List of algebraic geometry topics

Classical topics in projective geometry

Affine space

Projective space

Projective line, cross-ratio

Projective plane

Line at infinity

Complex projective plane

Complex projective space

Plane at infinity, hyperplane at infinity

Projective frame

Projective transformation

Fundamental theorem of projective geometry

Duality (projective geometry)

Real projective plane

Real projective space

Segre embedding of a product of projective spaces

Rational normal curve

Algebraic curves

Conics, Pascal's theorem, Brianchon's theorem

Twisted cubic

Elliptic curve, cubic curve

Elliptic function, Jacobi's elliptic functions, Weierstrass's elliptic functions

Elliptic integral

Complex multiplication

Weil pairing

Hyperelliptic curve

Klein quartic

Modular curve

Modular equation

Modular function

Modular group

Supersingular primes

Fermat curve

Bézout's theorem

Brill–Noether theory

Genus (mathematics)

Riemann surface

Riemann–Hurwitz formula

Riemann–Roch theorem

Abelian integral

Differential of the first kind

Jacobian variety

Generalized Jacobian

Moduli of algebraic curves

Hurwitz's theorem on automorphisms of a curve

Clifford's theorem on special divisors

Gonality of an algebraic curve

Weil reciprocity law

Goppa code

Algebraic surfaces

Enriques–Kodaira classification

List of algebraic surfaces

Ruled surface

Cubic surface

Veronese surface

Del Pezzo surface

Rational surface

Enriques surface

K3 surface

Hodge index theorem

Elliptic surface

Surface of general type

Zariski surface

Algebraic geometry: classical approach

Algebraic variety

Hypersurface

Quadric

Dimension of an algebraic variety

Hilbert's Nullstellensatz

Complete variety

Elimination theory

Gröbner basis

Projective variety

Quasiprojective variety

Canonical bundle

Complete intersection

Serre duality

Arithmetic genus, geometric genus, irregularity

Tangent space, Zariski tangent space

Function field of an algebraic variety

Ample line bundle

Ample vector bundle

Linear system of divisors

Birational geometry

Blowing up

Rational variety

Unirational variety

Ruled variety

Kodaira dimension

Canonical ring

Minimal model program

Intersection theory

Intersection number

Chow ring

Chern class

Serre's multiplicity conjectures

Albanese variety

Picard group

Modular form

Moduli space

Modular equation

J-invariant

Algebraic function

Algebraic form

Addition theorem

Invariant theory

Symbolic method of invariant theory

Geometric invariant theory

Toric variety

Deformation theory

Singular point, non-singular

Singularity theory

Newton polygon

Weil conjectures

Complex manifolds

Kähler manifold

Calabi–Yau manifold

Stein manifold

Hodge theory

Hodge cycle

Hodge conjecture

Algebraic geometry and analytic geometry

Mirror symmetry

Algebraic groups

Identity component

Linear algebraic group

Additive group

Multiplicative group

Algebraic torus

Reductive group

Borel subgroup

Parabolic subgroup

Radical of an algebraic group

Unipotent radical

Lie-Kolchin theorem

Haboush's theorem (also known as the Mumford conjecture)

Group scheme

Abelian variety

Theta function

Grassmannian

Flag manifold

Weil restriction

Differential Galois theory

Commutative algebra

Prime ideal

Valuation (algebra)

Krull dimension

Regular local ring

Regular sequence

Cohen–Macaulay ring

Gorenstein ring

Koszul complex

Spectrum of a ring

Zariski topology

Kähler differential

Generic flatness

Irrelevant ideal

Sheaf theory

Locally ringed space

Coherent sheaf

Invertible sheaf

Sheaf cohomology

Coherent sheaf cohomology

Hirzebruch–Riemann–Roch theorem

Grothendieck–Riemann–Roch theorem

Coherent duality

Dévissage

Schemes

Affine scheme

Scheme

Éléments de géométrie algébrique

Grothendieck's Séminaire de géométrie algébrique

Flat morphism

Finite morphism

Quasi-finite morphism

Proper morphism

Semistable elliptic curve

Grothendieck's relative point of view

Smooth scheme

Category theory

Grothendieck topology

Topos

Descent (category theory)

Grothendieck's Galois theory

Algebraic stack

Gerbe

Étale cohomology

Motive (algebraic geometry)

Motivic cohomology

A¹ homotopy theory

Homotopical algebra

Algebraic geometers

Niels Henrik Abel

Carl Gustav Jacob Jacobi

Jakob Steiner

Julius Plücker

Arthur Cayley

Bernhard Riemann

Max Noether

William Kingdon Clifford

David Hilbert

Italian school of algebraic geometry

Guido Castelnuovo

Federigo Enriques

Francesco Severi

Solomon Lefschetz

Oscar Zariski

W. V. D. Hodge

Kunihiko Kodaira

André Weil

Jean-Pierre Serre

Alexander Grothendieck

Igor Shafarevich

Heisuke Hironaka

David Mumford

Phillip Griffiths

Pierre Deligne

Shigefumi Mori

Vladimir Voevodsky