1st Edition

Pencils of Cubics and Algebraic Curves in the Real Projective Plane

By Séverine Fiedler - Le Touzé Copyright 2019
    256 Pages 107 B/W Illustrations
    by Chapman & Hall

    256 Pages 107 B/W Illustrations
    by Chapman & Hall

    256 Pages 107 B/W Illustrations
    by Chapman & Hall

    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.

    Rational pencils of cubics and configurations of six or seven points in RP²

    Points, lines and conics in the plane

    Configurations of six points

    Configurations of seven points

    Pencils of cubics with eight base points lying in convex position in RP²

    Pencils of cubics

    List of conics

    Link between lists and pencils

    Pencils with reducible cubics

    Classification of the pencils of cubics

    Tabulars

    Application to an interpolation problem

    Algebraic curves

    Hilbert’s 16th problem

    M-curves of degree 9

    M-curves of degree 9 with deep nests

    M-curves of degree 9 with four or three nests

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

    Curves of degree 11 with many nests

    Totally real pencils of curves

    Biography

    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.