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

Robin Hartshorne University of California, Berkeley, Berkeley, CA
Available Formats:
Electronic ISBN: 978-1-4704-0206-8
Product Code: MEMO/130/617.E
List Price: $46.00 MAA Member Price:$41.40
AMS Member Price: $27.60 Click above image for expanded view Families of Curves in${\mathbb P}^3$and Zeuthen’s Problem Robin Hartshorne University of California, Berkeley, Berkeley, CA Available Formats:  Electronic 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.

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
• Requests

Review Copy – for reviewers who would like to review an AMS book
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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.

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 reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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