| Description |
1 online resource (220 p.) |
| Series |
SISSA Springer series ; v.4 |
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SISSA Springer series ; v. 4.
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| Contents |
Intro -- 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 |
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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 |
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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 |
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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 |
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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 |
| Summary |
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 |
| Notes |
1.7.5 Exercise 5: Construction of Enriques Surfaces via a Reye Congruence |
| Bibliography |
Includes bibliographical references |
| Notes |
Online resource; title from PDF title page (SpringerLink, viewed May 15, 2023) |
| Subject |
Geometry, Algebraic.
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Geometry, Algebraic
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Geometria algebraica.
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| Genre/Form |
Llibres electrònics.
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| Form |
Electronic book
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| Author |
Catanese, F. (Fabrizio)
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Postinghel, Elisa.
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| ISBN |
9783031241512 |
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3031241517 |
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