Resolution of Curve and Surface Singularities: in Characteristic Zero
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| Main Author: | |
|---|---|
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Dordrecht
Springer Netherlands
2004
|
| Series: | Algebras and Applications
4 |
| Subjects: | |
| Online Access: | Volltext |
| Item Description: | 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} |
| Physical Description: | 1 Online-Ressource (XXII, 486 p) |
| ISBN: | 9781402020292 9789048165735 |
| ISSN: | 1572-5553 |
| DOI: | 10.1007/978-1-4020-2029-2 |
Staff View
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| 500 | |a 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} | ||
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Record in the Search Index
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| discipline | Mathematik |
| doi_str_mv | 10.1007/978-1-4020-2029-2 |
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| spelling | Kiyek, K. Verfasser aut Resolution of Curve and Surface Singularities in Characteristic Zero by K. Kiyek, J. L. Vicente Dordrecht Springer Netherlands 2004 1 Online-Ressource (XXII, 486 p) txt rdacontent c rdamedia cr rdacarrier Algebras and Applications 4 1572-5553 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} Mathematics Geometry, algebraic Algebra Field theory (Physics) Differential equations, partial Algebraic Geometry Commutative Rings and Algebras Field Theory and Polynomials Several Complex Variables and Analytic Spaces Mathematik Charakteristik Null (DE-588)4472918-2 gnd rswk-swf Fläche (DE-588)4129864-0 gnd rswk-swf Singularität Mathematik (DE-588)4077459-4 gnd rswk-swf Kurve (DE-588)4033824-1 gnd rswk-swf 1\p (DE-588)4006432-3 Bibliografie gnd-content Singularität Mathematik (DE-588)4077459-4 s Kurve (DE-588)4033824-1 s Fläche (DE-588)4129864-0 s Charakteristik Null (DE-588)4472918-2 s 2\p DE-604 Vicente, J. L. Sonstige oth https://doi.org/10.1007/978-1-4020-2029-2 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Kiyek, K. Resolution of Curve and Surface Singularities in Characteristic Zero Mathematics Geometry, algebraic Algebra Field theory (Physics) Differential equations, partial Algebraic Geometry Commutative Rings and Algebras Field Theory and Polynomials Several Complex Variables and Analytic Spaces Mathematik Charakteristik Null (DE-588)4472918-2 gnd Fläche (DE-588)4129864-0 gnd Singularität Mathematik (DE-588)4077459-4 gnd Kurve (DE-588)4033824-1 gnd |
| subject_GND | (DE-588)4472918-2 (DE-588)4129864-0 (DE-588)4077459-4 (DE-588)4033824-1 (DE-588)4006432-3 |
| title | Resolution of Curve and Surface Singularities in Characteristic Zero |
| title_auth | Resolution of Curve and Surface Singularities in Characteristic Zero |
| title_exact_search | Resolution of Curve and Surface Singularities in Characteristic Zero |
| title_full | Resolution of Curve and Surface Singularities in Characteristic Zero by K. Kiyek, J. L. Vicente |
| title_fullStr | Resolution of Curve and Surface Singularities in Characteristic Zero by K. Kiyek, J. L. Vicente |
| title_full_unstemmed | Resolution of Curve and Surface Singularities in Characteristic Zero by K. Kiyek, J. L. Vicente |
| title_short | Resolution of Curve and Surface Singularities |
| title_sort | resolution of curve and surface singularities in characteristic zero |
| title_sub | in Characteristic Zero |
| topic | Mathematics Geometry, algebraic Algebra Field theory (Physics) Differential equations, partial Algebraic Geometry Commutative Rings and Algebras Field Theory and Polynomials Several Complex Variables and Analytic Spaces Mathematik Charakteristik Null (DE-588)4472918-2 gnd Fläche (DE-588)4129864-0 gnd Singularität Mathematik (DE-588)4077459-4 gnd Kurve (DE-588)4033824-1 gnd |
| topic_facet | Mathematics Geometry, algebraic Algebra Field theory (Physics) Differential equations, partial Algebraic Geometry Commutative Rings and Algebras Field Theory and Polynomials Several Complex Variables and Analytic Spaces Mathematik Charakteristik Null Fläche Singularität Mathematik Kurve Bibliografie |
| url | https://doi.org/10.1007/978-1-4020-2029-2 |
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