Singular Points of Plane Curves C

Cambridge University Press 0521839041 - Singular Points of Plane Curves C. T. C. Wall Index More information

Index

ai (negative self-intersection), 192 rP (C) (number of branches at P ), Ak singularity, 26 157 th βi (Puiseux exponent), 40 Ti (i blown up surface), 43 C(i) (ith strict transform of C), 43 vi (valence of vertex Vi), 230 δ(B) (double point number), 79 V(Γ) (vertex set of graph), 57 Z δi (defect), 195 [ ] (fundamental cycle), 193 δi,j (Kronecker delta), 53 ∆K (t) (Alexander polynomial), 122 acnode, 169 Dk singularity, 87 Alexander polynomial, 122 ei = hcf(β0,... ,βi), 40 algebraic link, 224 E(Γ) (edge set of graph), 57 analytic, 2 E (semigroup of effective cycles), 196 arrowhead vertex, 188 th k (k curvette), 53 atoroidal (3-manifold), 222 th Ei (i exceptional curve), 43 augmented dual tree, 58 [Ei] (exceptional cycle), 188 bitangent, 162 [Ei] (class of total transform), 189 o blowing up, 40 E (punctured Ei), 230 i branch, 26 Ek singularity, 87 FC (decomposed Milnor fibre), 269 cable knot, 117 ΓE (C) (Eggers tree), 76 + carousel, 112 ΓE (C), 240 C centre (of blow up), 40 ΓR( ) (resolution tree), 58 class (of plane curve), 156 + C ΓR( ), 58 closed complement (of link), 135 ++ ΓR (C), 238 cluster, 328 ≤ (ordering of vanishing cycles), 196 coefficient, 2 th mi(B) (multiplicity of i blow up), complete (local ring), 31 50 cone point, 244 mi (multiplicity of curvette), 198 constructible function, 163 th Mi = Mi(C)(i order of f), 198 convergent (in mt-adic sense), 3 ME (h.c.f. (Mi,Mj )), 275 convergent power series, 2 hMi (h.c.f. Mr), 277 core (of dual graph), 238 m (maximal ideal in O0), 318 core (of dual tree of branch), 61 mP (C) (multiplicity at P ), 157 core (of E–N diagram), 250 µP (C) (Milnor number at P ), 157 crunode, 169 νi (coeff. in fundamental cycle), 193 curvette, 53 νi =1+νi, 199 cycle, 189 O0 (ring of germs at O), 317 th Oi (centre of i blow up), 43 dead branch, 240 n Pz(n)(=(t − 1)), 124 defect (of partial blow up), 195

368

Index 369

degree (of polynomial), 3 mapping torus, 132 determinacy, 341 maximal contact, 90 differentials, 350 meridian, 118 discriminant, 8 Milnor fibration, Milnor fibre, double point number (of branch), 138 79 Milnor number, 139 double point number (of curve), minimal (resolution), 49 151 monic (polynomial), 9 doubly augmented dual graph, 238 monodromy, 132 dual curve, 159 morsification, 146 dual graph (of resolution), 58 multiplicity, 6, 30 dual semigroup, 82 Newton polygon, 16 effective (cycle), 196 non-degenerate (singularity), 141 efficient resolution, 238 normal crossings, 47 Eggers tree, 76 normalisation, 49 equisingular (of branch), 86 normalisation (of ring), 337 equisingular (of curves), 87 NPND, 154 exceptional curve, 41 exceptional cycle, 189 open book decomposition, 139 exceptional fibre, 244 order (of power series), 2 exponent, 2 exponent of contact, 68 partition of unity, 106 plumbing, 230 fibration, fibre, 131 polar (polar curve), 13 formal power series, 2 polar discriminant, 259 fundamental cycle, 193 polar quotient, 241 polynomial, 2 geodesic (in graph), 58 primary ideal, 320 germ, 4 pro-branch, 27, 68 good (parametrisation), 4 projective space, 5 good resolution, 47 proximate, 50 graph, 57 proximity matrix, 52 hairy Eggers tree, 240 Puiseux characteristic, 40 Herbrand function, 74 Puiseux exponents, 70 Hessian, 14 Puiseux series, 26 holomorphic, 2 hyperflex, 175 ramification group, 94 reduced (curve), 27 incidence variety, 166 reduced (equation), 5 incompressible surface, 221 regular of order m,20 infinitely near point, 49 resolution, 43 integral closure (of ideal), 337 resultant, 8 integral closure (of ring), 336 root (of polynomial), 2 integral curve (of vector field), rupture point, 57 104 irreducible (3-manifold), 221 satellite point, 51 isomorphism of Eggers trees, 77 semi-algebraic set, 137 isotopy, 103 semi-analytic, 137 semigroup of branch, 79 Jacobian matrix, 10 simple point, 31 jet, 318 simple singularities, 87 smooth (curve), 4 latitude, 118 smooth (map), 10 linking number, 110 splice (of links), 248 local ring, 31 strict transform, 42 local ring of curve, 79 strict transform (cycle), 189

370 Index

tangent lines, 30 tree, 58 Tjurina number, 178 tubular neighbourhood, topological zeta function, 205 117 total transform, 42 total transform (cycle), 189 valence, 57 transverse polar, 14 vector ﬁeld, 104