UID:
almahu_9947362729102882
Umfang:
XXII, 486 p.
,
online resource.
ISBN:
9781402020292
Serie:
Algebras and Applications, 4
Inhalt:
The Curves The Point of View of Max Noether Probably the oldest references to the problem of resolution of singularities are found in Max Noether's works on plane curves [cf. [148], [149]]. And probably the origin of the problem was to have a formula to compute the genus of a plane curve. The genus is the most useful birational invariant of a curve in classical projective geometry. It was long known that, for a plane curve of degree n having l m ordinary singular points with respective multiplicities ri, i E {1, . . . , m}, the genus p of the curve is given by the formula = (n - l)(n - 2) _ ~ "r. (r. _ 1) P 2 2 L. . ,. •• . Of course, the problem now arises: how to compute the genus of a plane curve having some non-ordinary singularities. This leads to the natural question: can we birationally transform any (singular) plane curve into another one having only ordinary singularities? The answer is positive. Let us give a flavor (without proofs) 2 on how Noether did it • To solve the problem, it is enough to consider a special kind of Cremona trans formations, namely quadratic transformations of the projective plane. Let ~ be a linear system of conics with three non-collinear base points r = {Ao, AI, A }, 2 and take a projective frame of the type {Ao, AI, A ; U}.
Anmerkung:
I Valuation Theory -- 1 Marot Rings -- 2 Manis Valuation Rings -- 3 Valuation Rings and Valuations -- 4 The Approximation Theorem For Independent Valuations -- 5 Extensions of Valuations -- 6 Extending Valuations to Algebraic Overfields -- 7 Extensions of Discrete Valuations -- 8 Ramification Theory of Valuations -- 9 Extending Valuations to Non-Algebraic Overfields -- 10 Valuations of Algebraic Function Fields -- 11 Valuations Dominating a Local Domain -- II One-Dimensional Semilocal Cohen-Macaulay Rings -- 1 Transversal Elements -- 2 Integral Closure of One-Dimensional Semilocal Cohen-Macaulay Rings -- 3 One-Dimensional Analytically Unramified and Analytically Irreducible CM-Rings -- 4 Blowing up Ideals -- 5 Infinitely Near Rings -- III Differential Modules and Ramification -- 1 Introduction -- 2 Norms and Traces -- 3 Formally Unramified and Unramified Extensions -- 4 Unramified Extensions and Discriminants -- 5 Ramification For Quasilocal Rings -- 6 Integral Closure and Completion -- IV Formal and Convergent Power Series Rings -- 1 Formal Power Series Rings -- 2 Convergent Power Series Rings -- 3 Weierstraß Preparation Theorem -- 4 The Category of Formal and Analytic Algebras -- 5 Extensions of Formal and Analytic Algebras -- V Quasiordinary Singularities -- 1 Fractionary Power Series -- 2 The Jung-Abhyankar Theorem: Formal Case -- 3 The Jung-Abhyankar Theorem: Analytic Case -- 4 Quasiordinary Power Series -- 5 A Generalized Newton Algorithm -- 6 Strictly Generated Semigroups -- VI The Singularity Zq = XYp -- 1 Hirzebruch-Jung Singularities -- 2 Semigroups and Semigroup Rings -- 3 Continued Fractions -- 4 Two-Dimensional Cones -- 5 Resolution of Singularities -- VII Two-Dimensional Regular Local Rings -- 1 Ideal Transform -- 2 Quadratic Transforms and Ideal Transforms -- 3 Complete Ideals -- 4 Factorization of Complete Ideals -- 5 The Predecessors of a Simple Ideal -- 6 The Quadratic Sequence -- 7 Proximity -- 8 Resolution of Embedded Curves -- VIII Resolution of Singularities -- 1 Blowing up Curve Singularities -- 2 Resolution of Surface Singularities I: Jung’s Method -- 3 Quadratic Dilatations -- 4 Quadratic Dilatations of Two-Dimensional Regular Local Rings -- 5 Valuations of Algebraic Function Fields in Two Variables -- 6 Uniformization -- 7 Resolution of Surface Singularities II: Blowing up and Normalizing -- Appendices -- A Results from Classical Algebraic Geometry -- 1 Generalities -- 1.1 Ideals and Varieties -- 1.2 Rational Functions and Maps -- 1.3 Coordinate Ring and Local Rings -- 1.4 Dominant Morphisms and Closed Embeddings -- 1.5 Elementary Open Sets -- 1.6 Varieties as Topological Spaces -- 1.7 Local Ring on a Subvariety -- 2 Affine and Finite Morphisms -- 3 Products -- 4 Proper Morphisms -- 4.1 Space of Irreducible Closed Subsets -- 4.3 Proper Morphisms -- 5 Algebraic Cones and Projective Varieties -- 6 Regular and Singular Points -- 7 Normalization of a Variety -- 8 Desingularization of a Variety -- 9 Dimension of Fibres -- 10 Quasifinite Morphisms and Ramification -- 10.1 Quasifinite Morphisms -- 10.2 Ramification -- 11 Divisors -- 12 Some Results on Projections -- 13 Blowing up -- 14 Blowing up: The Local Rings -- B Miscellaneous Results -- 1 Ordered Abelian Groups -- 1.1 Isolated Subgroups -- 1.2 Initial Index -- 1.3 Archimedean Ordered Groups -- 1.4 The Rational Rank of an Abelian Group -- 2 Localization -- 3 Integral Extensions -- 4 Some Results on Graded Rings and Modules -- 4.1 Generalities -- 4.3 Homogeneous Localization -- 4.4 Integral Closure of Graded Rings -- 5 Properties of the Rees Ring -- 6 Integral Closure of Ideals -- 6.1 Generalities -- 6.2 Integral Closure of Ideals -- 6.3 Integral Closure of Ideals and Valuation Theory -- 7 Decomposition Group and Inertia Group -- 8 Decomposable Rings -- 9 The Dimension Formula -- 10 Miscellaneous Results -- 10.1 The Chinese Remainder Theorem -- 10.2 Separable Noether Normalization -- 10.3 The Segre Ideal -- 10.4 Adjoining an Indeterminate -- 10.5 Divisor Group and Class Group -- 10.6 Calculating a Multiplicity -- 10.7 A Length Formula -- 10.8 Quasifinite Modules -- 10.9 Maximal Primary Ideals -- 10.10 Primary Decomposition in Non-Noetherian Rings -- 10.11 Discriminant of a Polynomial -- Index of Symbols.
In:
Springer eBooks
Weitere Ausg.:
Printed edition: ISBN 9789048165735
Sprache:
Englisch
DOI:
10.1007/978-1-4020-2029-2
URL:
http://dx.doi.org/10.1007/978-1-4020-2029-2
URL:
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