# 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 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