عنوان

Singular algebraic curves :

(Number (ISBN

3030033503

(Number (ISBN

9783030033507

Erroneous ISBN

303003349X

Erroneous ISBN

9783030033491

Title Proper

Singular algebraic curves :

General Material Designation

[Book]

Other Title Information

with an appendix by Oleg Viro /

First Statement of Responsibility

Gert-Martin Greuel, Christoph Lossen and Eugenii Shustin.

Place of Publication, Distribution, etc.

Cham :

Name of Publisher, Distributor, etc.

Springer,

Date of Publication, Distribution, etc.

2018.

Specific Material Designation and Extent of Item

1 online resource

Series Title

Springer monographs in mathematics

Text of Note

Intro; Preface; Acknowledgements; References; Contents; Notations and Conventions; References; 1 Zero-Dimensional Schemes for Singularities; 1.1 Cluster and Zero-Dimensional Schemes; 1.1.1 Constellations and Cluster; 1.1.2 Cluster Schemes and Equisingularity; 1.1.3 The Hilbert Scheme of Cluster Schemes; 1.1.4 Zero-Dimensional Schemes for Analytic Types; 1.2 Non-classical Singularity Invariants; 1.2.1 Determinacy Bounds; 1.2.2 New Topological Invariants; 1.2.3 New Analytic Invariants; 1.3 Historical Notes and References; References; 2 Global Deformation Theory; 2.1 Classical Global Theorems

Text of Note

2.1.1 Divisors and Linear Systems2.1.2 Bézout's Theorem; 2.1.3 Bertini's and Noether's Theorems; 2.1.4 Polar and Dual Curves; 2.2 Equisingular Families of Singular Algebraic Varieties; 2.2.1 Families with Imposed Conditions on Singularities; 2.2.2 Hilbert Schemes of Singular Hypersurfaces; 2.2.3 T-Smooth Families of Isolated Hypersurface Singularities; 2.3 Construction via Deformation; 2.3.1 General Idea of the Patchworking Construction; 2.3.2 Polytopes and calS-transversality; 2.3.3 Gluing Singular Hypersurfaces; 2.3.4 SQH and NND Isolated Hypersurface Singularities

Text of Note

2.3.5 Lower Deformations of Hypersurface Singularities2.3.6 Cohomology Vanishing Conditions; 2.4 Appendix: The Patchworking Construction; 2.4.1 Elements of Toric Geometry; 2.4.2 Viro's Theorem for Hypersurfaces; 2.4.3 Viro's Method in Real Algebraic Geometry; 2.4.4 Viro's Theorem for Complete Intersections; 2.4.5 Other Examples of Patchworking; 2.5 Historical Notes and References; References; 3 H1-Vanishing Theorems; 3.1 Riemann-Roch Type H1-Vanishing; 3.2 Applications of Kodaira Vanishing; 3.3 Reider-Bogomolov Theory; 3.4 The Horace Method; 3.4.1 The Basic Horace Method

Text of Note

3.4.2 Applications of the Basic Horace Method3.4.3 Variations of the Horace Method; 3.5 The Castelnuovo Function; 3.6 H1-Vanishing for Generic Zero-Dimensional Schemes; 3.7 Historical Notes and References; References; 4 Equisingular Families of Curves; 4.1 Overview of New Results and Methods; 4.1.1 Nonemptiness; 4.1.2 T-Smoothness and Deformation Completeness; 4.1.3 Irreducibility; 4.1.4 Comments on the Methods; 4.2 Formulation of Problems, Discussion of Results, Examples; 4.2.1 Statement of the Problem; 4.2.2 Curves with Nodes and Cusps: from Severi to Harris

Text of Note

4.2.3 Examples of Obstructed and Reducible ESF4.3 T-Smoothness; 4.3.1 Linear Conditions; 4.3.2 Quadratic Conditions; 4.3.3 Obstructed Equisingular Families; 4.3.4 ESF of Hypersurfaces in mathbbPn; 4.4 Independence of Simultaneous Deformations; 4.4.1 Joint Versal Deformations; 4.4.2 1-Parametric Deformations; 4.5 Existence; 4.5.1 Conditions for the Patchworking Construction; 4.5.2 Plane Curves with Nodes and Cusps; 4.5.3 Curves and Hypersurfaces with Simple Singularities; 4.5.4 Hypersurfaces with Arbitrary Singularities; 4.5.5 Plane Curves with Arbitrary Singularities

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Text of Note

Singular algebraic curves have been in the focus of study in algebraic geometry from the very beginning, and till now remain a subject of an active research related to many modern developments in algebraic geometry, symplectic geometry, and tropical geometry. The monograph suggests a unified approach to the geometry of singular algebraic curves on algebraic surfaces and their families, which applies to arbitrary singularities, allows one to treat all main questions concerning the geometry of equisingular families of curves, and, finally, leads to results which can be viewed as the best possible in a reasonable sense. Various methods of the cohomology vanishing theory as well as the patchworking construction with its modifications will be of a special interest for experts in algebraic geometry and singularity theory. The introductory chapters on zero-dimensional schemes and global deformation theory can well serve as a material for special courses and seminars for graduate and post-graduate students. Geometry in general plays a leading role in modern mathematics, and algebraic geometry is the most advanced area of research in geometry. In turn, algebraic curves for more than one century have been the central subject of algebraic geometry both in fundamental theoretic questions and in applications to other fields of mathematics and mathematical physics. Particularly, the local and global study of singular algebraic curves involves a variety of methods and deep ideas from geometry, analysis, algebra, combinatorics and suggests a number of hard classical and newly appeared problems which inspire further development in this research area.--

Source for Acquisition/Subscription Address

Springer Nature

Stock Number

com.springer.onix.9783030033507

Title

Singular algebraic curves.

International Standard Book Number

9783030033491

Curves, Algebraic.

Curves, Algebraic.

MATHEMATICS-- Geometry-- General.

MAT-- 012000

PBMW

PBMW

Number

Edition

Class number

Greuel, G.-M., (Gert-Martin)

Lossen, Christoph

Shustin, Eugenii

Date of Transaction

20200823075310.0

Cataloguing Rules (Descriptive Conventions))

pn

Electronic name

[Book]

Y