An Introduction to Algebraic Geometry: Playing with Plane Curves and Projective Space
Notes (from "Undergraduate Algebraic Geometry" by Miles Reid)
Chapter 1.1-1.6: Parametrizations, Rational Points, and Classification of Conics
In this section we start by doing some examples of simple parametrizations of curves in R^2. In particular, curves which are cut out by degree two polynomial equations. One of the natural questions posed in number theory is: When given an expression, what are all the integral solutions? In particular, the first problem we explore is using parametrizations of the unit circle to find all integral Pythagorean triples. Via very simple calculations this problem can be solved very nicely using geometric methods. This method can then generalize to many other similar problems like integral points of ellipses. With this example to get us used to working in charts, we then turn to the slightly more complicated problem of classifying all spaces in R^3 which can be cut out by a quadratic polynomial in two variables. Using techniques of homogenization and translating the problem into the projective context we see that the tools of modern algebra, specifically the correspondence of symmetric bilinear forms and homogeneous quadratics in three variables, we find that the seemingly mysterious classification of conic sections from high school math is nothing but a classification by signatures of symmetric 3x3 matrices! This kicks off our exciting study of algebraic geometry with the solutions of two very different flavors of problems: 1) Number theoretic existence problems and 2) Classification of geometric objects using projective geometry.