# Algebraic Geometry
## Core Concepts
### Affine Varieties
1. **Polynomial Rings**
```math
k[x_1,...,x_n]/I
```
where:
- k is field
- I is ideal
- x_i are variables
2. **Zero Sets**
```math
V(I) = \{x \in k^n : f(x) = 0 \text{ for all } f \in I\}
```
where:
- I is ideal
- k is field
- V is variety
### Schemes
1. **Spectrum**
```math
\text{Spec}(R) = \{P \subset R : P \text{ is prime ideal}\}
```
where:
- R is ring
- P is prime ideal
2. **Structure Sheaf**
```math
\mathcal{O}_X(U) = \{s: U \to \coprod_{p \in U} \mathcal{O}_{X,p}\}
```
where:
- X is scheme
- U is open set
- O_{X,p} is local ring
## Advanced Concepts
### Cohomology Theory
1. **Sheaf Cohomology**
```math
H^i(X,\mathcal{F}) = R^i\Gamma(X,\mathcal{F})
```
where:
- F is sheaf
- Γ is global sections
- R^i is derived functor
2. **Čech Cohomology**
```math
\check{H}^i(X,\mathcal{F}) = \varinjlim_{\mathfrak{U}} \check{H}^i(\mathfrak{U},\mathcal{F})
```
where:
- U is open cover
- F is sheaf
### Intersection Theory
1. **Intersection Product**
```math
Z_1 \cdot Z_2 = \sum_i m_i[V_i]
```
where:
- Z_i are cycles
- m_i are multiplicities
- V_i are components
2. **Chow Ring**
```math
A^*(X) = \bigoplus_k A^k(X)
```
where:
- A^k is k-codimensional cycles
- X is variety
## Applications
### Moduli Spaces
1. **Moduli of Curves**
```math
\mathcal{M}_g = \{[C] : C \text{ is smooth curve of genus } g\}
```
where:
- g is genus
- [C] is isomorphism class
2. **Stable Maps**
```math
\overline{\mathcal{M}}_{g,n}(X,\beta)
```
where:
- g is genus
- n is marked points
- β is homology class
### Mirror Symmetry
1. **Hodge Numbers**
```math
h^{p,q}(X) = h^{n-p,q}(X^\vee)
```
where:
- X is Calabi-Yau
- X^\vee is mirror
- h^{p,q} are Hodge numbers
2. **Gromov-Witten Invariants**
```math
\langle τ_{a_1}(γ_1),...,τ_{a_n}(γ_n)\rangle_{g,β}
```
where:
- τ_a are descendants
- γ_i are cohomology classes
- β is curve class
## Implementation
### Computational Algebraic Geometry
```python
class AlgebraicVariety:
def __init__(self,
polynomials: List[str],
variables: List[str]):
"""Initialize algebraic variety.
Args:
polynomials: Defining polynomials
variables: Variable names
"""
self.polynomials = polynomials
self.variables = variables
self.ring = self._construct_ring()
def compute_groebner_basis(self,
ordering: str = 'lex') -> List[str]:
"""Compute Gröbner basis.
Args:
ordering: Monomial ordering
Returns:
basis: Gröbner basis
"""
# Convert to ideal
ideal = self._polynomials_to_ideal()
# Compute basis
basis = self._buchberger_algorithm(ideal, ordering)
return basis
def dimension(self) -> int:
"""Compute variety dimension.
Returns:
dim: Krull dimension
"""
# Get Gröbner basis
basis = self.compute_groebner_basis()
# Compute dimension
return self._krull_dimension(basis)
```
### Intersection Theory
```python
class IntersectionTheory:
def __init__(self,
variety: AlgebraicVariety):
"""Initialize intersection theory.
Args:
variety: Algebraic variety
"""
self.variety = variety
def intersection_product(self,
cycle1: Cycle,
cycle2: Cycle) -> Cycle:
"""Compute intersection product.
Args:
cycle1: First cycle
cycle2: Second cycle
Returns:
product: Intersection product
"""
# Compute proper intersection
components = self._proper_intersection(cycle1, cycle2)
# Get multiplicities
multiplicities = self._intersection_multiplicities(
cycle1, cycle2, components
)
return Cycle(components, multiplicities)
def chern_class(self,
bundle: VectorBundle) -> ChernClass:
"""Compute Chern class.
Args:
bundle: Vector bundle
Returns:
chern: Chern class
"""
return self._compute_chern_class(bundle)
```
## Advanced Topics
### Derived Categories
1. **Derived Functor**
```math
RF: D^b(A) \to D^b(B)
```
where:
- D^b is bounded derived category
- A,B are abelian categories
2. **Fourier-Mukai Transform**
```math
\Phi_\mathcal{P}: D^b(X) \to D^b(Y)
```
where:
- P is kernel
- X,Y are varieties
### Stacks
1. **Moduli Stack**
```math
\mathcal{M} = [X/G]
```
where:
- X is scheme
- G is group
- [/] is quotient stack
2. **Derived Stack**
```math
R\mathcal{M} = [RX/G]
```
where:
- RX is derived scheme
- G is group
## Future Directions
### Emerging Areas
1. **Derived Algebraic Geometry**
- Derived Schemes
- Higher Stacks
- Spectral Algebraic Geometry
2. **Motivic Theory**
- Motivic Integration
- Motivic Cohomology
- Virtual Classes
### Open Problems
1. **Theoretical Challenges**
- Hodge Conjecture
- Minimal Model Program
- Derived Categories
2. **Practical Challenges**
- Algorithm Efficiency
- Symbolic Computation
- Numerical Methods
## Related Topics
1. [[commutative_algebra|Commutative Algebra]]
2. [[category_theory|Category Theory]]
3. [[complex_geometry|Complex Geometry]]
4. [[number_theory|Number Theory]]