APPENDIX A Integral Bilinear Forms and Dynkin Diagrams (AI) Definition. A lattice (M, ( , )) is a pair consisting of a finitely generated free abelian group M together with a bilinear form ( , ): M x M -+ 7l. which is either: (i) symmetric, i.e., (x, y) = (y, x) for all x, y EM; (ii) skew-symmetric, i.e. (x, y) = -(y, x) for all x, y E M. For simplicity, we refer to M as a lattice when the bilinear form ( , ) is clear. The main example which we have in mind is the Milnor lattice Lx = (iin(F), < , » of an n-dimensional IHS X: f = 0 as defined in (3.3.6). (A2) Definition. A symmetric lattice M is even if (x, x) == 0 (mod 2) for any xEM. A symmetric lattic M is odd if it is not even. Note that the Milnor lattice Lx of an even dimensional IHS X is an even lattice by (3.3.7). (A3) Definition. The lattice M is nondegenerate if it satisfies the following two equivalent conditions: (i) Rad(M):= {x E M; (x, y) = 0 for all y EM} = 0; (ii) the natural group homomorphism iM : M -+ M' := Hom(M, 7l.), x 1--+ (x, .) is injective. The lattice M is unimodular if iM is an isomorphism. When the lattice M is non degenerate, we call the finite group D(M) = coker(iM ) 220 Appendix A. Integral Bilinear Forms and Dynkin Diagrams the discriminant group of M and denote its order ID(M)I by det(M}. The quo­ tient lattice M = M/Rad M is called the reduced lattice associated to M. (A4) Exercise. (i) Show that D(M) = D( -M), D(Ml Ef) M 2 ) = D(Ml) ® D(M2 ). Here - M is the lattice (M, - ( , » obtained by changing to signs of all the products (x, y) and Ml Ef) M2 is the direct sum lattice, i.e., (Xl + X2' Yl + Y2) = (Xl' yd + (x2, Y2) for all Xl' Yl E Ml and X2' Y2 E M 2· (ii) Show that det(Lx} = IA(I)I, for a nondegenerate Milnor lattice Lx, where A denotes the characteristic polynomial of the monodromy operator of X. Hint. Recall the proof of (3.4.7). (A5) Lemma. If N c M is a sub lattice in the nondegenerate lattice M such that rk M = rk M, then the quotient group M/N is finite and its order satisfies the relation IM/NI2 det M = det N. Proof Use the structure of the subgroup N as described in [La], p. 393. 0 (A6) Definition. Let M and N be two lattices. A group homomorphism qJ: M ~ N is called a lattice morphism if (x, y) = (qJ(x), cp(y)) for all x, Y E M. A lattice morphism qJ is called an embedding (resp. an isomorphism) if qJ is a group monomorphism (resp. isomorphism). (A 7) Theorem (Structure of Skew-Symmetric Lattices). Any skew-symmetric lattice M is isomorphic to a direct sum of "elementary" skew-symmetric lattices (Z2, ( )d,) 61'" EB (Z2, ( }dk ) Ef) (zm, ( )0) where Z2 = Ze l + Ze2 and (e l , e2 )d = d; for some integers d; > 0 and (x, y)o = 0 for all x, y E zm. Moreove'r the positive integers d; are uniquely determined if we ask in addition that dl ld 2 •• ·Idk • For a proof, see [La], p. 3S0. In particular, M is nondegenerate if and only if m = 0 and then det M = d i ... dt. And M is unimodular if and only if m = 0 and d 1 = ... = dt = 1. We use the notation 12k for this unimodular skew-symmetric lattice of rank 2k. (AS) Exercise. Let X: f = 0 be a reduced plane curve singularity with Milnor number Il and number of irreducible branches r. Then the Milnor lattice Lx is isomorphic to the direct sum Appendix A. Integral Bilinear Forms and Dynkin Diagrams 221 Hint. Use the Wang exact sequence (3.1.18) associated to the Milnor fibration for X. When M is a symmetric lattice we can obtain a real bilinear form tensoring by IKt All the usual terminology for the latter applies to M and hence we can speak about the signature sign M = (m_, mo, m+). For instance, M is nega­ tive definite when mo = m+ = 0, and M is indefinite when m_ > 0 and m+ > O. The difference m+ - m_ is called the index of the lattice. The classification of the indefinite unimodular lattices is due to Milnor [M2] and we do not recall it here since it is more complicated than (A 7). Moreover, the Milnor lattices Lx for the most familiar singularities X are neither indefinite nor unimodular, so the best way to introduce them to the reader is just by listing them. As remarked in (3.3.23), to each IHS X there are two naturally associated Milnor lattices, a symmetric one Lx and a skew­ symmetric one V;. In what follows, we list the Dynkin diagrams for several important classes of singularities, namely for all simple and unimodular sin­ gularities in Arnold's lists [AGVl]. These Dynkin diagrams determine at once the symmetric lattices L"x and they determine also the skew-symmetric lattices Lx via Gabrielov's result (3.3.22'). Recall that each vertex in a Dynkin diagram D corresponds to a vanishing cycle 11, with <l1i, l1i> = -2. Two distinct vertices, corresponding to vanishing cycles l1i and I1j , respectively, are joined by kedges (resp. k dotted edges) if their intersection number (l1 i, I1j ) is k (resp. -k). Hence j • • j • • Note also that the vertices in a Dynkin diagram are numbered (corresponding to the order of the vanishing cycles 111 ... , 11/l in the associated distinguished basis 11), unless any order is good (i.e., any order of l1/s corresponds to a distinguished basis). All the singularities discussed in what follows are stably equivalent to sur­ face singularities in (:3. The reader has already encountered the notations for (and the equations of) these singularities in Chapter 2, §4, so here we describe only their Dynkin diagrams and some of their properties. 222 Appendix A. Integral Bilinear Forms and Dynkin Diagrams (A9) Dynkin Diagrams for the Simple Singularities. Type Dykin diagram Ak,k~l --... - (kpoints) Dk,k~4 -- ... ---< (kpoints) • • I • • • • • I • • • • • • I • • We let All: denote the (symmetric) Milnor lattice of the singularity A" and use the same convention for all the other singularities discussed. A first striking fact about the Dynkin diagrams ofthe simple singularities is that they coincide with their resolution graphs, see (2.4.3). This remark gives us the first part ofthe following result (for a complete proof we refer to [Df4J). (AIO) Proposition. (i) The Milnor lattices A", Dk , E6 , E7 and Es are negative definite. (ii) Any IHS X whose symmetric Milnor lattice is negative definite is a simple singularity. (iii) det All: = k + 1, det D" = 4, det E, = 9 - I fori = 6,7,8. (All) Dynkin Diagrams for the Simple-Elliptic Singularities £6, £7' and Ea. First we associate to each of these simple-elliptic singularities £ a triple of positive integers (p, q, r) as follows £6 1--+ (3, 3, 3), £7 1--+ (2,4,4), £8 1--+ (2, 3, 6). A look at the equations for these singularities given in (2.4.9) will explain to the reader where these triples come from! See also (A13). Next to each triple Appendix A. Integral Bilinear Forms and Dynkin Diagrams 223 (p, q, r) we associate the following diagram: p-l q - 1 T(p,q,r) - ... >--..... - with a numbering such that the vertices denoted by i and i + 1 get consecutive indices. (AI2) Proposition. (i) The correspondence E 1--+ (p, q, r) 1--+ T(p, q, r) associates to each simple­ elliptic singularity E,,(k = 6, 7, 8) a Dynkin diagram. (ii) The lattice E" is negative semidefinite, its radical has rank 2 and the cor­ responding reduced Milnor lattice E" is exactly the Milnor lattice E". (iii) Any IHS X whose symmetric Milnor lattice is negative semidefinite is a simple-elliptic singularity. For a proof, see, for instance, [Df4]. For a proof of the following more complicated results we refer to Ebeling [EI] and [E2]. (A 13) Proposition. (i) A Dynkin diagram for the cusp singularity T",q,. is given by the diagram T(p, q, r) described above. (ii) sign(T",q,.) = (p + q + r - 3, 1, 1). (iii) det(J;"q,.) = pqr(l - lip - llq - llr). This is not a surprise, since we can regard the simple-elliptic singularities E" as special cases of the T",q,.-singularities for 1 1 1 -+-+-=1. p q r (AI4) Proposition. (i) A Dynkin diagram for the triangle singularity Dp,q,. is given by the follow­ ing diagram: 224 Appendix A. Integral Bilinear Forms and Dynkin Diagrams p' - 1 q' - 1 r T(p,q,r) _ ... ~-..... - with a numbering such that the vertices denoted by i and i + I get consecu­ tive indices and (p', q', r') are the Gabrielov numbers given in (2.4.7). (ii) sign(Dp,q,r) = (p' + q' + r' - 2,0, 2). (iii) det(Dp,q,r) = p' q'r'(l - lip' - l/q' - l/r'). Recall that Dp.q,r denotes in fact two singularities: one weighted homoge­ neous and the other one semiweighted homogeneous. By our discussion in (3.1.19) it follows that the two singularities have the same topological in­ variants, in particular, the same Dynkin diagrams. (A15) Exercise. Check that the Dolgacev numbers (p, q, r) and the Gabrielov numbers (p', q', r') associated to a given triangle singularity Dp,q,r in (2.4.7) satisfy the following relation pqr(l- ~ - ~ -~) = p'q'r'(l- ~ - ~ - ~).