Classical topics Affine space Projective space Projective plane Algebraic curve Elliptic curve Elliptic function Elliptic integral Hyperelliptic curve Klein quartic Bézout's theorem Genus (mathematics) Riemann surface Riemann-Roch theorem Algebraic variety Quadric Dimension of an algebraic variety Hilbert's Nullstellensatz Elimination theory Serre duality Tangent space Function field Birational geometry Algebraic group Abelian variety Grassmannian Flag manifold Algebraic torus Weil restriction Intersection number Modular form Moduli space Invariant theory Algebraic form Kähler manifold Calabi-Yau manifold Weil conjectures Hodge conjecture Foundations Commutative algebra Prime ideal Valuation (mathematics) Spectrum of a ring Affine scheme Scheme Zariski topology Sheaf Locally ringed space Grothendieck topology Topos The geometers Niels Henrik Abel Carl Gustav Jakob Jacobi Julius Plucker Bernhard Riemann William Kingdon Clifford Italian school of algebraic geometry Solomon Lefschetz Oscar Zariski Erich Kähler W. V. D. Hodge André Weil Jean-Pierre Serre Alexander Grothendieck Heisuke Hironaka Vladimir Voevodsky