nLab algebraic geometry - contents

Classical algebraic geometry

• algebraic equation, semialgebraic set
• maximal spectrum, Zariski topology
• locally ringed space
• affine variety, algebraic variety
• (quasi)projective variety
• algebraic curve, elliptic curve
• regular map, rational map
• birational geometry
• resolution of singularities
• quasicoherent sheaf, coherent sheaf
• Kähler manifold, Hodge theory
• normal variety
• blowup, tangent cone
• Kähler differential
• geometric invariant theory
• algebraic group, abelian variety

Algebraic schemes

• prime spectrum, affine scheme
• projective scheme
• open subscheme, closed subscheme
• reduced scheme, noetherian scheme
• regular scheme, quasiseparated scheme
• relative scheme, proper morphism
• separated morphism of schemes
• proper morphism of schemes
• flat topology, etale topology
• quasicompact morphism, fpqc topology
• Picard scheme, Quot scheme, Hilbert scheme
• formally smooth morphism, crystal

Cohomology theories

• Weil cohomology theory, Weil conjecture
• Hodge conjecture, standard conjectures
• motive, noncommutative motive
• homological mirror symmetry

Generalizations

• ind-scheme, formal scheme
• algebraic space
• Artin stack
• Deligne-Mumford stack
• tropical geometry
• absolute algebraic geometry
• noncommutative algebraic geometry
• noncommutative scheme
• derived algebraic geometry
• Landau-Ginzburg model
• semiorthogonal decomposition