### IV.4 Algebraic Geometry

### *János Kollár*

### 1 Introduction

Succinctly put, algebraic geometry is the study of geometry using polynomials and the investigation of polynomials using geometry.

Many of us were taught the beginnings of algebraic geometry in high school, under the name “analytic geometry.” When we say that *y* = *mx* + *b* is the equation of a line *L*, or that *x*^{2} + *y*^{2} = *r*^{2} describes a circle *C* of radius *r*, we establish a basic connection between geometry and algebra.

If we want to find the points where the line *L* and the circle *C* intersect, we just substitute *mx* + *b* for *y* in the circle equation to get *x*^{2} + (*mx* + *b*)^{2} = *r*^{2} and solve the resulting quadratic equation to obtain the *x* coordinates of the two intersection points.

This simple example ...