) → H ( V (1) , V ; Z ). Let∆ X ( t ) := det (cid:0) id − t · b ϕ ∗ | H ( V (1) ,V ; C ) (cid:1) be the characteristic polynomial of this operator. Proposition 6.
We have ∆ X ( t ) = (1 − t ) Y j =1 ζ X,j ( t ) . Proof.
We have the following commutative diagram of split short exact sequences:0 / / H ( V (1) ; Z ) / / ϕ (1) ∗ (cid:15) (cid:15) H ( V (1) , V ; Z ) / / b ϕ ∗ (cid:15) (cid:15) H ( V ; Z ) / / ϕ (2) ∗ (cid:15) (cid:15) / / H ( V (1) ; Z ) / / H ( V (1) , V ; Z ) / / H ( V ; Z ) / / X ( t ) = det (cid:0) id − t · ϕ (1) ∗ | H ( V (1) ; C ) (cid:1) det (cid:0) id − t · ϕ (2) ∗ | H ( V ; C ) (cid:1) = ζ X, ( t ) − ζ X, ( t ) = (1 − t ) Y j =1 ζ X,j ( t )since ζ X, ( t ) = (1 − t ) − . (cid:3) Let X be an ICIS with a Coxeter-Dynkin diagram of type Π γ ,γ ,γ ,γ . Then thepolynomial ∆ X ( t ) is equal to the characteristic polynomial ∆(Π γ ,γ ,γ ,γ )( t ) of the Coxeterelement corresponding to this Coxeter-Dynkin diagram. By [E1, Proposition 3.6.2], wehave ∆(Π γ ,γ ,γ ,γ )( t ) = (1 − t ) ∆( S γ ,γ ,γ ,γ )( t )where ∆( S γ ,γ ,γ ,γ )( t ) is the characteristic polynomial of the Coxeter element correspond-ing to the graph S γ ,γ ,γ ,γ depicted in Fig. 2. (Note that there is a slight mistake in theproof of [E1, Proposition 3.6.2] which was corrected in [E2].) By Proposition 6 we get Y j =1 ζ X,j ( t ) = ∆( S γ ,γ ,γ ,γ )( t ) . (7.1) • δ µ • ✤✤✤✤✤✤✤✤ ❂❂❂❂❂❂❂❂❂✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ δ µ − • δ · · · • ✁✁✁✁✁✁✁✁✁ δ γ − • ✁✁✁✁✁✁✁✁✁ ❂❂❂❂❂❂❂❂❂ δ µ − • δ γ − · · · • δ • ⑤⑤⑤⑤⑤⑤⑤⑤ δ γ − • ❇❇❇❇❇❇❇❇ δ γ − · · · ⑤⑤⑤⑤⑤⑤⑤⑤ · · · ❇❇❇❇❇❇❇❇ • δ • δ Figure 2.
The graph S γ ,γ ,γ ,γ This also gives an interpretation of the characteristic polynomial of the Coxeter element c ♭ considered in [E2].A k = 0 element of one of the series can again be given as the zero set of twoquasihomogeneous functions of weights w , w , w , w and degrees d , d . We are nowready to state the following analogue of [ET3, Theorem 6]: Theorem 7.
Let X be a virtual ICIS and e X be the k = 0 element of the dual series.Then we have Y j =1 ζ X,j ( t ) = P e X ( t ) · Or e X ( t ) = Y j =1 ζ ∗ e X ,j ( t ) . (7.2) Proof.
By Equation (7.1), the left-hand side of Equation (7.2) is equal to ∆( S γ ,γ ,γ ,γ )( t ).By [E1, p. 98], there is the following formula for this polynomial :∆( S γ ,γ ,γ ,γ )( t ) = ( t − t − t + 1) Y i =1 t γ i − t − t X i =1 t γ i − − t − Y j =1 ,j = i t γ j − t − . (7.3)Using this formula, we can compute the polynomial Q j =1 ζ X,j ( t ) in each case. The resultis given in Table 11. Under a certain non-degeneracy condition, the function ζ X, ( t ) canalso be computed by the formula of [Gu, Theorem 4] from the Newton polytope.On the other hand, we can compute the Poincar´e series from the weights and degreesof the dual ICIS given in Table 5. The polynomial Or e X ( t ) is given byOr e X ( t ) = (1 − t ) − Y i =1 (1 − t γ i ) TRANGE DUALITY BETWEEN COMPLETE INTERSECTION SINGULARITIES 19 where γ , γ ; γ , γ are the Gabrielov numbers of X which are the Dolgachev numbers of e X . Comparing these polynomials, we obtain the first equality of Equation (7.2). X Dol( X ) Gab( X ) Q j =1 ζ X,j ( t ) Weight system e X Dual J ′ , − ,
2; 2 , ,
2; 2 , · · / · · , , ,
4; 8 , J ′ , − K ′ , − ,
2; 4 , ,
2; 4 , · · / · , , ,
3; 6 , K ′ , − K ♭ , − ,
4; 2 , ,
2; 3 , · · / · , , ,
3; 7 , L , − L , − ,
2; 3 , ,
4; 2 , · · / · , , ,
3; 6 , K ♭ , − L ♯ , − ,
3; 2 , ,
3; 2 , · · / · , , ,
3; 7 , L ♯ , − M , − ,
3; 2 , ,
3; 2 , · / , , ,
3; 6 , M , − M ♯ , − ,
3; 3 , ,
3; 3 , · / , , ,
3; 6 , M ♯ , − I , − ,
3; 3 , ,
3; 3 , · / · , , ,
2; 6 , I , − Table 11.
Dolgachev numbers, Gabrielov numbers and monodromy zeta functionsThe second equality follows from [EG1]. (cid:3)
In each case, the polynomial Q j =1 ζ e X ,j ( t ) has already been indicated in [E2, Table 7]under the heading π ∗ . Remark 8.
The spectrum of an ICIS was defined in [ESt]. In a similar way one candefine the spectrum of a virtual ICIS X . Spectra for the series of ICIS above have beencalculated by Steenbrink [St2]. The spectrum of a virtual ICIS agrees with the spectrumdefined by setting k = − Q j =1 ζ X,j ( t ). Acknowledgements . This work has been partially supported by DFG. The secondnamed author is also supported by JSPS KAKENHI Grant Number JP16H06337.
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