| Description |
1 online resource (257 pages) |
| Contents |
Cover; Half Title; Title; Copyright; Dedication; Contents; Preface; List of Figures; List of Tables; Acknowledgments; Contributors; Symbols; Part I Rational cubics and configurations of six orseven points in RP2; 1 Points, lines and conics in the plane; 1.1 Configurations of points; 1.2 Definitions and results; 2 Configurations of six points; 2.1 Rational pencils of cubics; 2.2 Diagrams and codes; 3 Configurations of seven points; 3.1 Fourteen configurations; 3.2 Line-walls and conic-walls; 3.3 Refined line-walls |
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Part II Pencils of cubics with eight base points lying in convex position in RP24 Pencils of cubics; 4.1 Preliminaries; 4.2 Singular pencils; 5 Lists; 5.1 Points in convex position and conics; 5.2 Admissible lists; 5.3 Extremal lists; 5.4 Distances between points; 5.5 Isotopies of octuples of points; 5.6 Elementary changes; 6 Link between lists and pencils; 6.1 Nodal lists; 6.2 Pairs of distinguished cubics; 6.3 Changes of lists and of pencils; 7 Pencils with reducible cubics; 7.1 Two non-generic lists; 7.2 Pencil with six reducible cubics; 7.3 Symmetric lists |
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8 Classification of the pencils of cubics8.1 Nodal pencils; 8.2 Inductive constructions; 9 Tables; 10 Application to an interpolation problem; 10.1 A non-generic pencil of cubics; 10.2 Solution to the interpolation problem; Part III Algebraic curves; 11 Hilbert's 16th problem; 11.1 Real and complex schemes; 11.2 Classical restriction method and degree 7; 11.3 Orevkov's method; 11.4 M-curves of degree 9; 12 M-curves of degree 9 with deep nests; 12.1 Results and rigid isotopy invariants; 12.2 Curves without O1-jumps; 13 M-curves of degree 9 with four or three nests |
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13.1 Statement of the results and first proofs13.2 Inequalities; 13.3 M-curves with three nests and a jump; 13.4 End of the proof, using two Orevkov formulas; 14 More restrictions; 14.1 M-curves of degree 9 or 11 with one non-empty oval; 14.2 Curves of degree 11 with many nests; 15 Totally real pencils of cubics; 15.1 Two real schemes of sextics; 15.2 Nodal pencil again; Bibliography; Index |
| Summary |
Pencils of Cubics and Algebraic Curves in the Real Projective Plane thoroughly examines the combinatorial configurations of n generic points in RP². Especially how it is the data describing the mutual position of each point with respect to lines and conics passing through others. The first section in this book answers questions such as, can one count the combinatorial configurations up to the action of the symmetric group? How are they pairwise connected via almost generic configurations? These questions are addressed using rational cubics and pencils of cubics for n = 6 and 7. The book's second section deals with configurations of eight points in the convex position. Both the combinatorial configurations and combinatorial pencils are classified up to the action of the dihedral group D8. Finally, the third section contains plentiful applications and results around Hilbert's sixteenth problem. The author meticulously wrote this book based upon years of research devoted to the topic. The book is particularly useful for researchers and graduate students interested in topology, algebraic geometry and combinatorics. Features: Examines how the shape of pencils depends on the corresponding configurations of points Includes topology of real algebraic curves Contains numerous applications and results around Hilbert's sixteenth problem About the Author: Séverine Fiedler-le Touzé has published several papers on this topic and has been invited to present at many conferences. She holds a Ph. D. from University Rennes1 and was a post-doc at the Mathematical Sciences Research Institute in Berkeley, California |
| Notes |
Print version record |
| Subject |
Curves, Algebraic.
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Curves, Plane.
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Geometry, Projective.
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MATHEMATICS -- Geometry -- General.
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MATHEMATICS -- Number Theory.
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Algebraic Geometry.
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Combinatoirics.
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Curves.
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Hilbert.
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Pencils.
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Topology.
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Curves, Algebraic
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Curves, Plane
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Geometry, Projective
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| Form |
Electronic book
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| ISBN |
9780429838255 |
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0429838255 |
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9780429838248 |
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0429838247 |
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