Enumerative Algebraic Geometry of Conics

Abstract

In this expository paper, we describe the solutions to several enumerative problems involving conies, 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, / lines, and c conies in the plane, how many conies are there that contain the given points, are tangent to the given lines, and are tangent to the given conies? It is not even clear a priori that these questions are well-posed. The answers may depend on which points, lines, and conies 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.