This is a list of algebraic geometry topics, by Wikipedia page.

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
• Algebraic geometry codes

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 (algebraic geometry)
  • Dimension of an algebraic variety
  • Hilbert's Nullstellensatz
  • Complete variety
  • Elimination theory
  • Gröbner basis
  • Projective variety
  • Quasiprojective variety
  • Canonical bundle
  • Complete intersection
  • Serre duality
  • Spaltenstein variety
  • 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
  • Resolution of singularities
  • 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

• Linear algebraic group
  • Additive group
  • Multiplicative group
  • Algebraic torus
  • Reductive group
  • Borel 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

Contemporary foundations

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
• Fiber product of schemes
• Flat morphism
• Smooth scheme
• Finite morphism
• Quasi-finite morphism
• Proper morphism
• Semistable elliptic curve
• Grothendieck's relative point of view
• Hilbert scheme

Category theory

• Grothendieck topology
• Topos
• Derived category
• 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
• Sir Michael Atiyah
• Kunihiko Kodaira
• André Weil
• Jean-Pierre Serre
• Alexander Grothendieck
• Friedrich Hirzebruch
• Igor Shafarevich
• Heisuke Hironaka
• Shreeram S. Abhyankar
• Pierre Samuel
• C.P. Ramanujam
• David Mumford
• Michael Artin
• Phillip Griffiths
• Pierre Deligne
• Yuri Manin
• Shigefumi Mori
• Vladimir Drinfeld
• Vladimir Voevodsky
• Claire Voisin
• János Kollár
• Caucher Birkar
• Burt Totaro
• Patrick Brosnan
• Robin Hartshorne
• Joe Harris