This volume collects the lecture notes of the school TiME2019 (Treasures in Mathematical Encounters). The aim of this book is manifold, it intends to overview the wide topic of algebraic curves and surfaces (also with a view to higher dimensional varieties) from different aspects: the historical development that led to the theory of algebraic surfaces and the classification theorem of algebraic surfaces by Castelnuovo and Enriques; the use of such a classical geometric approach, as the one introduced by Castelnuovo, to study linear systems of hypersurfaces; and the algebraic methods used to find implicit equations of parametrized algebraic curves and surfaces, ranging from classical elimination theory to more modern tools involving syzygy theory and Castelnuovo-Mumford regularity. Since our subject has a long and venerable history, this book cannot cover all the details of this broad topic, theory and applications, but it is meant to serve as a guide for both young mathematicians to approach the subject from a classical and yet computational perspective, and for experienced researchers as a valuable source for recent applications.
Author(s): Laurent Busé, Fabrizio Catanese, Elisa Postinghel
Series: SISSA Springer Series, 4
Publisher: Springer-SISSA
Year: 2023
Language: English
Pages: 219
City: Trieste
Preface
Abridged History of the Theory of Curves and Surfaces
Contents of the Volume
Contents
1 The P12-Theorem: The Classification of Surfaces and Its Historical Development
1.1 Introduction
1.2 Lecture I: The Basic Set Up
1.2.1 First New Concepts Introduced by Enriques
1.2.1.1 Intersection Product
1.2.1.2 The Severi Group and the Neron-Severi Group
1.2.2 The Canonical Divisor and Riemann-Roch for Divisors on Surfaces
1.2.2.1 The Hurwitz Formula
1.2.3 The Arithmetic Genus of a Curve on a Surface
1.2.4 Linear Systems and Morphisms
1.2.5 Exceptional Curves of the First Kind and the Theorem of Castelnuovo-Enriques
1.2.6 Birational Invariants of S and the Albanese Variety
1.2.6.1 Irregular Surfaces and the Albanese Variety
1.2.7 Uniqueness Versus Non Uniqueness of Minimal Models
1.2.7.1 Elementary Transformations of Geometrically Ruled Surfaces
1.2.8 Castelnuovo's Key Theorem
1.2.9 Biregular Invariants of the Minimal Model
1.3 Lecture II: First Important Results for the Classification Theorem of Surfaces
1.3.1 A Basic Tool: Unramified Coverings
1.3.2 Castelnuovo's Theorem on Irregular Ruled Surfaces
1.3.3 Surfaces Fibred Over Curves
1.3.4 Castelnuovo's Criterion of Rationality
1.4 Lecture III: The Classification Theorem
1.4.1 Description of the Surfaces with 12 KS 0 (Case II, P12(S)=1)
1.4.2 Hyperelliptic Surfaces
1.5 Lecture IV: Isotriviality. Central Methods and Ideas in the Proof of the P12-Theorem
1.5.1 Structure of the Proof of the Classification Theorem
1.5.1.1 The Canonical Divisor Formula for Elliptic Fibrations
1.5.1.2 On the Existence of Elliptic Fibrations
1.5.1.3 P12 of Elliptic Fibrations
1.5.2 The Special Case KS nef, KS2=0, pg(S)=0, q(S)=1 and the Crucial Theorem
1.5.3 First Transcendental Proof of Isotriviality for Fibre Genus g = 1.
1.5.3.1 All the Fibres Smooth of Genus g=1
1.5.3.2 g=1 and there are multiple fibres.
1.5.4 Second Transcendental Proof of Isotriviality Using Teichmüller Space for Fibre Genus g ≥2
1.5.5 Modern Proof of Isotriviality Using Variation of Hodge Structures, and the Theorems of Fujita and Arakelov
1.5.5.1 Fujita's and Arakelov's Theorems
1.5.6 Algebraic Approaches by Castelnuovo-Enriques, Bombieri-Mumford
1.5.6.1 Lemma of Enriques and Mumford mum1
1.6 Appendix: Surfaces with Arithmetic Genus -1, Hyperelliptic Surfaces and Elliptic Surfaces According to Enriques
1.6.1 Analysis of Enriques' Argument
1.6.2 An Explicit Example of Surfaces of Type (2.0,0)
1.7 Some Exercises
1.7.1 Exercise 1 : Exceptional Curves of the First Kind
1.7.2 Exercise 2 : Fibred Surfaces with Fibre Genus g=0
1.7.3 Exercise 3 : Minimal K3 Surfaces, Surfaces with KS 0 (KS is Trivial), q(S) = 0
1.7.4 Exercise 4: Enriques' Construction of Enriques Surfaces
1.7.5 Exercise 5: Construction of Enriques Surfaces via a Reye Congruence
References
2 Linear Systems of Hypersurfaces with Singularities and Beyond
2.1 Introduction
2.2 Plane Singular Curves
2.2.1 Polynomial Interpolation Problems
2.2.2 Geometric Formulation: Linear Systems with Multiple Base Points
2.2.3 Algebraic Formulation: Ideals of Powers of Linear Forms
2.2.4 Plane Curves
2.2.4.1 The Segre-Harbourne-Gimigliano-Hirschowitz Conjecture
2.2.4.2 A Conjecture by Nagata
2.2.5 Standard Cremona Involution
2.2.6 Historical Readings
2.2.6.1 Linear Systems of Plane Curves
2.2.6.2 Castelnuovo-Mumford Regularity
2.2.7 Further Readings
2.2.7.1 Birkhoff Interpolation
2.2.7.2 Classification of Surfaces and (-1)-Curves
2.2.8 Exercises
2.3 Nodal Hypersurfaces
2.3.1 The Exceptional Cases of the Alexander-Hirschowitz Theorem
2.3.1.1 Quadrics in Pn
2.3.1.2 Quartics in Pn, with n≤4
2.3.1.3 Cubics in P4 with Seven Nodes
2.3.2 Degeneration Techniques in the Proofs of Theorem 2.15
2.3.2.1 Specialisation of Points: Horace Methods
2.3.2.2 Degenerations of the Space
2.3.3 Secant Varieties of Veronese Varieties
2.3.4 Further Readings
2.3.4.1 Waring's Problem for Polynomials
2.3.5 Historical Readings
2.3.5.1 Degenerations and Castelnuovo's Exact Sequences
2.3.6 Exercises
2.4 Results and Conjectures in Higher Dimension: Base Loci
2.4.1 Laface-Ugaglia Conjecture for P3
2.4.2 The Toric Case: n+1 Points of Pn
2.4.2.1 A Combinatorial Trick
2.4.3 Linear Base Locus: The Case s≤n+2
2.4.3.1 Connections to the Fröberg-Iarrobino Conjecture
2.4.4 Base Locus Rational Normal Curves Through n+3 points of Pn and Their Secant Varieties
2.4.4.1 Divisorial Joins
2.4.5 Base Locus Weyl Cycles Through n+4 Points of Pn,n=3,4
2.4.6 Concluding Remarks
2.4.7 Further Readings
2.4.7.1 Compactifications of the Moduli Space of Rational Curves
2.4.8 Exercises
2.5 Birational Properties of Blow-Ups
2.5.1 Ampleness, Very Ampleness and l-Very Ampleness
2.5.2 Cones of Divisors
2.5.2.1 Nef Cones
2.5.2.2 Mori Dream Spaces, Effective and Movable Cones of Divisors and Their Mori Chamber Decomposition
2.5.3 Further Readings
2.5.3.1 Jet Ampleness of Line Bundles
2.5.3.2 Blow-Ups of Projective Spaces, Moduli Theory and Hilbert's 14th Problem
2.5.3.3 Classification of Mori Dream Space Blow-Ups of Products of Projective Spaces
2.5.4 Exercises
References
3 Implicit Representations of Rational Curves and Surfaces
3.1 Introduction
3.2 Plane Rational Curve Parameterizations
3.2.1 The Classical Implicitization Method
3.2.2 Syzygies of Curve Parameterizations
3.2.3 Matrix Representations of Curve Parameterizations
3.2.4 Intersection of Two Rational Curves
3.2.5 Singular Points
3.2.6 Exercises
3.3 Matrix Representations in Elimination Theory
3.3.1 Fitting Elimination Ideals
3.3.1.1 Elimination Ideals
3.3.1.2 Fitting Invariants and Elimination Matrices
3.3.2 Finite Fibers and Elimination Matrices
3.3.3 Koszul-Type Elimination Matrices
3.3.3.1 The Ideal of Inertia Forms
3.3.3.2 Upper Bound on the Regularity
3.3.3.3 Elimination Matrices
3.3.4 Hybrid Elimination Matrices
3.3.4.1 Sylvester Forms
3.3.4.2 Elimination Matrices
3.3.5 Exercises
3.4 Rational Curve Parameterizations
3.4.1 Syzygies and the Equations of the Graph
3.4.2 Fibers by Means of Matrix Representations
3.4.3 Using Quadratic Relations
3.4.3.1 Sylvester Forms
3.4.3.2 Matrix Representations by Means of Linear and Quadratic Relations
3.4.4 Application to Intersection Problems
3.4.4.1 Inversion of a Point
3.4.4.2 Intersection of Two Rational Curves
3.4.5 Exercises
3.5 Rational Hypersurface Parameterizations
3.5.1 The Degree of Parameterized Hypersurfaces
3.5.2 Graph and Blowup Algebras
3.5.2.1 The Rees Algebra
3.5.2.2 The Symmetric Algebra
3.5.2.3 Approximation Complexes
3.5.3 Matrix Representations of Hypersurface Parameterizations
3.5.4 Applications to Intersection Problems
3.5.5 Further Readings
References
