Bridging Algebra, Geometry, and Topology (Springer Proceedings in Mathematics & Statistics Book 96)

Algebra, geometry and topology cover a variety of different, but intimately related research fields in modern mathematics. This book focuses on specific aspects of this interaction. The present volume contains refereed papers which were presented at the International Conference __“Experimental and Theoretical Methods in Algebra, Geometry and Topology”__, held in Eforie Nord (near Constanta), Romania, during 20-25 June 2013. The conference was devoted to the 60th anniversary of the distinguished Romanian mathematicians Alexandru Dimca and Ştefan Papadima. The selected papers consist of original research work and a survey paper. They are intended for a large audience, including researchers and graduate students interested in algebraic geometry, combinatorics, topology, hyperplane arrangements and commutative algebra. The papers are written by well-known experts from different fields of mathematics, affiliated to universities from all over the word, they cover a broad range of topics and explore the research frontiers of a wide variety of contemporary problems of modern mathematics. Preface 6 Contents 12 Solving via Modular Methods 14 1 Introduction 14 2 Preliminary Technicalities 15 3 Modular Methods 17 4 Examples and Timings 19 References 21 Lazarsfeld–Mukai Bundles and Applications: II 23 1 Introduction 23 2 Lazarsfeld–Mukai Bundles on Surfaces with q=0 24 3 Lazarsfeld–Mukai Bundles of Rank Two on Rational Surfaces 27 4 Syzygies of Curves 30 References 31 References 31 Multinets in P2 33 1 Introduction 33 2 Preliminaries 34 2.1 Pencils of Curves and Multinets in P2 34 2.2 Properties of Multinets and Examples That Have Been Known 35 2.3 Constructions of Nets 36 3 Construction of Multinets 36 3.1 Multinets in Higher Dimensions 36 3.2 Construction of Multinets 37 4 Multinets Induced by Qn 37 4.1 General Position and Survey of Possibilities 37 4.2 Heavy Induced Multinets (Lines of Multiplicity n) 38 4.3 Heavy Induced Multinets (Lines of Multiplicity 2) 38 4.4 Light Induced Multinets (Points of Multiplicity n) 39 4.5 Light Induced Multinets (Points of Multiplicity 2) 39 4.6 Light Induced Multinets (Several Points of Multiplicity 2) 40 4.7 Fixed Components 42 4.8 Summary of Properties of Induced Multinets from Qn 45 5 Combinatorics Inside Blocks 46 6 Conjectures and Open Problems 46 References 47 A More General Framework for CoGalois Theory 48 1 Introduction 49 2 Preliminaries 51 2.1 Profinite and Spectral Spaces 51 2.2 Profinite Groups 51 2.3 The Lattice of Closed Subgroups of a Profinite Group 52 2.4 Profinite Operator Groups 52 2.4.1 The Lattice of Ideals of a Profinite Operator Group 53 2.5 Galois and CoGalois Connections 53 2.5.1 Standard Galois Connections 54 3 An Abstract Framework for CoGalois Theory 56 3.1 Generating Cocycles 56 3.1.1 Universal Cocycles 57 3.2 The CoGalois Connection Associated with a Generating Cocycle 58 4 Continuous Actions on Discrete Abelian Groups 61 4.1 The CoGalois Group of a Discrete Module 61 4.2 The Galois Connection Associated with a Torsion Module 62 4.3 The Associated Profinite Module and Generating Cocycle 64 4.4 A Nondegenerate Pairing and the Induced Galois Connection 66 4.5 Examples 68 5 Kneser and Minimal Non-Kneser Triples 72 5.1 Surjective Cocycles 72 5.1.1 Surjectivity Criteria for Cocycles 74 5.1.2 Bijective Cocycles Induced by Self-Actions 75 5.1.3 Examples 76 5.2 Kneser Ideals 78 5.3 A General Kneser Criterion 80 6 CoGalois and Minimal Non-coGalois Triples 81 7 Partial Answers to Classification Problem 5.24 84 7.1 The Abelian Case of Problem 5.24 86 References 94 Connectivity and a Problem of Formal Geometry 96 1 Introduction 96 2 Background Material 98 3 Extending Formal-Rational Functions 100 References 104 Hodge Invariants of Higher-Dimensional Analogues of KodairaSurfaces 106 1 Introduction 106 2 Complex Principal Torus Bundles 107 3 Computation of Hodge Numbers 111 4 Structure of the Kodaira Manifolds 115 References 116 An Invitation to Quasihomogeneous Rigid Geometric Structures 118 1 Introduction 118 2 Rigid Geometric Structures and Killing Fields 120 3 Quasihomogeneous Real-Analytic Connections 124 4 Holomorphic Geometric Structures 128 References 132 Koszul Binomial Edge Ideals 135 1 Introduction 135 2 Koszul Graphs Are Chordal and Claw Free 136 3 Gluing of Koszul Graphs Along a Vertex 140 References 145 On the Fundamental Groups of Non-generic R-Join-Type Curves 147 1 Introduction 147 2 The Groups G(p;q) and G(p;q;r) 150 3 Special Pencil Lines 152 4 Bifurcation Graph 153 5 Proof of Theorem 1 156 6 Applications 165 6.1 Maximal Irreducible Nodal Curves 165 6.2 Curves with Node and Cusp Singularities 166 References 166 Some Remarks on the Realizability Spaces of (3,4)-Nets 168 1 (3,4)-Nets 170 References 174 Critical Points of Master Functions and the mKdV Hierarchy of Type A(2)2 176 1 Introduction 176 2 Kac–Moody Algebras of Types A2(2) and A2(1) 178 2.1 Kac–Moody Algebra of Type A2(2) 178 2.1.1 Definition 178 2.1.2 Realizations of g(A(2)2) 179 2.1.3 Element Λ(2) 181 2.2 Kac–Moody Algebra of Type A2(1) 182 2.2.1 Definition 182 2.2.2 Realizations of g(A2(1)) 183 2.2.3 Element Λ(1) 184 2.2.4 Lie Algebra g(A(2)2) as a Subalgebra of g(A2(1)) 185 3 mKdV Equations 185 3.1 mKdV Equations of Type A(2)2 185 3.2 mKdV Equations of Type A(1)2 186 3.3 Comparison of mKdV Equations of Types A(2)2 and A(1)2 187 3.4 KdV Equations of Type A(1)2 188 3.5 Miura Maps 188 4 Critical Points of Master Functions and Generation of Pairs of Polynomials 189 4.1 Master Function 189 4.2 Polynomials Representing Critical Points 190 4.3 Elementary Generation 190 4.4 Degree Increasing Generation 192 4.5 Degree-Transformations and Generation of Vectors of Integers 193 4.6 Multistep Generation 194 5 Critical Points of Master Functions and Miura Opers 195 5.1 Miura Oper Associated with a Pair of Polynomials, MV2 195 5.2 Deformations of Miura Opers of Type A(2)2, MV2 196 5.3 Miura Opers Associated with the Generation Procedure 197 6 Vector Fields 199 6.1 Statement 199 6.2 Proof of Theorem 6.1 for m=1 200 6.3 Proof of Theorem 6.1 for m>1 201 6.4 Proof of Theorem 6.4 202 6.5 Critical Points and the Population Generated from y 204 References 204 Gauss–Lucas and Kuo–Lu Theorems 205 1 Introduction 205 2 Convexity 205 3 Results 206 References 209 Fibonacci Numbers and Self-Dual Lattice Structures for PlaneBranches 210 1 Introduction 210 2 The Enriques Diagram and the Multiplicity Sequence 212 3 The Number of Combinatorial Types of Branches with Fixed Blow-Up Complexity 217 4 Multiplicity Increasing Operators 218 5 Extremal Multiplicities and Milnor Numbers for Fixed Complexity 222 6 The Self-Dual Lattice Structure on En 224 7 Basic Facts About Posets and Lattices 228 8 The Uniqueness of the Duality of Plane Branches 231 References 236 Four Generated, Squarefree, Monomial Ideals 238 1 Introduction 238 2 Depth and Stanley Depth 240 3 A Special Case of r=4 243 4 Proof of Theorem 6 244 References 254 The Connected Components of the Space of Alexandrov Surfaces 256 1 Introduction and Results 256 2 Proofs 257 References 261 Motivic Milnor Fibre for Nondegenerate Function Germson Toric Singularities 262 1 Introduction 262 2 Motivic Nearby Fibre 263 2.1 Grothendieck Groups of Varieties 263 2.2 Nearby and Milnor Fibre 264 2.3 Toroidal Embeddings 266 2.4 Motivic Nearby Fibre in Toroidal Setting 267 3 Nondegenerate Laurent Polynomials 269 3.1 Toric and Toroidal Singularities 269 3.2 Newton Polyhedron and Nondegeneracy 269 3.3 Newton Modification 270 3.4 Motivic Milnor Fibre 271 4 Example: Weighted Homogeneous Laurent Polynomials 272 References 272 References 273 Non-Abelian Resonance: Product and Coproduct Formulas 275 1 Introduction 275 2 Flat Connections and Holonomy Lie Algebras 276 2.1 Differential Graded Algebras and Lie Algebras 276 2.2 Flat, g-Valued Connections 277 2.3 Holonomy Lie Algebra 277 3 Resonance Varieties 278 3.1 Twisted Differentials 278 3.2 Resonance Varieties of a cdga 279 3.3 Resonance Varieties of a Lie Algebra 279 4 Products 280 4.1 Holonomy Lie Algebra and Products 280 4.2 Flat Connections and Products 281 4.3 Resonance and Products 282 4.4 Product Formulas for Resonance 282 5 Coproducts 283 5.1 Holonomy Lie Algebras and Coproducts 284 5.2 Resonance and Coproducts 284 5.3 A Coproduct Formula for Degree 1 Resonance 285 References 286 Complements of Hypersurfaces, Variation Maps, and Minimal Models of Arrangements 287 1 Introduction 287 2 Complements of Hypersurfaces 288 3 Variation Maps of Pencils of Affine Hypersurfaces 289 3.1 Pencils with Isolated Singularities 289 3.2 Variation Maps 290 3.3 Number of Cells and Polar Invariants 292 4 Proof of Theorem 1 293 References 294 Front Matter....Pages i-xii Solving via Modular Methods....Pages 1-9 Lazarsfeld–Mukai Bundles and Applications: II....Pages 11-20 Multinets in P 2 $$\mathbb{P}^{2}$$ ....Pages 21-35 A More General Framework for CoGalois Theory....Pages 37-84 Connectivity and a Problem of Formal Geometry....Pages 85-94 Hodge Invariants of Higher-Dimensional Analogues of Kodaira Surfaces....Pages 95-106 An Invitation to Quasihomogeneous Rigid Geometric Structures....Pages 107-123 Koszul Binomial Edge Ideals....Pages 125-136 On the Fundamental Groups of Non-generic R $$\mathbb{R}$$ -Join-Type Curves....Pages 137-157 Some Remarks on the Realizability Spaces of (3,4)-Nets....Pages 159-166 Critical Points of Master Functions and the mKdV Hierarchy of Type A 2 (2) ....Pages 167-195 Gauss–Lucas and Kuo–Lu Theorems....Pages 197-201 Fibonacci Numbers and Self-Dual Lattice Structures for Plane Branches....Pages 203-230 Four Generated, Squarefree, Monomial Ideals....Pages 231-248 The Connected Components of the Space of Alexandrov Surfaces....Pages 249-254 Motivic Milnor Fibre for Nondegenerate Function Germs on Toric Singularities....Pages 255-267 Non-Abelian Resonance: Product and Coproduct Formulas....Pages 269-280 Complements of Hypersurfaces, Variation Maps, and Minimal Models of Arrangements....Pages 281-289 Solving Via Modular Methods -- Lazarsfeld-mukai Bundles And Applications : Ii -- Multinets In P2 -- A More General Framework For Cogalois Theory -- Connectivity And A Problem Of Formal Geometry -- Hodge Invariants Of Higher-dimensional Analogues Of Kodaira Surfaces -- An Invitation To Quasihomogeneous Rigid Geometric Structures -- Koszul Binomial Edge Ideals -- On The Fundamental Groups Of Non-generic R-join-type Curves -- Some Remarks On The Realizability Spaces Of (3,4)-nets -- Critical Points Of Master Functions And The Mkdv Hierarchy Of Type A22 -- Gauss-lucas And Kuo-lu Theorems -- Fibonacci Numbers And Self-dual Lattice Structures For Plane Branches -- Four Generated, Squarefree, Monomial Ideals -- The Connected Components Of The Space Of Alexandrov Surfaces -- Motivic Milnor Fibre For Nondegenerate Function Germs On Toric Singularities -- Non-abelian Resonance : Product And Coproduct Formulas -- Complements Of Hypersurfaces, Variation Maps And Minimal Models Of Arrangements. Denis Ibadula, Willem Veys, Editors. Includes Bibliographical References.