Enumerative Algebraic Geometry: Counting Conics

摘要

Projectiles follow parabolic paths and planets move in elliptical orbits. Circles, hyperbolas, parabolas and ellipses are curves that are so abundant in nature, engineering, and art that we cannot help but notice them. Each of these curves is an example of a conic. In 1848, the mathematician Jacob Steiner posed a famous question: 'How many conics are tangent to five fixed conics.' Steiner claimed to have solved the problem and he gave the answer 7776. The solution was accepted as valid for sixteen years. When the problem was revisited in 1864, the mathematician Michel Chasles realized that Steiner had miscounted the true number of conics that satisfied the conditions. Not all conics are smooth plane curves. Singular conics are curves whose defining polynomials are reducible to the product of two linear factors. These conics can be represented as either a pair of crossed lines or a line of multiplicity two. Steiner failed to account for the degenerate conics that can be represented as double line. He fell victim to what algebraic geometers call excess intersection. This Trident project is centered on understanding how excess intersection affects problems of enumeration involving plane conics. Research was focused on finding the solutions to twenty-one variations of Steiner's problem. These problems were solved by examining the blowup of the space of conics along the set of double lines and excuting computations in what is known as the Chow Ring. These methods provide not only tangible numerical results but help to illuminate the rich underlying geometry of these fundamental problems.