| Description |
Bitangents are lines which are tangent to a curve at two points. The bitangents of a classical quartic are well understood, and a result originally due to Cayley tells us that there are always precisely 28 bitangents to a generic quartic plane curve. When looking at Tropical Geometry, the situation is different. We can still define a bitangent line, but a generic quartic curve has only 7 bitangents. The goal of this paper is to prove these results, and to provide an exposition of the necessary language and results on which they rely. To do this, we will first introduce the basic notions of Tropical Geometry. From there, we will give a proof of the classical result, giving special attention to the development the language and theory of divisors on a curve. Section 3 will introduce divisors on a tropical curve, focusing on chip-firing and the development of algorithms necessary for computation. From here, we have all the tools required to prove that a Tropical Curve has precisely 7 bitangent lines. The final section focuses on the relationship between the two results, areas of further possible research, and open questions in the field. |
