Pencils of Cubics and Algebraic Curves in the Real Projective Plane

Séverine Fiedler - Le Touzé

• Publisher: Chapman and Hall/CRC
• Year: 2018
• Print ISBN: 9781138590519
• eBook ISBN: 9780429838248
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

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.

Contents

Preface

List of Figures

List of Tables

Acknowledgments

Contributors

Symbols

I Rational cubics and configurations of six or seven 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

II Pencils of cubics with eight base points lying inconvex position in RP2

4 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

8 Classification of the pencils of cubics

8.1 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

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

13.1 Statement of the results and first proofs

13.2 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

Preface

“C’est à la fois très simple et très compliqué.” Capitaine Haddock, Tintin au Pays de L’or Noir

Two points in the real projective plane determine a line, five points determine a conic, and nine points a cubic. What objects do n ≤ 9 generic points determine? In addition to non-singular curves, let us allow ...

Table of contents

• Cover
• Halftitle
• Title Page
• Copyright Page
• Table of Contents