《Enumerative Algebraic Geometry of Conics AMM awarded paper》.pdf

Enumerative Algebraic Geometry of Conics Andrew Bashelor, Amy Ksir, and Will Traves 1. INTRODUCTION. In 1848 Jakob Steiner, professor of geometry at the Univer- sity 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 enumera- tive problems in algebraic geometry. The tools developed to solve these problems have been used in many other situations and reveal deep and beautiful geometric phenom- ena. 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. Enumerative problems have a long history: many such problems were posed by the ancient Greeks. Enumerative geometry is also currently one of the most active areas of research in algebraic geometry, mainly due to a recent influx of ideas from string theory. For instance, mirror symmetry and Gromov-Witten theory are two hot m