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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
- 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
- Hyperelliptic curve
- Klein quartic
- Modular curve
- 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
- 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
- Intersection theory
- Albanese variety
- Picard group
- Modular form
- Moduli space
- Modular equation
- Algebraic function
- Algebraic form
- Addition theorem
- Invariant theory
- Geometric invariant theory
- Toric variety
- Deformation theory
- Singular point, non-singular
- Singularity theory
- 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
- 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)
- 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
- 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
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