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Families of Curves in \({\mathbb P}^3\) and Zeuthen's Problem
Robin Hartshorne, University of California, Berkeley, CA

Memoirs of the American Mathematical Society
1997; 96 pp; softcover
Volume: 130
ISBN-10: 0-8218-0648-3
ISBN-13: 978-0-8218-0648-7
List Price: US$46
Individual Members: US$27.60
Institutional Members: US$36.80
Order Code: MEMO/130/617
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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.


Graduate students and research mathematicians interested in algebraic geometry.

Table of Contents

  • Introduction
  • Preliminaries
  • Families of quadric surfaces
  • Degenerations of cubic surfaces
  • Standard form for certain deformations
  • Local Picard group of some normal hypersurface singularities
  • Solution of Zeuthen's problem
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