Intro

Algebraic Geometry: Intro

Meeting algebraic geometry through polynomials and curves (high-school to first-year university level)

What you will learn

Algebraic geometry is the field that "captures shapes as the zero sets of polynomial equations". In the Intro level we begin with concrete plane curves and practice describing circles, ellipses, hyperbolas, parabolas, and the like in the language of polynomials. The "second-degree curves" learned in high school are in fact all entrances to algebraic geometry.

Prerequisites

  • High-school Mathematics II, expressions and equations (products of polynomials, factoring)
  • High-school Mathematics II, figures and equations (equations of lines and circles)
  • High-school Mathematics III, second-degree curves (ellipse, hyperbola, parabola) — optional

Contents

  1. Chapter 1 What Algebraic Geometry Is

    The viewpoint of capturing shapes by zero sets of polynomials, and the relation between classical and modern AG

  2. Chapter 2 Polynomials and Plane Algebraic Curves

    The style of writing lines, circles, cubic curves, etc. as a polynomial $f(x, y) = 0$

  3. Chapter 3 The Algebraic Description of Conic Sections

    Treating ellipses, hyperbolas, and parabolas uniformly as second-degree curves $ax^2 + bxy + cy^2 + \cdots = 0$

Goals

  • Understand the basic viewpoint of algebraic geometry: "zero set of a polynomial = a shape"
  • Be able to express lines, circles, and second-degree curves in the plane by polynomials
  • Understand that the classification of conic sections is determined by the discriminant $b^2 - 4ac$
  • Feel how the figures of high-school mathematics become an entrance to algebraic geometry