What is Algebraic Geometry?

Assume you want to find all numbers x and y such that the cube of x equals the square of y. In mathematics, this is stated as finding the values for x and y that satisfy x3 – y2 = 0.

For example, x = 2 and y = 3 do not satisfy the equation since 23 – 32 = 8 – 9 = -1, which is different from 0. However, x = 0 and y = 0 satisfy the equation since 0*3 – 0*2 = 0. Another solution is x = 1 and y = 1. Assume that we are able to find all such solutions. These solutions, a set of pairs of numbers (x,y), called an algebraic curve, can be plotted in a coordinate system as in Figure 1. This simple example shows that these solutions often have an interesting geometry. The specific solutions (0,0), (1,1), and (1,-1) are marked as dots. The point (0,0) on this curve is discontinuous. Such points are interesting and are called singularities.

Figure 1: The algebraic curve of f(x) = x3 – y2

Remark. The reader who is familiar with complex numbers will see that we get complex solutions for negative x values. Due to the obvious difficulty of drawing complex numbers, we will restrict ourselves to drawing real solutions only.

Algebraic geometry is the study of curves like these. The name algebraic geometry is due to the fact that we study these curves using algebraic techniques. Geometry has been one of the main mathematical disciplines since its use in ancient Greece. Today, algebraic geometry is a vast discipline with a sophisticated technical language.