NeuralBlitz name: algebraic-geometry
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Algebraic Geometry — Varieties, Schemes, and Cohomology
Covers: Affine Varieties · Projective Varieties · Schemes · Sheaves · Cohomology · Dimension Theory · Algebraic Curves
Affine Algebraic Geometry
Affine Space
n-dimensional affine space over algebraically closed field k:
A^n(k) = k^n
Points correspond to n-tuples (a₁, ..., a_n).
Affine Varieties
An affine variety is an irreducible algebraic set:
V(I) = {p ∈ A^n | f(p) = 0 for all f ∈ I}
Where I is an ideal in k[x₁, ..., x_n].
Hilbert's Nullstellensatz:
I(V(J)) = √J for any ideal J
The correspondence between radical ideals and varieties is bijective.
Coordinate Ring
k[V] = k[x₁, ..., x_n]/I(V)
The coordinate ring captures the algebraic information of the variety.
Morphisms of Affine Varieties
A morphism φ: V → W corresponds to k-algebra homomorphism:
φ* : k[W] → k[V]
Finite morphisms: Integral extension Dominant morphisms: Dense image
Dimension
Krull dimension: Length of longest chain of prime ideals.
For variety V:
dim V = dim k[V]
Properties:
- dim A^n = n
- dim of hypersurface = n - 1
- Points have dimension 0
Hilbert Function
H_V(t) = dim_k k[V]_t
For projective varieties, this is eventually polynomial (Hilbert polynomial).
Regular Functions
For affine variety V, regular functions are elements of coordinate ring:
O_V(V) = k[V]
On open set U, regular functions are locally given by fractions with denominator non-zero on U.
Projective Geometry
Projective Space
n-dimensional projective space:
P^n(k) = (k^{n+1} \ {0}) / ~
Where (a₀, ..., a_n) ~ λ(a₀, ..., a_n) for λ ≠ 0.
Homogeneous coordinates: [x₀ : x₁ : ... : xₙ]
Projective Varieties
A projective variety is a closed subset of Pⁿ(k) defined by homogeneous polynomials.
Projective Nullstellensatz:
I(V(I)) = √I for homogeneous ideal I
V(I) = ∅ ⇔ √I contains all x_i
Homogeneous Coordinates
Ring of polynomials graded by total degree:
k[x₀, ..., x_n] = ⊕_{d≥0} k[x₀, ..., x_n]_d
A polynomial is homogeneous of degree d if all monomials have degree d.
Homogenization
For polynomial f ∈ k[x₁, ..., x_n] of degree d:
f^h = x₀^d f(x₁/x₀, ..., x_n/x₀)
Dehomogenization
Setting x₀ = 1:
f^h|_{x₀=1} = f
Projective Coordinate Ring
k[V] = k[x₀, ..., x_n]/I(V)
Graded by degree: k[V] = ⊕_{d≥0} k[V]_d
Proj Construction
For graded ring S:
Proj(S) = {prime ideals not containing S_+}
This is the universal projective variety with homogeneous coordinate ring S.
Affine Cover
Pⁿ has (n+1) affine open sets:
U_i = {x ∈ Pⁿ | x_i ≠ 0} ≅ Aⁿ
Transition maps given by rational functions.
Morphisms and Rational Maps
Regular (Morphic) Maps
A morphism f: X → Y of varieties is locally given by regular functions:
f = (f₀, ..., f_m) with f_i ∈ O_X(U)
Dominant Maps
f is dominant if f(X) is dense in Y. This corresponds to:
k(Y) ⊂ k(X) (field extension)
Rational Maps
A rational map is morphism defined on open subset:
φ: X ⇢ Y = (f₀, ..., f_m) with f_i rational functions
Domain of definition: Where denominators nonzero.
Birational Maps
A birational map is rational map with rational inverse:
k(Y) ≅ k(X)
Varieties are birationally equivalent if such map exists.
Theorem: Every smooth variety is birationally equivalent to a hypersurface in Pⁿ for sufficiently large n.
Resolution of Singularities
For any variety V, there exists birational morphism:
π: V' → V
Where V' is smooth (non-singular). This is Hironaka's theorem (characteristic 0).
Sheaves and Local Theory
Presheaf
For topological space X, presheaf F assigns:
- To each open U: group/ring F(U)
- To each inclusion V ⊂ U: restriction map F(U) → F(V)
With properties:
- F(∅) = 0
- Restriction is functorial
Sheaf
A presheaf is a sheaf if:
- Locality: Sections determined by restrictions to opens
- Gluing: Locally compatible sections glue uniquely
Sheaf of regular functions O_X on variety X.
Stalk
Stalk at point x:
F_x = lim_{x∈U} F(U)
Germs: equivalence classes of sections near x.
Sheaf Homomorphism
Map of sheaves φ: F → G gives maps on all stalks:
φ_x: F_x → G_x
Kernel and Cokernel
For sheaf homomorphism φ: F → G:
- Kernel: (ker φ)(U) = ker(F(U) → G(U))
- Cokernel: (coker φ)(U) = coker(F(U) → G(U))
Sheaf Cohomology
Derived functors of global sections:
H^i(X, F) = R^iΓ(X, F)
Properties:
- H⁰(X, F) = Γ(X, F) (sections)
- H^i(X, F) = 0 for i > dim X on affine varieties (Serre)
- Flasque sheaves give exact global sections
Čech Cohomology
For open cover U = {U_i}:
Č^p(F) = ∏ F(U_{i₀}∩...∩U_{i_p})
Differential: δ: Č^p → Č^{p+1} H^(U, F) = H^(Č^*(F))
Cohomology of Sheaves
Invertible Sheaves (Line Bundles)
Picard group:
Pic(X) = H¹(X, O_X*)
Line bundles correspond to divisors modulo linear equivalence.
Divisors
Weil divisor: formal sum of codimension 1 subvarieties
D = Σ n_i V_i
Cartier divisor: locally principal codimension 1 subscheme
For Noetherian normal variety: Weil ↔ Cartier
Canonical Divisor
For smooth variety X:
K_X = div(ω_X) where ω_X = Ω^n_{X/k}
sheaf O(D)
For divisor D:
Γ(U, O(D)) = {f ∈ k(X)* | div(f) + D ≥ 0 on U}
Adjunction Formula
For smooth subvariety Y ⊂ X:
K_Y = (K_X + Y)|_Y
Riemann-Roch Theorem (Curves)
For line bundle L on curve C:
h⁰(C, L) - h⁰(C, K_C ⊗ L⁻¹) = deg L + 1 - g
Where g = genus of C.
Hodge Diamond (Surfaces)
For smooth projective surface:
h^{p,q} = dim H^p(X, Ω^q_X)
With h^{p,q} = h^{q,p}.
sheaf Cohomology Computations
Cech to Derived Functor
For good open cover, Čech cohomology computes sheaf cohomology:
H^*(U, F) ≅ H^*(X, F)
Serre Duality
For smooth projective variety X of dimension n:
H^i(X, F) ≅ H^{n-i}(X, K_X ⊗ F⁻¹)*
Leray Spectral Sequence
For map f: X → Y:
E²_{p,q} = H^p(Y, R^q f_* F) ⇒ H^{p+q}(X, F)
Vanishing Theorems
Kodaira vanishing: For ample L on smooth projective:
H^i(X, K_X ⊗ L) = 0 for i > 0
Serre vanishing: For ample L, for large m:
H^i(X, O(mL)) = 0 for i > 0
Hirzebruch-Riemann-Roch
For line bundle L on X:
χ(X, L) = χ(O_X) + (L^{n})/n! + higher (Todd genus)
Classical Enumerative Geometry
Chern Classes
Total Chern class:
c(E) = 1 + c₁(E) + c₂(E) + ...
For line bundle: c₁(L) = divisor class of L.
Schubert Calculus
Intersection theory on Grassmannians:
- Flags, conditions on subspaces
- Giambelli-Soudry-Thom formula
Hilbert Scheme
Parametrizes all subschemes of Pⁿ with given Hilbert polynomial.
For 0-dimensional schemes of degree d:
Hilb^d(Pⁿ) dimension = n·d
Counting Curves
Gromov-Witten invariants: Count curves of given genus through generic points.
Donaldson-Thomas invariants: Count subschemes satisfying ideal sheaf conditions.
Severi Varieties
Parametrizes plane curves of degree d with δ nodes:
V_{d,δ} ⊂ P^{d(d+3)/2}
Modern Scheme Theory
Schemes
Scheme X is locally ringed space locally isomorphic to affine scheme:
Spec R = {prime ideals of R}
Structure sheaf O_X on Spec R.
Fiber Product
For morphisms X → S and Y → S:
X ×_S Y = Spec (O_X ⊗_{O_S} O_Y)
Separated Morphisms
Morphism f: X → Y is separated if diagonal is closed.
Proper Morphisms
Proper = separated + universally closed. Projective morphisms are proper.
Flat Morphisms
Locally: Tor^O_{Y, f(x)}(O_X,x, k(f(x))) = 0 for i > 0
Properties:
- Fibers vary "continuously"
- Base change for cohomology
Descent Theory
Grothendieck's fpqc descent:
- Quasi-coherent sheaves descend
- Effective descent for morphisms
Algebraic Stacks
Generalizes moduli spaces with automorphisms:
- DM stacks: Deligne-Mumford
- Artin stacks: general
Dimension Theory
Dimension of Affine Variety
dim V = transcendence degree of k(V) over k
Equivalently: length of generic chain of irreducible subvarieties.
Dimension of Scheme
dim X = sup length of chain of irreducible closed subsets
Krull's Principal Ideal Theorem
For Noetherian ring:
height P ≤ n if P contains n elements of a system of parameters
Height and Depth
- Height of prime: length of longest chain to (0)
- Depth of module: length of maximal regular sequence
Regular Local Rings
Local ring is regular if:
dim = minimal number of generators of maximal ideal
Geometrically: non-singular point.
Singular Locus
Sing(X) = {x ∈ X | O_{X,x} not regular}
For variety over characteristic 0, singular locus is proper closed subset.
Normal Varieties
Normal: Integral, locally integral closure in function field.
- Normalization: birational morphism from normal variety
- Normality ⇒ Serre's R1 + S2 conditions
Algebraic Curves
Riemann Surface
Smooth algebraic curve = compact Riemann surface. Genus g relates to topology:
- g = (2 - #faces + #edges - #vertices)/2 for triangulation
- g = b_1/2 for Betti number
Canonical Embedding
For g ≥ 2, complete linear system |K_C| embeds C into P^{g-1}.
Theorem (Riemann): Complex algebraic curves = compact Riemann surfaces.
Elliptic Curves
g = 1: Smooth cubic in P².
Group law: chord-tangent process.
- Identity: point at infinity
- Associativity: geometric
Complex tori: C/Λ ≅ elliptic curve.
Hyperelliptic Curves
g ≥ 2: Double cover of P¹ branched at 2g+2 points.
Model: y² = f(x) where deg f = 2g+1 or 2g+2.
Moduli of Curves
M_g: moduli space of genus g curves.
- Dimension: 3g - 3
- Compactification M̅_g by stable curves
Jacobian
Pic⁰(C) ≅ Alb(C) = H⁰(K_C)*/H¹(O_C)
Complex torus J(C) = H⁰(K_C)*/H¹(O_C)
Torelli Theorem
C determined by J(C) with polarization.
Common Errors to Avoid
- Confusing affine and projective varieties
- Forgetting to homogenize/dehomogenize when switching
- Ignoring that projective closure adds points at infinity
- Confusing dimension of variety with dimension of ambient space
- Not understanding sheaf vs. presheaf distinction
- Applying algebraic results without checking hypotheses (e.g., normality)
- Forgetting that field must be algebraically closed for Nullstellensatz
- Confusing regular and rational functions
- Misunderstanding sheaf cohomology vs. singular cohomology
Key References
- Algebraic Geometry by Hartshorne — Standard text
- Algebraic Geometry by Griffiths & Harris — Classical methods
- Algebraic Geometry by Shafarevich — Introduction
- Principles of Algebraic Geometry by Harris — More accessible