List of algebraic geometry topics

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, multi-way projective space
• 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
• Edwards curve
• Genus (mathematics)
• Riemann surface
• Riemann–Hurwitz formula
• Riemann–Roch theorem
• Abelian integral
• Differential of the first kind
• Jacobian variety
  • Generalized Jacobian
• Hurwitz's automorphisms theorem
• Clifford's theorem
• Gonality of an algebraic curve
• Weil's 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
  • Quasiprojective variety
    • Gröbner basis
  • Canonical bundle
  • Complete intersection
  • Serre duality
  • Arithmetic genus, geometric genus, irregularity
• Tangent space, Zariski tangent space
• Function field
• Ample vector bundle
• Linear system of divisors
• Birational geometry
  • Blowing up
  • Rational variety
  • Unirational variety
• Intersection number
  • Intersection theory
  • Serre's multiplicity conjectures
• Albanese variety
• Picard group
• Pluricanonical ring
• 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 geometry
• 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
  • Borel subgroup
  • Parabolic subgroup
  • Radical of an algebraic group
  • Unipotent radical
  • Lie-Kolchin theorem
  • Mumford conjecture
• Abelian variety
  • Theta function
• Grassmannian
• Flag manifold
• Algebraic torus
• Weil restriction
• Differential Galois theory

Contemporary foundations

Main article glossary of scheme theory

Commutative algebra

• Prime ideal
• Valuation (mathematics)
• Regular local ring
• Regular sequence (algebra)
• 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
• Hirzebruch–Riemann–Roch theorem
• Grothendieck–Riemann–Roch theorem
• Coherent duality
• Dévissage

Schemes

• Affine scheme
• Scheme
• Glossary of scheme theory
• Éléments de géométrie algébrique
• Grothendieck's Séminaire de géométrie algébrique
• Flat morphism
• Finite morphism
• Quasi-finite morphism
• Group scheme
• Semistable elliptic curve
• Grothendieck's relative point of view

Category theory

• Grothendieck topology
• Topos
• Descent (category theory)
  • Grothendieck's Galois theory
• Algebraic stack
• Gerbe
• Étale cohomology
• Motive (mathematics)
• Motivic cohomology
• Homotopical algebra

Algebraic geometers

• Niels Henrik Abel
• Carl Gustav Jakob Jacobi
• Jakob Steiner
• Julius Plücker
• Bernhard Riemann
• William Kingdon Clifford
• Italian school of algebraic geometry
  • Guido Castelnuovo
  • Francesco Severi
• Solomon Lefschetz
• Oscar Zariski
• Erich Kähler
• W. V. D. Hodge
• Kunihiko Kodaira
• André Weil
• Jean-Pierre Serre
• Alexander Grothendieck
• David Mumford
• Igor Shafarevich
• Heisuke Hironaka
• Shigefumi Mori
• Vladimir Voevodsky