In high school algebra 1 and 2, we study the theory of single variable polynomial equations. In linear algebra, we study systems of linear equations, or polynomial equations with no exponents greater than 1. Algebraic Geometry combines these disciplines to study polynomial equations (in particular, their solutions) in many variables. This theory is useful as it is specific enough that we can compute with it, yet general enough that it applies to many problems we care about. The promised invitation will be extended by way of plane conics and cubics. After some definitions and preliminaries, we will review a family of classical results regarding plane conics (quadratic polynomials in two variables.
We will begin by classifying conics into families who’s members are “alike”. We will then leverage this classification to study the intersections of two conics. Our conclusion to this first act will be a detailed discussion of the number of conics passing through n points in the plane.
A follow up talk may consist of an attempt to generalize this theory to the case of plane cubics; polynomials of degree three in two variables. Through this endeavor we will introduce results from classical algebraic geometry; these may (if time permits) include Bézout’s theorem, intersection theory, and blowups; of course, for these developments, we must introduce projective geometry and some more advanced language. Finally, our third act will discuss in vague terms the position of stacks and moduli spaces in the picture, together with the categorical and algebraic foundation they necessitate. As such, these notes are incomplete.