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