Milnor Fiber Boundary of a Non-isolated Surface Singularity (Lecture Notes in Mathematics Book 2037)

In the study of algebraic/analytic varieties a key aspect is the description of the invariants of their singularities. This book targets the challenging non-isolated case. Let f be a complex analytic hypersurface germ in three variables whose zero set has a 1-dimensional singular locus. We develop an explicit procedure and algorithm that describe the boundary M of the Milnor fiber of f as an oriented plumbed 3-manifold. This method also provides the characteristic polynomial of the algebraic monodromy. We then determine the multiplicity system of the open book decomposition of M cut out by the argument of g for any complex analytic germ g such that the pair (f, g) is an ICIS. Moreover, the horizontal and vertical monodromies of the transversal type singularities associated with the singular locus of f and of the ICIS (f, g) are also described. The theory is supported by a substantial amount of examples, including homogeneous and composed singularities and suspensions. The properties peculiar to M are also emphasized 001Download PDF (161.3 KB)front-matter 1 Milnor Fiber Boundary of a Non-isolated Surface Singularity 3 Acknowledgements 7 Contents 9 002Download PDF (113.4 KB)fulltext 13 Chapter 1 Introduction 13 1.1 Motivations, Goals and Results 13 1.2 List of Examples with Special Properties 18 003Download PDF (107.0 KB)fulltext 23 Chapter 2 The Topology of a Hypersurface Germ f in Three Variables 23 2.1 The Link and the Milnor Fiber F of Hypersurface Singularities 23 2.2 Germs with 1-Dimensional Singular Locus: Transversal Type 25 2.3 The Decomposition of the Boundary of the Milnor Fiber 26 004Download PDF (130.8 KB)fulltext 29 Chapter 3 The Topology of a Pair (f,g) 29 3.1 Basics of ICIS: Good Representatives 29 3.2 The Milnor Open Book Decompositions of F 32 3.3 The Decomposition of F Revisited 32 3.4 Relation with the Normalization of the Zero Locus of f 34 005Download PDF (224.4 KB)fulltext 37 Chapter 4 Plumbing Graphs and Oriented Plumbed 3-Manifolds 37 4.1 Oriented Plumbed Manifolds 37 4.2 The Plumbing Calculus 42 4.3 Examples: Resolution Graphs of Surface Singularities 47 4.4 Examples: Multiplicity Systems and Milnor Fibrations 52 006Download PDF (175.3 KB)fulltext 57 Chapter 5 Cyclic Coverings of Graphs 57 5.1 The General Theory of Cyclic Coverings 57 5.2 The Universal Cyclic Covering of (X,f) 59 5.3 The Resolution Graph of f(x,y)+zN=0 63 007Download PDF (125.5 KB)fulltext 67 Chapter 6 The Graph C of a Pair (f,g): The Definition 67 6.1 The Construction of the Curve C and Its Dual Graph 67 6.2 Summary of Notation for C and Local Equations 69 6.3 Assumption A 72 008Download PDF (212.0 KB)fulltext 75 Chapter 7 The Graph C: Properties 75 7.1 Why One Should Work with C? 75 7.2 A Partition of C and Cutting Edges 77 7.3 The Graph C1 78 7.4 The Graph C2 80 7.5 Cutting Edges Revisited 85 009Download PDF (114.3 KB)fulltext 91 Chapter 8 Examples: Homogeneous Singularities 91 8.1 The General Case 91 8.2 Line Arrangements 93 010Download PDF (174.6 KB)fulltext 95 Chapter 9 Examples: Families Associated with Plane Curve Singularities 95 9.1 Cylinders of Plane Curve Singularities 95 9.2 Germs of Type f=zf'(x,y) 96 9.3 Double Suspensions 97 9.4 The Ta,*,*-Family 107 011Download PDF (190.5 KB)fulltext 113 Chapter 10 The Main Algorithm 113 10.1 Preparations for the Main Algorithm 113 10.2 The Main Algorithm: The Plumbing Graph of F 114 10.3 Plumbing Graphs of 1F and 2F 119 10.4 First Examples of Graphs of F, 1F and 2F 122 012Download PDF (219.1 KB)fulltext 129 Chapter 11 Proof of the Main Algorithm 129 11.1 Preliminary Remarks 129 11.2 The Guiding Principle and the Outline of the Proof 130 11.3 The First Step: The Real Varieties Sk 131 11.4 The Strict Transform k of k Via r 133 11.5 Local Complex Algebraic Models for the Points of k 134 11.6 The Normalization Snorm k of k 135 11.7 The “Resolution” k of k 139 11.8 The Plumbing Graph: The End of the Proof 140 11.9 The ``Extended'' Monodromy Action 141 013Download PDF (138.0 KB)fulltext 143 Chapter 12 The Collapsing Main Algorithm 143 12.1 Elimination of Assumption B 143 12.2 The Collapsing Main Algorithm 148 12.3 The Output of the Collapsing Main Algorithm 149 014Download PDF (197.0 KB)fulltext 151 Chapter 13 Vertical/Horizontal Monodromies 151 13.1 The Monodromy Operators 151 13.2 General Facts 152 13.3 Characters: Algebraic Preliminaries 153 13.4 The Divisors Div, Divj and Div'j in Terms of C 157 13.5 Examples 160 13.6 Vertical Monodromies and the Graph G 161 015Download PDF (109.8 KB)fulltext 165 Chapter 14 The Algebraic Monodromy of H1(F): Starting Point 165 14.1 The Pair (F,F \ Vg) 165 14.2 The Fibrations arg(g) 166 016Download PDF (107.5 KB)fulltext 169 Chapter 15 The Ranks of H1(F) and H1(F\Vg) via Plumbing 169 15.1 Plumbing Homology and Jordan Blocks 169 15.2 Bounds for corank A and corank (A,I) 171 017Download PDF (131.7 KB)fulltext 173 Chapter 16 The Characteristic Polynomial of F Via P# and P#j 173 16.1 The Characteristic Polynomial of G and G 173 16.2 The Characteristic Polynomial of F 174 018Download PDF (167.4 KB)fulltext 179 Chapter 17 The Proof of the Characteristic Polynomial Formulae 179 17.1 Counting Jordan Blocks of Size 2 179 17.2 Characters 183 019Download PDF (102.5 KB)fulltext 185 Chapter 18 The Mixed Hodge Structure of H1(F) 185 18.1 Generalities: Conjectures 185 020Download PDF (261.9 KB)fulltext 191 Chapter 19 Homogeneous Singularities 191 19.1 The First Specific Feature: Mver = (Mhor)-d 191 19.2 The Second Specific Feature: The Graphs G2,j 192 19.3 The Third Specific Feature: The d-Covering 194 19.4 The Characteristic Polynomial of F 196 19.5 M'j,hor, M'j,ver, Mj,hor, Mj,ver, M,hor and M,ver 198 19.6 When is F a Rational/Integral Homology Sphere? 199 19.7 Cases with d Small 200 19.8 Rational Unicuspidal Curves with One Puiseux Pair 203 19.9 The Weight Filtration of the Mixed Hodge Structure 206 19.10 Line Arrangements 209 021Download PDF (114.1 KB)fulltext 213 Chapter 20 Cylinders of Plane Curve Singularities: f = f'(x,y) 213 20.1 Using the Main Algorithm: The Graph G 213 20.2 Comparing with a Different Geometric Construction 215 20.3 The Mixed Hodge Structure on H1(F) 216 022Download PDF (119.3 KB)fulltext 217 Chapter 21 Germs f of Type zf'(x,y) 217 21.1 A Geometric Representation of F and F 217 023Download PDF (85.6 KB)fulltext 221 Chapter 22 The T*,*,*-Family 221 22.1 The Series Ta,∞,∞ 221 22.2 The Series Ta,2,∞ 221 024Download PDF (122.8 KB)fulltext 223 Chapter 23 Germs f of Type f (xayb,z): Suspensions 223 23.1 f of Type f (xy,z) 223 23.2 f of Type f (xayb,z) 224 025Download PDF (119.0 KB)fulltext 229 Chapter 24 Peculiar Structures on F: Topics for Future Research 229 24.1 Contact Structures 229 24.2 Triple Product: Resonance Varieties 230 24.3 Relations with the Homology of the Milnor Fiber 231 24.4 Open Problems 232 026Download PDF (216.4 KB)back-matter 235 List of Examples 235 List of Notations 237 References 243 Index 249 Front Matter....Pages i-xii Introduction....Pages 1-7 Front Matter....Pages 9-9 The Topology of a Hypersurface Germ f in Three Variables....Pages 11-15 The Topology of a Pair ( f,g )....Pages 17-23 Plumbing Graphs and Oriented Plumbed 3-Manifolds....Pages 25-43 Cyclic Coverings of Graphs....Pages 45-54 The Graph $$\mathit\Gamma_{\mathcal{C}}$$ of a pair ( f,g ): The Definition....Pages 55-61 The Graph $$\mathit\Gamma_{\mathcal{C}}$$ : Properties....Pages 63-77 Examples: Homogeneous Singularities....Pages 79-82 Examples: Families Associated with Plane Curve Singularities....Pages 83-97 Front Matter....Pages 99-99 The Main Algorithm....Pages 101-115 Proof of the Main Algorithm....Pages 117-130 The Collapsing Main Algorithm....Pages 131-138 Vertical/Horizontal Monodromies....Pages 139-151 The Algebraic Monodromy of H 1 (∂ F ): Starting Point....Pages 153-156 The Ranks of H1(∂ F) and H 1 (∂ F \ Vg) via plumbing....Pages 157-160 The Characteristic Polynomial of ∂F via P # and $${P}^\sharp_{j}$$ ....Pages 161-166 The Proof of the Characteristic Polynomial Formulae....Pages 167-172 The Mixed Hodge Structure of H 1 ( ∂F )....Pages 173-176 Front Matter....Pages 177-177 Homogeneous Singularities....Pages 179-199 Cylinders of Plane Curve Singularities: $$f= f^{\prime}(x,y)$$ ....Pages 201-204 Front Matter....Pages 177-177 Germs ƒ of Type $$ z f^{\prime}(x,y)$$ ....Pages 205-208 The $$ T_{\ast,\ast,\ast}$$ –family....Pages 209-210 Germs ƒ of Type $$ \tilde{f}(x^{a}y^{b},z)$$ : Suspensions....Pages 211-214 Front Matter....Pages 215-215 Peculiar Structures on ∂F : Topics for Future Research....Pages 217-222 Back Matter....Pages 223-240 In the study of algebraic/analytic varieties a key aspect is the description of the invariants of their singularities. This book targets the challenging non-isolated case. Let f be a complex analytic hypersurface germ in three variables whose zero set has a 1-dimensional singular locus. We develop an explicit procedure and algorithm that describe the boundary M of the Milnor fiber of f as an oriented plumbed 3-manifold. This method also provides the characteristic polynomial of the algebraic monodromy. We then determine the multiplicity system of the open book decomposition of M cut out by the argument of g for any complex analytic germ g such that the pair (f, g) is an ICIS. Moreover, the horizontal and vertical monodromies of the transversal type singularities associated with the singular locus of f and of the ICIS (f, g) are also described. The theory is supported by a substantial amount of examples, including homogeneous and composed singularities and suspensions. The peculiar properties of M are also emphasized