Enumerative Algebraic Geometry of Conics
Abstract
In 1848 Jakob Steiner, professor of geometry at the University of Berlin, posed the following problem [19]: Given five conics in the plane, are there any conics that are tangent to all five? If so, how many are there? Problems that ask for the number of geometric objects with given properties are known as enumerative problems in algebraic geometry. The tools developed to solve these problems have been used in many other situations and reveal deep and beautiful geometric phenomena. In this expository paper, we describe the solutions to several enumerative problems involving conics, including Steiner's problem. The results and techniques presented here are not new; rather, we use these problems to introduce and demonstrate several of the key ideas and tools of algebraic geometry. The problems we discuss are the following: Given p points, l lines, and c conics in the plane, how many conics are there that contain the given points, are tangent to the given lines, and are tangent to the given conics? It is not even clear a priori that these questions are well-posed. The answers may depend on which points, lines, and conics we are given. Nineteenth and twentieth century geometers struggled to make sense of these questions, to show that with the proper interpretation they admit clean answers, and to put the subject of enumerative algebraic geometry on a firm mathematical foundation. Indeed, Hilbert made this endeavor the subject of his fifteenth challenge problem.
Document Details
- Document Type
- Technical Report
- Publication Date
- Oct 01, 2008
- Accession Number
- ADA518580
Entities
People
- Amy Ksir
- Andrew Bachelor
- Will Traves
Organizations
- United States Naval Academy