Algebraic Geometry, An Introduction

Introduction

As the name suggests, algebraic geometry is a marriage of algebra and geometry.  This should not be confused with analytic geometry, which is also a marriage of algebra and geometry.  The latter starts with a shape, like a circle, and asks how to describe that shape as an equation, e.g. x2+y2 = 1.  High school students build quadratic equations in two variables to represent various conic sections, thus dabbling in analytic geometry.  In contrast, algebraic geometry, an advanced topic in graduate mathematics, begins with an equation and builds the corresponding shape in n space.  The geometry of the shape often constrains the solutions to the equation.  Furthermore, the "shape" need not exist in real space.  Any field is fair game.  For instance, we might be finding solutions to x2+y2 = 1 over the finite field of order 343.  The solution space is still a circle, of sorts, existing in another world.  It is one dimensional, in a two dimensional space, and we can make that rigorous.  This will become clearer as we proceed.