Algebraic Geometry

Algebraic Geometry — where polynomials and shapes meet

About this series

Algebraic geometry is the field of mathematics that views shapes as the zero sets of polynomial equations. Beginning with the conic sections of ancient Greece, through 19th-century projective geometry, the classical theory of the Italian school, and Grothendieck's scheme theory in the latter half of the 20th century, it occupies a central place in mathematics as a whole.

This series starts from the conic sections of high-school mathematics, passes through affine and projective varieties, and advances step by step to Grothendieck's scheme theory. Four levels:

Learn by level

Intro

High-school to first-year university level. Polynomials and plane algebraic curves, the algebraic description of conic sections.

  • What algebraic geometry is
  • Polynomials and plane algebraic curves
  • The algebraic description of conic sections (ellipse, hyperbola, parabola)

Basic

First- to second-year university level. An overview of classical algebraic geometry.

  • Classical algebraic geometry

Intermediate

Second- to third-year university level. Algebraic varieties and their correspondence with the ideals and coordinate rings of polynomial rings.

  • Affine algebraic varieties
  • Projective space and projective varieties
  • The Zariski topology and irreducible decomposition

Advanced

Graduate level. Scheme theory and modern algebraic geometry.

  • Grothendieck's algebraic geometry (schemes)

Learning roadmap

  1. Intro: first get used to the viewpoint of capturing shapes by polynomials $f(x, y) = 0$. It reads as an extension of Japanese high-school mathematics II/III (conic sections moved to Mathematics C in the curriculum in force since 2022).
  2. Basic: survey the vocabulary of classical algebraic geometry (affine varieties, coordinate rings). The Intermediate level builds on Intro, so this stage is optional and may be skipped.
  3. Intermediate: rigorously study the dual correspondence between ideals of polynomial rings and algebraic varieties (Hilbert's basis theorem, the Nullstellensatz), projective space, and the Zariski topology.
  4. Advanced: with Grothendieck's scheme theory, move on to the framework "any commutative ring → a geometric object". A springboard to arithmetic geometry.

Prerequisites:

  • Intro → Intermediate: linear algebra + the basics of ring theory (Algebra, Intermediate) + the rudiments of topology (open and closed sets, irreducibility)
  • Intermediate → Advanced: commutative ring theory + category theory + topology