Pencils of Cubics and Algebraic Curves in the Real Projective Plane / Edition 1

by Séverine Fiedler - Le Touzé

Overview

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.

Product Details

ISBN-13:	9781138322578
Publisher:	Taylor & Francis
Publication date:	11/26/2018
Pages:	256
Product dimensions:	6.12(w) x 9.19(h) x (d)

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.

Table of Contents

I Rational pencils of cubics and configurations of six or seven points in RP 2

1 Points, lines and conics in the plane

2 Configurations of six points

3 Configurations of seven points

II Pencils of cubics with eight base points lying in convex position in RP 2

4 Pencils of cubics

5 Lists of conics

6 Link between lists and pencils

7 Pencils with reducible cubics

8 Classification of the pencils of cubics

9 Tabulars

10 Application to an interpolation problem

III Algebraic curves

11 Hilbert’s 16th problem

12 M -curves of degree 9

13 M -curves of degree 9 with deep nests

14 M -curves of degree 9 with four or three nests

15 M -curves of degree 9 or 11 with one non-empty oval

16 Curves of degree 11 with many nests

17 Totally real pencils of curves