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

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<(T)
			4.3.3 Base Cases
		4.4 Cauwbergs' Example
		4.5 Schrauwen, Steenbrink and Stevens Example
	5 Generalized Kouchnirenko's Formula
		5.1 Euler Characteristic of the Milnor Fiber of a Quasi Homogeneous Polynomial
		5.2 Generalized Kouchnirenko's Formula
	6 Monodromy Zeta Function
		6.1 The Invariant Monodromy Zeta Function
		6.2 The Monodromy Zeta Function of the Milnor Fiber
	References
Nash Modification on Toric Curves
	1 Introduction
	2 Nash Modification of a Toric Variety
		2.1 Combinatorial Algorithm for Affine Toric Varieties
		2.2 Combinatorial Algorithm for Affine Toric Curves
	3 Resolution of Toric Curves and Number of Iterations
		3.1 Resolution of Toric Curves
		3.2 Number of Iterations
	4 Counting Division Algorithms
	5 Some Other Features
		5.1 Hilbert-Samuel Multiplicity
		5.2 Embedding Dimension
		5.3 Zero Locus of an Ideal Defining the Nash Modification
		5.4 Possible Generalizations for Toric Surfaces
	References
A Recursive Formula for the Motivic Milnor Fiber of a Plane Curve
	1 Introduction
	2 The Theorem
	3 Proof of Theorem 1
	4 Proof of Theorem 2
	5 A Recursive Formula for the Spectrum
	6 An Example
	References
On Hasse–Schmidt Derivations: The Action of Substitution Maps
	1 Introduction
	2 Rings and (Bi)modules of Formal Power Series
	3 Substitution Maps
	4 The Action of Substitution Maps
	5 Multivariate Hasse-Schmidt Derivations
	6 The Action of Substitution Maps on HS-Derivations
	7 Generating HS-Derivations
	References
An Introduction to Resolution of Singularities via the Multiplicity
	1 Introduction
	2 An Overview of the Main Results
		2.1 Pairs, Transformations of Pairs and Closed Subsets
		2.2 The Role of Pairs in the Simplification of Singularities: Hironaka's Approach
		2.3 The Role of Pairs in the Simplification of the Multiplicity
		2.4 The Role of Pairs in the Simplification of the Multiplicity in Characteristic Zero
	3 Multiplicity, Projection on Smooth Schemes and Blow-Ups
		3.1 Multiplicity and Integral Closure of Ideals
		3.2 On the Class T
	4 On Elimination in Characteristic Zero and the Proof of Theorem 2.25
	5 Proof of Theorem 2.22
	6 On Some Properties of Constructive Resolution and the Proof of Theorem 1.3
	References
Platonic Surfaces
	1 Introduction
	2 Preliminaries
		2.1 General Notions
		2.2 Constellations
		2.3 Platonic Surfaces
	3 Geometry of Platonic Rational Surfaces
		3.1 X2 Has Only a Finite Number of (-1)-Curves
		Case 1:
		Case 2:
		Case 3:
		Case 4:
		3.2 X2 Has Only a Finite Number of (-2)-Curves
		3.3 Regularity of nef Divisors on X2
	4 Finite Generation of the Cox Rings of Platonic Surfaces
	References
Coverings of Rational Ruled Normal Surfaces
	1 Introduction
	2 Preliminaries
		2.1 Quotient Singularities and Weighted Blow-Ups
			2.1.1 Weighted Blow-Ups of Smooth Points
			2.1.2 Weighted Blow-Ups of Singular Points: Special Case
		2.2 The Projection Formula
		2.3 Picard Group for Normal Surfaces
		2.4 The Canonical Cycle for Q-Resolutions and the Riemann-Roch Formula
		2.5 The Correction Term Delta_S
	3 Rational Ruled Toric Surfaces
		3.1 Weighted Nagata Transformations
		3.2 Construction of the Surfaces S(d1,d2,p1,p2,r)
		3.3 The Picard Group of S
		3.4 The Canonical Cycle of S
	4 Rational Biruled Toric Surfaces
	5 Rational Uniruled Toric Surfaces
		5.1 Case k=0
		5.2 Case k>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