Algebraic Geometry Foundations

Algebraic geometry studies geometric objects defined by polynomial equations and the maps between them. Classically the objects were varieties — zero loci of polynomials in affine or projective space over a field. Grothendieck’s mid-twentieth-century reformulation generalised them to schemes, geometric objects locally modelled on the spectrum of an arbitrary commutative ring, and reorganised the theory around sheaves and their cohomology. The modern subject is the central language of large parts of number theory (arithmetic geometry, Galois representations, the Langlands program), of complex geometry, of representation theory (geometric Langlands, geometric representation theory), of mirror symmetry and string theory, and of the homotopical algebra surrounding derived and ∞-categories. This note covers the classical foundations through schemes, sheaves, divisors, line bundles, cohomology, intersection theory, the Riemann-Roch theorem, moduli, and an overview of derived categories — enough to support reading Hartshorne, the Stacks Project, and current research.

1. Commutative algebra preliminaries

Algebraic geometry rests on commutative ring theory. A working command of these objects is assumed:

Commutative ring with unity, ring homomorphism, ideal, prime ideal, maximal ideal, quotient ring.
Modules, exact sequences, tensor product, Hom, flatness.
Noetherian conditions. A ring is Noetherian if every ideal is finitely generated, equivalently every ascending chain of ideals stabilises. The Hilbert basis theorem (Hilbert 1890): if R is Noetherian, so is R[x]. Applied iteratively, polynomial rings over fields or over Z in finitely many variables are Noetherian.
Local rings. A ring with a unique maximal ideal m; the residue field is R/m. Localisation R_p of a ring R at a prime p inverts the multiplicative set R \ p, producing a local ring with maximal ideal pR_p. Geometrically, localisation at p corresponds to passing to the stalk at the point p ∈ Spec R.
Integral closure and normal rings. A ring R is normal if it equals its integral closure in its field of fractions.
Krull dimension. The supremum of lengths of strict chains of prime ideals. For a polynomial ring k[x_1,…,x_n] over a field, Krull dimension equals n.
Regular local rings. Local ring (R, m, k) with dim_k(m/m^2) = Krull dim R. Geometrically: nonsingular points.
Cohen-Macaulay, Gorenstein, complete intersection. A hierarchy of regularity-style conditions on rings.

References: Atiyah-Macdonald Introduction to Commutative Algebra, Eisenbud Commutative Algebra with a View toward Algebraic Geometry, Matsumura Commutative Ring Theory.

2. Affine varieties

Fix an algebraically closed field k (often C). Affine n-space is A^n_k = k^n as a set; the polynomial ring k[x_1,…,x_n] is its ring of regular functions.

2.1 Algebraic sets

For a subset T ⊆ k[x_1,…,x_n], the vanishing locus is V(T) = {p ∈ A^n : f(p) = 0 for all f ∈ T}. V(T) depends only on the ideal generated by T. An algebraic set is V(I) for some ideal I.

For S ⊆ A^n, the ideal of S is I(S) = {f ∈ k[x_1,…,x_n] : f(p) = 0 for all p ∈ S}.

2.2 Hilbert’s Nullstellensatz

The cornerstone theorem (Hilbert 1893). Over algebraically closed k, for any ideal I in k[x_1,…,x_n]:

I(V(I)) = √I = {f : f^n ∈ I for some n ≥ 1}.

Consequences:

Weak Nullstellensatz. A proper ideal I ⊊ k[x_1,…,x_n] has nonempty vanishing locus V(I) ≠ ∅. Equivalently, the maximal ideals of k[x_1,…,x_n] correspond exactly to points of A^n: m_p = (x_1 - a_1, …, x_n - a_n) for p = (a_1,…,a_n).
Correspondence. Bijection {radical ideals of k[x_1,…,x_n]} ↔ {algebraic sets in A^n}, with prime ideals ↔ irreducible algebraic sets (called affine varieties), maximal ideals ↔ points.

An affine variety is an irreducible algebraic set; equivalently, V(p) for a prime ideal p. The coordinate ring of V(p) is A(V) = k[x_1,…,x_n]/p; it is a finitely generated reduced k-algebra that is an integral domain.

2.3 Zariski topology

The Zariski topology on A^n has closed sets the algebraic sets. Open sets are complements of vanishing loci. Properties:

Zariski topology is Noetherian (descending chains of closed sets stabilise).
Highly non-Hausdorff: every two non-empty Zariski-open sets intersect.
Quasi-compact (every open cover has a finite subcover).
The Zariski topology on a variety differs starkly from the classical/Euclidean topology when k = C: the classical topology is finer and Hausdorff.

2.4 Morphisms of affine varieties

A regular map φ: V → W between affine varieties is given by polynomials in the coordinates of W, viewed as functions on V. Equivalently, φ corresponds to a k-algebra homomorphism φ*: A(W) → A(V) (contravariant). This sets up an antiequivalence of categories:

{affine varieties over k} ↔ {finitely generated reduced k-algebras that are integral domains}^op.

The function field of V is K(V) = Frac(A(V)). The dimension of V is the transcendence degree of K(V) over k, equal to the Krull dimension of A(V).

2.5 Local rings and tangent spaces

The local ring at p ∈ V is A(V)_m_p, the localisation. Its maximal ideal is m_p · A(V)_m_p. The Zariski cotangent space at p is m_p / m_p^2; the tangent space is its k-dual. Dimension comparison: V is smooth (regular) at p if dim_k T_p V = dim V; otherwise singular.

The Jacobian criterion: if V = V(f_1,…,f_r) and p ∈ V, then p is smooth iff the Jacobian matrix (∂f_i/∂x_j)|_p has rank n - dim V.

3. Projective varieties

Projective n-space P^n_k is the set of lines through the origin in k^{n+1}, parametrised by homogeneous coordinates [x_0 : x_1 : … : x_n] (defined up to nonzero scalar). The graded ring k[x_0,…,x_n] is its coordinate ring; homogeneous ideals (generated by homogeneous polynomials) cut out closed subsets.

3.1 Projective Nullstellensatz

The maximal homogeneous ideal m_+ = (x_0,…,x_n) is the irrelevant ideal. For a homogeneous ideal I, V_+(I) ⊆ P^n is empty iff √I ⊇ m_+. Projective Nullstellensatz: I_+(V_+(I)) = √I if √I ≠ m_+. The Proj construction generalises: Proj of a graded ring is a topological space with a sheaf, giving a scheme structure.

3.2 Affine charts and the Segre embedding

P^n is covered by n+1 affine charts U_i = {x_i ≠ 0} each isomorphic to A^n via dehomogenisation. Products: P^m × P^n embeds in P^{(m+1)(n+1)-1} via the Segre map [x_i] × [y_j] ↦ [x_iy_j]. Products of projective varieties are projective.

3.3 Veronese embedding

The d-th Veronese embedding P^n ↪ P^{N} (with N = ((n+d) choose d) - 1) is given by all monomials of degree d in the homogeneous coordinates. The image is the Veronese variety; it realises the degree-d Plücker-like map.

3.4 Projective curves and intersection in P^2

A projective curve in P^2 is the vanishing locus of a homogeneous polynomial of degree d. Two distinct irreducible curves of degrees d_1 and d_2 intersect in exactly d_1 d_2 points counted with multiplicity (Bézout’s theorem, in P^2 over an algebraically closed field).

3.5 Grassmannians, flag varieties

The Grassmannian Gr(k,n) parametrises k-dimensional subspaces of an n-dimensional vector space. It embeds in P^{(n choose k)-1} via the Plücker embedding (sending a subspace to the wedge product of a basis). Cut out by the Plücker relations (quadratic). Generalisation: flag varieties G/P parametrise flags in a vector space; for G = GL_n, P = parabolic stabilising a partial flag.

3.6 Toric varieties

A toric variety is a normal variety with an open dense torus action whose action extends to the whole variety. Fulton Introduction to Toric Varieties. Constructed from fans (combinatorial collections of rational polyhedral cones) in a lattice. The connection between geometry and combinatorics makes toric geometry a major source of computable examples; underlies tropical geometry and parts of mirror symmetry.

4. Schemes

Grothendieck’s redefinition (Éléments de Géométrie Algébrique, 1960–1967, with the Séminaire de Géométrie Algébrique 1960–1969 lecture series) replaces varieties with schemes — geometric objects locally modelled on the spectrum of an arbitrary commutative ring. The shift permitted geometry with nilpotents, geometry over arbitrary base rings (including Z), and geometry with non-reduced subschemes (“fat points”). The theory powered the proofs of the Weil conjectures (Deligne 1974) and most subsequent number theory.

4.1 Spec of a ring

For a commutative ring R, Spec R is the set of prime ideals of R. The Zariski topology has closed sets V(I) = {p : p ⊇ I}. A basis of open sets is the principal opens D(f) = {p : f ∉ p}. The structure sheaf O = O_Spec R assigns to D(f) the localisation R_f = R[1/f]; the stalk at p is the localisation R_p.

The pair (Spec R, O) is a locally ringed space — each stalk is a local ring. The functor R ↦ Spec R from commutative rings (opposite category) to locally ringed spaces is fully faithful.

4.2 Schemes

A scheme is a locally ringed space (X, O_X) covered by open subsets U_i each isomorphic (as locally ringed spaces) to Spec R_i for some ring R_i. A morphism of schemes is a morphism of locally ringed spaces.

Affine schemes Spec R are the building blocks; general schemes are glued together from them. Examples:

Spec Z. The arithmetic line; closed points are (p) for primes p; the generic point is (0).
Spec Z[x]. A 2-dimensional arithmetic surface; closed points are (p, f(x)) for f irreducible mod p.
Proj of a graded ring. Gives projective schemes. Proj k[x_0,…,x_n] = P^n_k.
Schemes over Z, S = Spec A. Relative schemes. A morphism X → S realises X as a family over S; fibres X_s are schemes over the residue field k(s).
Nonreduced schemes. Spec k[ε]/(ε^2) = “dual numbers” is a fat point; maps into a scheme X capture tangent vectors of X.

4.3 Properties of schemes and morphisms

Many adjectives propagate:

Reduced. No nilpotents in O_X.
Irreducible. Underlying space is irreducible.
Integral. Reduced + irreducible. ⇔ O_X(U) integral domain for all nonempty open U.
Noetherian. Locally Spec of a Noetherian ring; finitely many irreducible components.
Separated. Diagonal X → X × X is a closed immersion (the algebraic analogue of Hausdorff).
Proper. Universally closed + separated + finite type (the algebraic analogue of compact).
Smooth, étale, unramified. Properties of morphisms generalising “submersion” and “covering.”
Flat. Geometrically, “fibres vary continuously”; algebraically, the structure sheaf is flat over the base.
Quasi-coherent, coherent. Properties of sheaves of modules.

A finite-type morphism over a Noetherian base produces good behaviour; “finite type, separated, integral, normal, smooth over Spec k” is the algebraic-geometric analogue of “smooth manifold.”

4.4 Fibre products

In schemes (or any category), the fibre product X ×S Y exists. For affine schemes, Spec A ×{Spec R} Spec B = Spec(A ⊗_R B). Geometrically: pullback of families, intersections of subschemes, base change to a smaller or extended ring.

4.5 The functor of points

A scheme X over S is determined by its functor of points h_X: (S-schemes)^op → Sets sending T ↦ Hom_S(T, X). Yoneda’s lemma: this functor recovers X. Convention: write X(T) for h_X(T). Working with the functor of points makes many constructions cleaner — moduli problems naturally produce functors first, schemes (or stacks) second.

5. Sheaves

Sheaves encode local-to-global data. A presheaf F on a topological space X assigns an abelian group (or ring, set, module, …) F(U) to each open U with restriction maps F(U) → F(V) for V ⊆ U. A sheaf additionally satisfies the gluing axiom: compatible local sections glue uniquely to a global section.

5.1 Sheaves of modules

The structure sheaf O_X is a sheaf of rings on a scheme. An O_X-module is a sheaf F with each F(U) an O_X(U)-module compatibly with restriction. Operations: direct sum, tensor product ⊗_O, internal Hom, kernel, image, cokernel.

Quasi-coherent sheaf. Locally the sheaf M̃ associated to a module M over Spec R. Quasi-coherent sheaves on Spec R correspond exactly to R-modules.
Coherent sheaf. Quasi-coherent + finitely generated over each affine open + finitely presented kernel of any map between coherent sheaves.
Locally free sheaf of rank r. Locally isomorphic to O_X^r. Equivalent to a vector bundle of rank r over X.

5.2 Operations on sheaves

Pushforward f_*. For f: X → Y, (f_* F)(V) = F(f^{-1}(V)).
Pullback f^*. For an O_Y-module G, f^* G = f^{-1} G ⊗_{f^{-1} O_Y} O_X.
Cohomology H^i(X, F). Right-derived functors of global sections Γ(X, -); see §7.

5.3 Locally constant and étale sheaves

For étale cohomology and arithmetic, sheaves on the étale site of a scheme — built from étale morphisms rather than open immersions — provide the right setting. Étale cohomology with Q_l-coefficients underlies the proof of the Weil conjectures.

6. Divisors and line bundles

A divisor on a smooth variety encodes a codimension-one subvariety with multiplicities; the formalism connects geometry to the arithmetic of meromorphic functions and to line bundles.

6.1 Weil divisors

For an integral, separated, Noetherian scheme X regular in codimension 1, a Weil divisor is a finite formal Z-linear combination of irreducible codimension-one subvarieties (prime divisors). Div(X) is the free abelian group on prime divisors. For a nonzero rational function f ∈ K(X)^*, the principal divisor (f) = ∑ ord_D(f) · D, where ord_D is the discrete valuation associated to the local ring O_X,D. The divisor class group Cl(X) = Div(X) / {principal divisors}.

6.2 Cartier divisors and Picard group

A Cartier divisor is locally given by a single rational function up to nonzero regular function; equivalently a global section of the sheaf K^/O_X^. For smooth varieties, Cartier and Weil divisors coincide. The Picard group Pic(X) = {Cartier divisors} / {principal divisors} = {line bundles on X}/iso = H^1(X, O_X^*).

Examples:

Pic(P^n_k) = Z, generated by the hyperplane class O(1). Sections of O(d) are degree-d homogeneous polynomials, an (n+d choose d)-dimensional space.
Pic of a smooth projective curve of genus g: an extension 0 → Pic^0(C) → Pic(C) → Z → 0, where Pic^0(C) is an abelian variety (the Jacobian) of dimension g.
Pic of a Calabi-Yau variety: free abelian group of rank h^{1,1}.

6.3 Sections, sub-spaces, and maps to projective space

A line bundle L with global sections s_0,…,s_n (a basepoint-free linear system) defines a morphism X → P^n by x ↦ [s_0(x):…:s_n(x)]. If the s_i are a basis of H^0(X, L) we get the morphism associated to |L|. The image of X under the complete linear system of a sufficiently positive line bundle is an embedding (very ample line bundle).

Conversely every projective embedding X ↪ P^n comes from a very ample line bundle (the pullback of O(1)).

7. Cohomology

Sheaf cohomology measures the obstructions to lifting local sections to global. For an abelian sheaf F on X, the cohomology groups H^i(X, F) are the right-derived functors of Γ(X, -).

7.1 Čech cohomology

For an open cover {U_i} of X, Čech p-cochains are ∏*{i_0 < … < i_p} F(U*{i_0…i_p}). Čech cohomology Ĥ^p(X, F) coincides with sheaf cohomology under mild hypotheses (e.g., F a quasi-coherent sheaf on a separated scheme, with affine open covers). Computationally tractable for many examples.

7.2 Coherent cohomology of projective space

A foundational computation (Serre 1955, Annals of Math 61, FAC):

H^0(P^n, O(d)) = degree-d homogeneous polynomials in x_0,…,x_n; dimension (n+d choose n) for d ≥ 0.
H^n(P^n, O(d)) = dual to degree (-d-n-1) homogeneous polynomials; dimension (-d-1 choose n) for d ≤ -n-1.
All other H^i = 0.

The Euler characteristic χ(O(d)) = ∑ (-1)^i h^i(O(d)) is a polynomial in d of degree n — the Hilbert polynomial.

7.3 Finiteness, vanishing, duality

For a coherent sheaf F on a proper scheme over a field:

Finiteness. Each H^i(X, F) is finite-dimensional.
Serre vanishing. For ample L and i > 0, H^i(X, F ⊗ L^n) = 0 for n ≫ 0.
Kodaira vanishing. For smooth projective X over C and ample L, H^i(X, K_X ⊗ L) = 0 for i > 0 (canonical sheaf K_X = ∧^{dim X} Ω_X).
Serre duality. For smooth projective X of dimension n over a field, H^i(X, F)^∨ ≅ H^{n-i}(X, F^∨ ⊗ K_X).

7.4 Higher direct images and spectral sequences

For f: X → Y, the right-derived functors R^i f_generalize pushforward. The Leray spectral sequence E_2^{p,q} = H^p(Y, R^q f_ F) ⇒ H^{p+q}(X, F) reduces global cohomology of X to data over Y.

7.5 Étale cohomology

For schemes the Zariski topology is too coarse for many purposes (it produces zero cohomology in positive degree for constant sheaves on smooth schemes over algebraically closed fields). Grothendieck topologies — including the étale topology — refine the situation. Étale cohomology H^i_{ét}(X, F) with Q_l-coefficients (for l different from characteristic) gives Weil-cohomology theories. Used by Deligne to prove the Weil conjectures (Deligne 1974, Publications Mathématiques de l’IHÉS 43; 1980, Publications Mathématiques de l’IHÉS 52, “Weil II”).

8. Differentials and the canonical sheaf

The sheaf of Kähler differentials Ω_{X/Y} encodes infinitesimal information. For a morphism f: X → Y, Ω_{X/Y} fits in a short exact sequence (the cotangent sequence) and represents derivations. On a smooth scheme X of relative dimension n over Y, Ω_{X/Y} is locally free of rank n. The canonical sheaf K_X = det Ω_{X/k} = ∧^n Ω_{X/k} for n = dim X.

The canonical class plays a central role in birational geometry: the Kodaira dimension κ(X) = sup of dimensions of images of rational maps |K_X^{⊗m}|, classifying varieties by κ = -∞, 0, 1, …, dim X. The Minimal Model Program (MMP, Mori 1988; Birkar-Cascini-Hacon-McKernan 2010 Journal of the AMS 23 — proving existence of minimal models for varieties of general type) seeks canonical representatives in birational equivalence classes.

9. Riemann-Roch

The Riemann-Roch theorem computes Euler characteristics of coherent sheaves in terms of topological invariants.

9.1 Riemann-Roch for curves

For a smooth projective curve C of genus g over k and a divisor D on C:

h^0(D) - h^1(D) = deg D + 1 - g.

By Serre duality h^1(D) = h^0(K - D). The genus is g = h^0(K) = h^1(O) = dim H^0(C, Ω).

Consequences:

For deg D > 2g - 2, h^0(D) = deg D + 1 - g.
For deg D ≥ 2g, the linear system |D| gives an embedding into P^{deg D - g}.
Canonical embedding: |K| embeds C as a curve of degree 2g - 2 in P^{g-1} when g ≥ 3 and C is non-hyperelliptic.

9.2 Hirzebruch-Riemann-Roch

For a coherent sheaf F on a smooth projective n-dimensional variety X:

χ(X, F) = ∫_X ch(F) · td(T_X),

where ch is the Chern character and td the Todd class. The integral is the degree of the top-dimensional component in the cohomology of X (using cycle/Chow class formalism).

9.3 Grothendieck-Riemann-Roch

For a proper morphism f: X → Y of smooth schemes:

ch(f_! F) · td(T_Y) = f_*(ch(F) · td(T_X)),

where f_! is the K-theoretic pushforward. This generalises Hirzebruch-Riemann-Roch (Y = pt) and Riemann-Roch for curves.

10. Intersection theory

For a smooth projective variety X, the Chow group A^k(X) is the free abelian group on codimension-k subvarieties modulo rational equivalence. A^(X) = ⊕ A^k(X) is a graded ring under intersection product. For complex varieties there is a cycle class map to singular cohomology A^(X) → H^{2*}(X(C), Z).

Foundational reference: Fulton 1984 Intersection Theory (Springer).

10.1 Chern classes

For a rank-r vector bundle E on X, the Chern classes c_i(E) ∈ A^i(X) are characteristic classes satisfying:

Whitney sum formula: c(E ⊕ F) = c(E) · c(F), where c = 1 + c_1 + c_2 + … is the total Chern class.
Naturality: c_i(f^E) = f^ c_i(E).
Normalisation: For a line bundle L, c_1(L) ∈ A^1(X) = Pic(X) is the divisor class corresponding to L.

The splitting principle: formally, c(E) = ∏ (1 + α_i) for “Chern roots” α_i, allowing computation as symmetric polynomials in roots.

10.2 Schubert calculus

Cohomology ring of the Grassmannian Gr(k, n): A^*(Gr(k,n)) is freely generated by Schubert classes σ_λ indexed by Young diagrams λ inside a k × (n-k) rectangle. The product structure is given by the Littlewood-Richardson coefficients. Classical enumerative problems (e.g., the number of lines on a smooth cubic surface = 27) reduce to intersections of Schubert cycles.

10.3 Enumerative geometry and Gromov-Witten theory

Modern enumerative geometry counts curves on a variety satisfying incidence conditions. Gromov-Witten invariants of X are intersection numbers on the moduli space of stable maps (Kontsevich-Manin 1994 Communications in Mathematical Physics 164; Witten 1991). The Gromov-Witten potential and the WDVV equations connect to mirror symmetry and to topological field theories.

11. Moduli spaces

A moduli problem assigns to a parameter space S a set (or groupoid) of “families over S” of geometric objects (curves, vector bundles, sheaves, …). A moduli space M is a fine moduli space if the corresponding functor is representable; a coarse moduli space is a scheme with the universal property up to natural isomorphism.

11.1 Moduli of curves M_g

For g ≥ 2, M_g is the moduli space of smooth projective curves of genus g. It is an algebraic variety (in fact an irreducible Deligne-Mumford stack) of dimension 3g - 3. Its Deligne-Mumford compactification M̄_g (Deligne-Mumford 1969 Publications Mathématiques de l’IHÉS 36) adds stable nodal curves; M̄_g is a smooth proper Deligne-Mumford stack. The boundary M̄_g \ M_g is a normal-crossing divisor with combinatorial structure indexed by graphs.

11.2 Moduli of vector bundles

M_C(r, d) parametrises (semistable) vector bundles of rank r and degree d on a curve C. Geometric Invariant Theory (GIT, Mumford 1965 Geometric Invariant Theory) constructs the quotient. The Narasimhan-Seshadri theorem (1965 Annals of Math 82) identifies stable vector bundles with irreducible representations of the fundamental group; Donaldson 1985 Journal of Differential Geometry 18 supplied an analytic proof via Yang-Mills theory.

11.3 Moduli of sheaves and Donaldson-Thomas theory

Hilbert schemes Hilb^n(X) parametrise zero-dimensional subschemes of length n. Donaldson-Thomas invariants count ideal sheaves; connected to Gromov-Witten invariants by the MNOP correspondence (Maulik-Nekrasov-Okounkov-Pandharipande 2006).

11.4 Stacks

Moduli problems with nontrivial automorphisms (e.g., elliptic curves admit Z/2 inversion) fail to have a fine moduli space; the right object is a stack. Deligne-Mumford stacks (1969) for problems with finite automorphism groups; Artin stacks (1974 Inventiones Mathematicae 27) for more general ones (e.g., the moduli stack of all curves of genus g including stable curves, M̄_g). The Stacks Project (stacks.math.columbia.edu, started 2005 by Aise Johan de Jong) is the open-source reference textbook for stacks; ~7 800 pages as of 2024.

12. Birational geometry

Two varieties are birationally equivalent if they have isomorphic open subsets, equivalently, isomorphic function fields. Classification of varieties up to birational equivalence is a central problem:

Smooth projective curves. Birational classification = isomorphism classification (function field determines the curve). Stratified by genus.
Smooth projective surfaces. Castelnuovo’s contraction theorem (1901) allows blowing down (-1)-curves; minimal surfaces classified into rational, ruled, abelian, K3, Enriques, elliptic, surfaces of general type (Enriques-Kodaira classification).
Higher dimensions. Mori’s minimal model program (MMP). For varieties of general type, existence of minimal models proved by Birkar-Cascini-Hacon-McKernan 2010; Birkar 2016 Annals of Math 184 for boundedness of Fano varieties. The Sarkisov program for relations among MMP outputs.
Calabi-Yau threefolds. Active research: mirror symmetry, derived categories, MMP for Calabi-Yau threefolds with terminal singularities.

13. Derived categories

The derived category D(X) of an abelian category (e.g., coherent sheaves on X) localises chain complexes at quasi-isomorphisms. The bounded derived category D^b(X) is the natural home for many homological invariants: Ext groups, Tor, hyperext, derived pushforward, etc.

13.1 Triangulated structure

D(X) is triangulated, with shift functor [1] and distinguished triangles A → B → C → A[1]. Many homological theorems become cleaner in the triangulated language.

13.2 Derived functors

Lf^, Rf_. Derived pullback and pushforward of complexes of sheaves. Composition behaviour is much cleaner than the abelian versions.
L⊗, RHom. Derived tensor and Hom.
R Γ. Derived global sections, computing sheaf cohomology.
Grothendieck duality. For a proper morphism f: X → Y, an isomorphism RHom(Rf_F, G) ≅ Rf_ RHom(F, f^! G) for a dualising complex f^! G. Generalises Serre duality (when Y = Spec k and X smooth proper).

13.3 Bondal-Orlov reconstruction

For a smooth projective variety X with ample (or anti-ample) canonical bundle, the derived category D^b(X) determines X up to isomorphism (Bondal-Orlov 2001 Compositio Mathematica 125). Without the canonical hypothesis, distinct varieties may have equivalent derived categories — they are then called Fourier-Mukai partners. Examples: K3 surfaces; abelian varieties and their duals.

13.4 Stability conditions

Bridgeland 2007 Annals of Math 166 — a stability condition on a triangulated category D gives an analog of slope-semistability with continuous variation. The space Stab(D) is a complex manifold; its structure on D^b(X) has been investigated for many varieties. The connection to wall-crossing formulas, mirror symmetry, and physics is active.

13.5 Categorical and noncommutative geometry

Algebraic geometry has been categorified extensively: A_∞-categories, ∞-categories (Lurie’s Higher Topos Theory), derived algebraic geometry (Toën-Vezzosi, Lurie’s Spectral Algebraic Geometry). Noncommutative algebraic geometry replaces commutative rings with associative algebras or differential graded categories. Substantial bodies of work exist; outside the foundational scope of this note.

14. Curves: a closer look

Smooth projective curves are the bottom of the dimensional hierarchy and the most thoroughly understood. Quick taxonomy:

Genus 0. P^1 only (over algebraically closed k). Hilbert series 1/((1-t)(1-t)).
Genus 1. Elliptic curves; smooth plane cubics or smooth projective 1-dimensional group varieties. j-invariant classifies up to isomorphism. Pic^0(E) ≅ E.
Genus ≥ 2. Moduli space M_g has dimension 3g - 3. Canonical embedding into P^{g-1} when C is non-hyperelliptic; double cover of P^1 when hyperelliptic.

Arithmetic of curves over Q intersects deeply with the Langlands program: modular curves parametrise elliptic curves with extra structure; modularity theorem (Wiles, Taylor-Wiles, Breuil-Conrad-Diamond-Taylor) connects elliptic curves over Q to modular forms; the Birch-Swinnerton-Dyer conjecture relates the L-function of an elliptic curve to its Mordell-Weil rank.

15. Software and computational tools

Algebraic geometry has substantial computational infrastructure for Gröbner bases and beyond.

Macaulay2 (Grayson-Stillman). General-purpose system for commutative algebra and algebraic geometry. Standard tool for explicit examples.
Singular (Decker-Greuel-Pfister-Schönemann). Specialised in commutative algebra, polynomial computations.
SageMath (Stein, since 2005). Python-front-ended general mathematical system; includes Macaulay2 / Singular / GAP bindings.
Magma (University of Sydney). Commercial; strong in number theory and group theory; substantial algebraic geometry capabilities.
PARI/GP. Number theory and elliptic-curve computations.
Polymake. Polyhedral and toric geometry.
CoCoA. Commutative algebra.

Theoretical computer-verifications: Lean 4 mathlib has an active stream porting algebraic geometry foundations (sheaves, schemes, varieties); Coq’s MathComp; Isabelle/HOL’s AFP entries on commutative algebra and varieties.

16. Open problems

Hodge conjecture. Every Hodge class on a smooth projective variety over C is a rational linear combination of fundamental classes of algebraic subvarieties. Open in general.
Standard conjectures (Grothendieck). A series of conjectures about algebraic cycles and motives that would imply the Weil conjectures via a “purely algebraic” proof. Still open.
Tate conjecture. Analogue of Hodge for l-adic étale cohomology. Open.
Birational geometry of varieties of general type in higher dimensions. Minimal model program completed in dimensions ≤ 3; many open problems remain in higher dimensions and characteristic p.
Resolution of singularities in positive characteristic. Hironaka 1964 Annals of Math 79 proved characteristic zero; characteristic p > 0 in dimension ≥ 4 remains open (some recent progress by Cossart-Piltant, Temkin).
Mirror symmetry / geometric Langlands. Programs connecting algebraic geometry to representation theory and mathematical physics. Recent dramatic progress (Gaitsgory-Raskin-Faergeman-Lin-Chen-Lurie-Arinkin 2024 announcement of a proof of geometric Langlands over C).

17. Connections to other fields

Number theory. Schemes over Z and over Spec O_K are central to modern number theory. Faltings 1983 proof of the Mordell conjecture; modularity of elliptic curves over Q; the Langlands program.
Mathematical physics. Mirror symmetry, topological string theory, Donaldson-Thomas theory, quantum cohomology, integrable systems.
Representation theory. Geometric Langlands, geometric Satake, Springer theory, perverse sheaves.
Algebraic statistics. Maximum likelihood degree and ideals of statistical models.
Coding theory. Algebraic-geometric Goppa codes; Tsfasman-Vlăduţ-Zink 1982 Math. Nachr. 109 surpassed the Gilbert-Varshamov bound using modular curves.
Cryptography. Elliptic-curve cryptography; isogeny-based cryptography (SIDH, CSIDH, SQISign).
Computer vision and robotics. Algebraic geometry of multiview geometry, the trifocal tensor, polynomial systems for inverse kinematics.
Machine learning. Algebraic geometry of neural networks (Mehta-Wenzel-Trager 2022); algebraic statistics of graphical models.

Compendium

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Algebraic Geometry Foundations

Algebraic Geometry Foundations

See also

1. Commutative algebra preliminaries

2. Affine varieties

2.1 Algebraic sets

2.2 Hilbert’s Nullstellensatz

2.3 Zariski topology

2.4 Morphisms of affine varieties

2.5 Local rings and tangent spaces

3. Projective varieties

3.1 Projective Nullstellensatz

3.2 Affine charts and the Segre embedding

3.3 Veronese embedding

3.4 Projective curves and intersection in P^2

3.5 Grassmannians, flag varieties

3.6 Toric varieties

4. Schemes

4.1 Spec of a ring

4.2 Schemes

4.3 Properties of schemes and morphisms

4.4 Fibre products

4.5 The functor of points

5. Sheaves

5.1 Sheaves of modules

5.2 Operations on sheaves

5.3 Locally constant and étale sheaves

6. Divisors and line bundles

6.1 Weil divisors

6.2 Cartier divisors and Picard group

6.3 Sections, sub-spaces, and maps to projective space

7. Cohomology

7.1 Čech cohomology

7.2 Coherent cohomology of projective space

7.3 Finiteness, vanishing, duality

7.4 Higher direct images and spectral sequences

7.5 Étale cohomology

8. Differentials and the canonical sheaf

9. Riemann-Roch

9.1 Riemann-Roch for curves

9.2 Hirzebruch-Riemann-Roch

9.3 Grothendieck-Riemann-Roch

10. Intersection theory

10.1 Chern classes

10.2 Schubert calculus

10.3 Enumerative geometry and Gromov-Witten theory

11. Moduli spaces

11.1 Moduli of curves M_g

11.2 Moduli of vector bundles

11.3 Moduli of sheaves and Donaldson-Thomas theory

11.4 Stacks

12. Birational geometry

13. Derived categories

13.1 Triangulated structure

13.2 Derived functors

13.3 Bondal-Orlov reconstruction

13.4 Stability conditions

13.5 Categorical and noncommutative geometry

14. Curves: a closer look

15. Software and computational tools

16. Open problems

17. Connections to other fields

Further reading

Adjacent

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