**Memoirs of the American Mathematical Society**

1997;
96 pp;
Softcover

MSC: Primary 14;

Print ISBN: 978-0-8218-0648-7

Product Code: MEMO/130/617

List Price: $46.00

Individual Member Price: $27.60

**Electronic ISBN: 978-1-4704-0206-8
Product Code: MEMO/130/617.E**

List Price: $46.00

Individual Member Price: $27.60

# Families of Curves in $\mathbb P^{3}$ and Zeuthen’s Problem

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*Robin Hartshorne*

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.

#### Table of Contents

# Table of Contents

## Families of Curves in $\mathbb P^{3}$ and Zeuthen's Problem

- Contents vii8 free
- §0. Introduction 110 free
- §1. Preliminaries 817 free
- §2. Families of Quadric Surfaces 1625
- §3. Degenerations of Cubic Surfaces 2837
- §4. Standard Form for Certain Deformations 4655
- §5. Local Picard Group of Some Normal Hypersurface Singularities 6776
- §6. Solution of Zeuthen's Problem 7887

#### Readership

Graduate students and research mathematicians interested in algebraic geometry.