Search

3 results

14 - Algebraic Families, Dimension Counting, and Ruled Surfaces
Adam Sheffer, Bernard M. Baruch College, City University of New York
Book:

Polynomial Methods and Incidence Theory

Published online:

17 March 2022

Print publication:

24 March 2022, pp 204-224
- Chapter
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation
Summary

In this chapter we discuss advanced tools and techniques, which rely on additional concepts from algebraic geometry. These tools could be helpful for people who do research work in incidence theory and related topics. A reader who is new to this field might prefer to skip this chapter.
We sometimes wish to consider families of varieties, such as the set of circles in the plane or the set of planes in R^3 that not are incident to the origin. In this chapter, we rigorously define such families. We also generalize the idea of point-line duality to every family of varieties. We then see how these notions could be used to prove various results. In particular, we derive a new incidence bound and prove various properties of surfaces in R^3 and C^3.

Developable surfaces in Euclidean space
Part of
- Classical differential geometry
Vitaly Ushakov
Journal:

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics / Volume 66 / Issue 3 / June 1999

Published online by Cambridge University Press:

09 April 2009, pp. 388-402

Print publication:

June 1999
- Article
- - You have access
- PDF
- Export citation
The classical notion of a two-dimensional develpable surface in Euclidean three-space is extended to the case of arbitrary dimension and codimension. A collection of characteristic properties is presented. The theorems are stated with the minimal possible integer smoothness. The main tool of the investigation is Cartan's moving frame method.

Enumerating Rational Curves: The Rational Fibration Method
LUCIA CAPORASO, JOE HARRIS
Journal:

Compositio Mathematica / Volume 113 / Issue 2 / September 1998

Published online by Cambridge University Press:

04 December 2007, pp. 209-237

Print publication:

September 1998
- Article
- - You have access
- PDF
- Export citation
A new method to approach enumerative questions about rational curves on algebraic varieties is described. The idea is to reduce the counting problems to computations on the Néron-Severi group of a ruled surface. Applications include a short proof of Kontsevich‘s formula for plane curves and the solution of the analogous problem for the Hirzebruch surface F$_3$.