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
• Real projective plane
• Real projective space
• Segre embedding of a product of projective spaces
• Rational normal curve

  [Algebraic curve]s

• 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
• 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 surface]s

• 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 manifold]s

• Kähler manifold
• Calabi–Yau manifold
• Stein manifold
• Hodge theory
• Hodge cycle
• Hodge conjecture
• Algebraic geometry and analytic geometry
• Mirror symmetry

  [Algebraic group]s

• 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
• Group scheme
• Abelian variety
• *Theta function
• Grassmannian
• Flag manifold
• Weil restriction
• Differential Galois theory

  Contemporary foundations

[Commutative algebra]

• Prime ideal
• Valuation
• 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
• Finite morphism
• Quasi-finite morphism
• Proper morphism
• Semistable elliptic curve
• Grothendieck's relative point of view
• Smooth scheme

  [Category theory]

• Grothendieck topology
• Topos
• Descent
• *Grothendieck's Galois theory
• Algebraic stack
• Gerbe
• Étale cohomology
• Motive
• 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