Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics : Festschrift for Antonio Campillo on the Occasion of His 65th Birthday

This volume brings together recent, original research and survey articles by leading experts in several fields that include singularity theory, algebraic geometry and commutative algebra. The motivation for this collection comes from the wide-ranging research of the distinguished mathematician, Antonio Campillo, in these and related fields. Besides his influence in the mathematical community stemming from his research, Campillo has also endeavored to promote mathematics and mathematicians' networking everywhere, especially in Spain, Latin America and Europe. Because of his impressive achievements throughout his career, we dedicate this book to Campillo in honor of his 65th birthday. Researchers and students from the world-wide, and in particular Latin American and European, communities in singularities, algebraic geometry, commutative algebra, coding theory, and other fields covered in the volume, will have interest in this book. Preface Contents Contributors Acronyms Antonio Campillo 1 Prelude 2 Mathematical Education and Academic Positions 3 PhD Students 4 Collaborators 5 Fruits and Seeds 5.1 Books 5.2 Equisingularity and Deformations 5.3 Toric Geometry 5.4 Clusters, Proximity, Complete Ideals and Cones 5.5 Algebraic Geometry Codes 5.6 Algebraic Foliations and Polarity 5.7 Poincaré Series and Zeta Functions 5.8 Achivements in Headlines 6 Services and Influence 6.1 University Administration 6.2 Cooperation with Mathematical Societies 6.3 Scientific Policy 6.4 Conference of Spanish Deans of Mathematics 6.5 Responsibilities in the RSME 6.6 Coda References Singularities in Positive Characteristic: Equisingularity, Classification, Determinacy 1 Historical Review 2 Equisingularity 2.1 Hamburger–Noether Expansions 2.2 Equisingularity Strata 2.3 Pathologies and Open Problems 3 Classification of Singularities 3.1 Classification in Characteristic Zero 3.2 Classification in Positive Characteristic 3.3 Pathologies and a Conjecture 4 Finite Determinacy and Tangent Image 4.1 Finite Determinacy for Hypersurfaces 4.2 Finite Determinacy for Ideals and Matrices 4.3 Pathology and a Problem References Ultrametric Spaces of Branches on Arborescent Singularities 1 Introduction 2 A Reminder on Intersection Theory for Normal Surface Singularities 2.1 The Determinant of a Normal Surface Singularity 2.2 Mumford's Rational Intersection Number of Branches 3 Generalities on Ultrametrics and Trees 3.1 Trees, Rooted Trees, Arborescent Partial Orders and Hierarchies 3.2 Ultrametric Spaces and Dated Rooted Trees 3.3 Additive Distances on Trees 4 Arborescent Singularities and Their Ultrametric Spaces of Branches 4.1 Determinant Products for Arborescent Singularities 4.2 The Ultrametric Associated to a Branch on an Arborescent Singularity 4.3 Płoski's Theorem and the Ultrametric Nature of Eggers-Wall Trees 5 Valuative Considerations 5.1 Basic Types of Valuations and Semivaluations 5.2 The Valuative Partial Order for Arborescent Singularities 6 Perspectives on Non-arborescent Singularities 6.1 Non-arborescent Examples 6.2 Some Open Problems References Two Points of the Boundary of Toric Geometry 1 Using Toric Degeneration to Avoid Wild Ramification 2 Additive Preorders and Orders on Zr and Projective Limits of Toric Varieties References On the Milnor Formula in Arbitrary Characteristic 1 Introduction 2 Semi-quasihomogeneous Singularities 3 Teissier's Lemma in Characteristic p≥0 4 Tame Singularities 5 The Milnor Number of Plane Irreducible Singularities Appendix References Foliations in the Plane Uniquely Determined by Minimal Subschemes of its Singularities 1 Introduction and Statement of the Results 2 Foliations in the Projective Plane 2.1 Special Subschemes 3 The Proofs 4 Work in Progress 4.1 Drop Points from a Type C 4.2 Add Points to a Type B References Newton Transformations and the Motivic Milnor Fiber of a PlaneCurve 1 Introduction 2 Motivic Milnor Fibers 2.1 Motivic Setting 2.1.1 Grothendieck Rings 2.1.2 Rational Series 2.2 Arcs 2.2.1 Arc Spaces 2.2.2 Origin, Order, Angular Component and Action 2.3 The Motivic Milnor Fiber 2.4 Motivic Zeta Function and Differential Form 3 Newton Algorithm 3.1 Newton Polygons 3.2 Newton Algorithm 4 Motivic Milnor Fibers and Newton Algorithm 4.1 Decomposition of the Zeta Function Along N(f) 4.2 Rationality and Limit of Zγ(T), for a Face γ of N(f) Contained in a Non Compact Face 4.3 Rationality and Limit of Zγ(T) for a Face γ of N(f) Not Contained in a Non Compact Face 4.3.1 Rationality and Limit of Zγ=(T) 4.3.2 Rationality and Limit of Zγ 0 6 Cyclic Branched Coverings on Rational Ruled Toric Surfaces 6.1 The Divisor Lk 6.2 The First Cohomology Group of OSLk 6.3 Decomposition of H1(S,OS) 7 Examples References Ulrich Bundles on Veronese Surfaces 1 Introduction 2 Preliminaries 3 Ulrich Bundles on Veronese Surfaces References Multiple Structures on Smooth on Singular Varieties References Smoothness in Some Varieties with Dihedral Symmetry and the DFT Matrix 1 Introduction 2 Jacobi Formula for the Co-rank 3 Minors of the Vandermonde Vn on the n-th Roots of Unity 3.1 First Results 3.2 The Complementarity Theorem 3.3 The Case n=p Prime: Chebotaryov's Theorem 3.4 Some Cases Where n Is a Prime Power, n=pk, p Odd 3.5 The Murty-Whang Criterion 4 Some Complex Varieties with Cyclic and Dihedral Symmetry 4.1 Some Complex Varieties with Cyclic Symmetry 4.2 Some Complex Varieties with Dihedral Symmetry 5 Intersections of Real Affine Ellipsoids with Dihedral Symmetry References The Greedy Algorithm and the Cohen-Macaulay Property of Rings, Graphs and Toric Projective Curves 1 Introduction 2 Greedy Algorithms and Simplicial Complexes 3 Matroids and Cohen-Macaulay Complexes 3.1 Greedy on Locally Cohen-Macaulay Complexes 4 Maximum Independent Set and Cohen-Macaulay Graphs 4.1 The Edge Ideal and Correctness of VertexSelect 5 Chordal Graphs and Shellable Simplicial Complexes 6 Coin-Exchange Problems and Cohen-Macaulay Toric Projective Curves 6.1 The Algebraic Framework 6.2 Finding an Optimal Solution for Coin-Exchange Effectively Within the Algebraic Framework References Binomial Ideals and Congruences on Nn 1 Preliminaries 1.1 Binomial Ideals 1.2 Graded Algebras 2 Congruences on Monoids and Binomial Ideals 3 Toric, Lattice and Mesoprime Ideals 3.1 Toric Ideals and Toric Congruences 3.2 Lattice Ideals and Cancellative Congruences 3.3 Mesoprime Ideals and Prime Congruences 4 Cellular Binomial Ideals 4.1 Cellular Decomposition of Binomial Ideals 5 Mesoprimary Ideals References The K-Theory of Toric Schemes Over Regular Rings of Mixed Characteristic 1 Introduction 2 Free A-Sets 3 Normal Flatness for Monoid Schemes 4 A Descent Theorem for Functors via Realizations 5 Presheaves of Spectra and dg Categories 6 Ω"0365Ω and Dilated Cyclic Homologies 7 Descent for Hochschild and Cyclic Homology 8 Main Theorem References On Finite and Nonfinite Generation of Associated Graded Rings of Abhyankar Valuations 1 Introduction 2 Proofs of the Finite Generation Results 3 Examples of Nonfinite Generation References Symbolic Powers of Monomial Ideals and Cohen-Macaulay Vertex-Weighted Digraphs 1 Introduction 2 Irreducible Decompositions and Symbolic Powers 3 Cohen–Macaulay Weighted Oriented Trees References Asymptotics of Reduced Algebraic Curves Over Finite Fields 1 Introduction 2 A(q) for Non-irreducible Curves 3 Properties of Ar(q) and Ar(q) 4 Geometric Goppa Codes from Non-irreducible Curves 5 Geometric Goppa Codes from Reduced Plane Models 6 Conclusions References The Poincaré Polynomial of a Linear Code 1 Introduction 2 The Poincaré Polynomial 3 The Poincaré Polynomial in the Binary Case References The Metric Structure of Linear Codes 1 Introduction 2 Metric Structure of Generalized Toric Codes 3 Bilinear Forms on Vector Spaces Over Finite Fields 3.1 Characteristic Different from 2 3.2 Characteristic 2 4 Geometric Decompositions of Linear Codes 4.1 Characteristic Different from 2 4.2 Characteristic 2 5 Linear Codes and Bilinear Algebra 5.1 Stabilizer Quantum Codes 5.2 LCD Codes 5.3 Minimum Distance of a Linear Code References On Some Properties of A Inherited by Cb(X,A) 1 Introduction 2 Cb(X,A) 3 The Generalized Arens-Michael Decomposition 4 Inversion 5 Involution 6 Metrizability References A Fractional Partial Differential Equation for Theta Functions 1 Introduction 2 Differentiation Matrices for Trigonometric Polynomials 3 A Fractional Equation for Theta Functions 4 Computing Jacobi Elliptic Functions 5 Final Remark References On Continued Fractions 1 Introduction: The Golden Number 1.1 On Relations Between φ, Fibonacci, Polygones and Polyhedra 2 Continued Fractions of Rational and Irrational Numbers 3 Geometric Interpretation of Continued Fractions 3.1 Some Proofs and Corollaries 4 Jung-Hirzebruch Continued Fractions 5 Continued Fractions in Higher Dimension 5.1 Some Results in Dimension 3 5.2 Example: The Cubic Root of 2 References