Study of solution sets of systems of polynomial equations. Classical (Zariski-topological) algebraic geometry over a field k: affine/projective varieties, the ideal-variety correspondence (Nullstellensatz), regular and rational maps.…
algebraic-geometry
Affine variety
Common zero-locus of a family of polynomials: for S ⊆ k[x₁,…,x_n], V(S) = {p ∈ k^n : f(p) = 0 for all f ∈ S}. Equivalently, V(I) where I =…
Zariski topology
Topology on k^n (or on Spec R) whose closed sets are exactly the zero-loci V(I) of ideals I ⊆ k[x₁,…,x_n]. Coarser than any metric…
Hilbert's Nullstellensatz
For an algebraically-closed field k and an ideal J ⊆ k[x₁,…,x_n]: I(V(J)) = √J, the radical of J. In particular, V(J) = ∅ iff 1 ∈ J, and…
Projective variety
Common zero-locus of homogeneous polynomials in projective space ℙⁿ(k) = (k^{n+1} \ {0}) / k*. Homogeneity ensures the zero-set is…
Scheme
Locally-ringed space locally isomorphic to Spec(A) (affine scheme). Grothendieck EGA 1960s. Unifies varieties over ℂ with arithmetic…
Sheaf cohomology
Derived functors Hⁱ(X, 𝓕) of global sections; measures obstructions to extending local → global. Čech + Grothendieck resolutions; Serre…
Divisors & line bundles
Weil divisors = ℤ-combinations of codim-1 subvarieties; Cartier divisors = line bundles modulo trivial. Pic(X) group. Riemann-Roch theorem…
Étale cohomology
Grothendieck topology on schemes generalizing étale maps. ℓ-adic cohomology Hⁱ(X, ℚℓ); tool for Weil conjectures (Deligne 1974).
Algebraic stack
Generalization of scheme allowing moduli with automorphisms (moduli of curves ℳg, moduli of vector bundles). Deligne-Mumford + Artin stacks.
Moduli spaces
Parameter spaces for algebro-geometric objects (curves, vector bundles, Higgs bundles). Mumford GIT construction; Kontsevich moduli of…
Hilbert basis theorem
Polynomial ring k[x₁…xₙ] is Noetherian: every ideal is finitely generated.
Spec(A) as affine scheme
Contravariant equivalence CommRing^op ≃ AffSch sending A to its prime spectrum with structure sheaf. Points = prime ideals; generic points…
Proj(S) graded-ring construction
Projective scheme from graded ring S = ⊕ Sₙ: points = homogeneous primes not containing S₊. Yields ℙⁿ = Proj k[x₀…xₙ].
Serre duality
On a smooth projective variety X of dim n with dualising sheaf ωₓ: Hⁱ(X,F)* ≅ Hⁿ⁻ⁱ(X, F* ⊗ ωₓ). Specializes to Poincaré duality.
Riemann-Roch theorem (curves)
For divisor D on smooth projective curve X of genus g: ℓ(D) − ℓ(K−D) = deg D − g + 1. Controls linear systems; foundation of curve theory.
Hirzebruch-Riemann-Roch
χ(X, E) = ∫_X ch(E) · td(Tₓ). Generalises R-R to all dimensions via Chern character + Todd class.
Grothendieck-Riemann-Roch
For proper morphism f: X→Y: ch(f_* F) · td(Tᵧ) = f_*(ch(F) · td(Tₓ)). Relative R-R; birth of K-theory.
Chow ring A*(X)
Cycles mod rational equivalence, graded by codimension; intersection pairing A^p · A^q → A^{p+q}. Algebraic analogue of cohomology.
Intersection theory
Bezout's theorem, excess intersection, Gysin pullback, refined cycles (Fulton). Computes products in Chow ring geometrically.
Minimal model program (MMP)
Birational classification in dim ≥ 3 via K_X-negative extremal contractions and flips. Mori, Kawamata-Shokurov-Kollár; BCHM 2010 finite…
Hodge theory (algebraic)
For smooth projective X/ℂ: Hᵏ(X,ℂ) = ⊕_{p+q=k} H^{p,q}(X). Hodge structures, polarised variations, period map. Deligne's mixed Hodge…
Toric variety
Normal variety X with open dense torus T ↪ X and T-action extending translation. Encoded by fans in lattice N. Combinatorial geometry:…
Motives (Grothendieck)
Universal cohomology for algebraic varieties; Chow motives (effective/numerical). Voevodsky's derived category of mixed motives. Standard…
Weil conjectures
For smooth projective X/𝔽_q: Z(X,t) rational, functional equation, Riemann hypothesis |αᵢ| = q^{i/2}, Betti numbers from ℓ-adic. Deligne…
GAGA (Serre)
For projective X/ℂ: coherent algebraic and analytic sheaf categories equivalent; cohomology agrees. Serre 1956. Bridges algebra and complex…
Blowing up a subvariety
Bl_Z(X) replaces Z ⊂ X by its projectivized normal cone ℙ(N_{Z/X}). Resolves singularities (Hironaka 1964 char 0); fundamental birational…
Quasi-coherent & coherent sheaves
O_X-modules locally isomorphic to M̃ for A-module M (quasi-coherent) or f.g. M (coherent). Category Qcoh(X) is abelian; Coh(X) Noetherian…
Kähler differentials Ω^1_{X/k}
Universal module of derivations; sheaf of relative 1-forms. Smooth X: Ω^1 locally free of rank dim X. Cotangent complex L_{X/k} (Illusie).
Flat morphism / flat module
f: X → Y flat iff O_{X,x} flat over O_{Y,f(x)} (pure of relative dim). Preserves exact sequences; generic freeness; deformation-theoretic…
Smooth / étale / unramified morphism
Smooth = flat + geometrically regular fibres of equal dim; étale = smooth of rel dim 0; unramified = étale after base change to closed…
Gröbner basis
Finite generating set G ⊂ I with LT(G) generating LT(I) under a term order. Buchberger's algorithm 1965; effective ideal membership,…
Hodge conjecture (open)
On a smooth projective complex variety, every rational (p,p)-Hodge class is a ℚ-linear combination of classes of algebraic subvarieties.…
Bezout's theorem (plane curves)
Two plane projective curves of degrees d_1 and d_2 with no common component intersect in exactly d_1 · d_2 points, counted with…
Genus-degree formula
A smooth irreducible plane projective curve of degree d has geometric genus g = (d−1)(d−2)/2. Quadrics have g=0 (rational), cubics have…
Veronese embedding
The degree-d Veronese embedding sends P^n into P^N (with N = C(n+d, d)−1) by all degree-d monomials. A closed immersion realising O(d) as…
Segre embedding
Embeds the product P^m × P^n into P^{(m+1)(n+1)−1} via the outer-product coordinates x_i y_j. Image is cut out by the 2×2 minors of the…
Plücker embedding
Embeds the Grassmannian Gr(k, n) as a smooth projective variety inside P^{C(n,k)−1} via a basis wedge ([v_1 ∧ … ∧ v_k]). Image is cut out…
Noether normalization (algebraic geometry)
Every finitely generated reduced k-algebra A of Krull dimension d is a finite module over a polynomial subring k[t_1, …, t_d].…
Krull dimension (algebraic varieties)
For a topological space (taking irreducible closed subsets) or a ring (prime ideals), the Krull dimension is the supremum of lengths of…
Bertini's theorem
In characteristic zero (or more generally in separable settings), a generic hyperplane section of a smooth projective variety X is smooth. …