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Families of Curves in ${\mathbb P}^3$ and Zeuthen’s Problem
 
Robin Hartshorne University of California, Berkeley, Berkeley, CA
Families of Curves in P^3 and Zeuthen's Problem
eBook ISBN:  978-1-4704-0206-8
Product Code:  MEMO/130/617.E
List Price: $46.00
MAA Member Price: $41.40
AMS Member Price: $27.60
Families of Curves in P^3 and Zeuthen's Problem
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Families of Curves in ${\mathbb P}^3$ and Zeuthen’s Problem
Robin Hartshorne University of California, Berkeley, Berkeley, CA
eBook ISBN:  978-1-4704-0206-8
Product Code:  MEMO/130/617.E
List Price: $46.00
MAA Member Price: $41.40
AMS Member Price: $27.60
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1301997; 96 pp
    MSC: Primary 14;

    This book provides a negative solution to Zeuthen's problem, which was proposed as a prize problem in 1901 by the Royal Danish Academy of Arts and Sciences. The problem was to decide whether every irreducible family of smooth space curves admits limit curves which are stick figures, composed of lines meeting only two at a time.

    To solve the problem, the author makes a detailed study of curves on cubic surfaces in \({\mathbb P}^3\) and their possible degenerations as the cubic surface specializes to a quadric plus a plane or the union of three planes.

    Readership

    Graduate students and research mathematicians interested in algebraic geometry.

  • Table of Contents
     
     
    • Chapters
    • 0. Introduction
    • 1. Preliminaries
    • 2. Families of quadric surfaces
    • 3. Degenerations of cubic surfaces
    • 4. Standard form for certain deformations
    • 5. Local Picard group of some normal hypersurface singularities
    • 6. Solution of Zeuthen’s problem
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1301997; 96 pp
MSC: Primary 14;

This book provides a negative solution to Zeuthen's problem, which was proposed as a prize problem in 1901 by the Royal Danish Academy of Arts and Sciences. The problem was to decide whether every irreducible family of smooth space curves admits limit curves which are stick figures, composed of lines meeting only two at a time.

To solve the problem, the author makes a detailed study of curves on cubic surfaces in \({\mathbb P}^3\) and their possible degenerations as the cubic surface specializes to a quadric plus a plane or the union of three planes.

Readership

Graduate students and research mathematicians interested in algebraic geometry.

  • Chapters
  • 0. Introduction
  • 1. Preliminaries
  • 2. Families of quadric surfaces
  • 3. Degenerations of cubic surfaces
  • 4. Standard form for certain deformations
  • 5. Local Picard group of some normal hypersurface singularities
  • 6. Solution of Zeuthen’s problem
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.