List of algebraic geometry topics

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

Contents

1 Classical topics in projective geometry 2 Algebraic curves 3 Algebraic surfaces 4 Algebraic geometry: classical approach 5 Complex manifolds 6 Algebraic groups 7 Contemporary foundations 7.1 Commutative algebra 7.2 Sheaf theory 7.3 Schemes 7.4 Category theory 8 The geometers

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

Algebraic curves

• Conics, Pascal's theorem, Brianchon's theorem
• 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 prime
• 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
• Hurwitz's automorphisms theorem

Algebraic surfaces

• Ruled surface
• Cubic surface
• Veronese surface
• Del Pezzo surface
• K3 surface
• Hodge index theorem

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
  • Serre duality
• Tangent space, Zariski tangent space
• Function field
• Ample vector bundle
• Linear system of divisors
• Birational geometry
• Intersection number
  • Intersection theory
  • Serre's multiplicity conjectures
• 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
  • 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

Sheaf theory

• Locally ringed space
• Coherent sheaf
• Invertible sheaf
• Sheaf cohomology
• Grothendieck-Riemann-Roch theorem

Schemes

• Affine scheme
• Scheme
• Glossary of scheme theory
• Éléments de géométrie algébrique
• Grothendieck's Séminaire de géométrie algébrique
• Flat morphism
• Group scheme
• Semistable elliptic curve

Category theory

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

The 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
• André Weil
• Jean-Pierre Serre
• Alexander Grothendieck
• Igor Shafarevich
• Heisuke Hironaka
• Shigefumi Mori
• Vladimir Voevodsky