Introduction: What is Algebraic Geometry and Why It Matters
Algebraic geometry studies geometric objects defined by polynomial equations. It represents a powerful fusion of algebra and geometry where:
- Geometric objects (curves, surfaces, higher-dimensional varieties) are defined by polynomial equations
- Algebraic techniques provide tools to study these geometric objects systematically
- Applications span pure mathematics (number theory, topology), theoretical physics (string theory), and applied fields (cryptography, coding theory)
At its core, algebraic geometry establishes a two-way correspondence:
- Geometry → Algebra: Geometric objects correspond to algebraic structures (ideals, rings)
- Algebra → Geometry: Algebraic equations define geometric objects
Core Concepts and Principles
Affine and Projective Varieties
| Concept | Definition | Key Properties |
|---|---|---|
| Affine Variety | Set of solutions to polynomial equations in affine space (ℂⁿ or algebraically closed field) | • Local properties clearly visible<br>• May have “points at infinity” missing |
| Projective Variety | Set of solutions to homogeneous polynomial equations in projective space (ℙⁿ) | • Compact (in complex case)<br>• No “points at infinity” missing<br>• Better behaved for intersection theory |
| Quasi-Projective Variety | Open subset of a projective variety | • Generalization that includes both affine and projective varieties |
Fundamental Correspondence: Varieties and Ideals
| Geometric Side | Algebraic Side |
|---|---|
| Affine variety V(I) | Ideal I in polynomial ring k[x₁,…,xₙ] |
| Coordinate ring k[V] | Quotient ring k[x₁,…,xₙ]/I |
| Points on variety | Maximal ideals in coordinate ring |
| Irreducible components | Prime ideals in coordinate ring |
| Functions on variety | Elements of coordinate ring |
| Rational functions | Elements of function field |
Modern Foundations: Schemes
| Concept | Definition | Advantages |
|---|---|---|
| Scheme | Locally ringed space that is locally the spectrum of a ring | • Unifies various settings (varieties, number theory)<br>• Handles singularities naturally<br>• Incorporates nilpotent elements |
| Spectrum of a ring (Spec R) | Set of prime ideals of R with Zariski topology and structure sheaf | • Algebraic interpretation of geometric concepts<br>• Bridge between algebra and geometry |
| Sheaf | Assignment of algebraic structures to open sets with compatibility conditions | • Organizes local-to-global principles<br>• Tracks functions and their properties |
Key Structures and Properties
Morphisms and Maps
| Type | Definition | Key Properties |
|---|---|---|
| Regular Map | Map defined by rational functions where denominator doesn’t vanish | • Preserves algebraic structure<br>• Continuous in Zariski topology |
| Morphism of Varieties | Regular map between varieties | • Pullback of regular functions gives regular functions |
| Morphism of Schemes | Map of locally ringed spaces | • More general concept<br>• Local rings map compatibly |
| Isomorphism | Morphism with two-sided inverse | • Preserves all algebraic-geometric properties |
Divisors and Line Bundles
| Concept | Definition | Application |
|---|---|---|
| Divisor | Formal sum of codimension-1 subvarieties with integer coefficients | • Track zeros and poles of functions<br>• Describe intersection theory |
| Line Bundle | Locally free sheaf of rank 1 | • Geometric realization of divisor classes<br>• Tool for cohomology |
| Canonical Divisor (K) | Divisor associated with differential forms | • Key to classification theory<br>• Central in Riemann-Roch theorem |
Dimensions and Singularities
| Concept | Definition | Significance |
|---|---|---|
| Dimension | Maximum length of chain of prime ideals in coordinate ring | • Intrinsic measure of “size”<br>• Determined by Krull dimension |
| Smooth Point | Point where local ring is regular | • Tangent space well-defined<br>• Differential calculus works |
| Singular Point | Non-smooth point | • Requires special treatment<br>• Focus of resolution of singularities |
| Tangent Space | Vector space of derivations at a point | • Linear approximation to variety<br>• Dimension ≥ algebraic dimension |
Cohomology and Intersection Theory
Cohomology
| Cohomology Type | Description | Applications |
|---|---|---|
| Sheaf Cohomology | Measures obstructions to extending sections | • Riemann-Roch theorem<br>• Hodge theory |
| Čech Cohomology | Computed via open covers | • Practical calculations<br>• Connects to sheaf cohomology |
| de Rham Cohomology | Based on differential forms | • Topological invariants<br>• Links to singular cohomology |
Intersection Theory
| Concept | Definition | Key Results |
|---|---|---|
| Intersection Product | Product in Chow ring | • Generalizes intersection of subvarieties |
| Bézout’s Theorem | In ℙⁿ, the degree of intersection of hypersurfaces equals the product of their degrees | • Fundamental result on intersections |
| Chow Ring | Ring of algebraic cycles modulo rational equivalence | • Algebraic version of cohomology<br>• Captures intersection properties |
Classification of Curves and Surfaces
Curves (Dimension 1)
| Invariant | Meaning | Classification Role |
|---|---|---|
| Genus g | “Number of holes” or dimension of H¹(X,𝒪ₓ) | • Complete invariant for smooth projective curves |
| g = 0 | Rational curves (isomorphic to ℙ¹) | • Simplest case<br>• Rational parameterization |
| g = 1 | Elliptic curves | • Group structure<br>• Central in number theory |
| g ≥ 2 | Higher genus curves | • Moduli spaces<br>• Hyperbolic geometry |
Surfaces (Dimension 2)
| Classification | Examples | Key Properties |
|---|---|---|
| Rational Surfaces | ℙ², Hirzebruch surfaces | • Birational to ℙ² |
| Ruled Surfaces | Products of curve with ℙ¹ | • Fiber bundles over curves |
| K3 Surfaces | Quartic surfaces in ℙ³ | • Trivial canonical bundle<br>• Rich geometry |
| Abelian Surfaces | Complex tori | • Group structure |
| Minimal Models | Various | • No (-1)-curves<br>• Building blocks for classification |
Step-by-Step Methods
Computing Properties of a Variety
- Start with equations: Write polynomial equations defining the variety
- Check irreducibility: Determine if the ideal is prime
- Find singularities: Calculate Jacobian matrix and find where it drops rank
- Compute dimension: Find Krull dimension or transcendence degree
- Determine degree: For projective varieties, count points of intersection with generic linear space
- Calculate cohomology: Use Čech cohomology or spectral sequences
Blowing Up Process
- Identify center: Select subvariety to blow up
- Construct blow-up space: Introduce exceptional divisor
- Define projection: Map from blow-up to original variety
- Analyze exceptional divisor: Study the new component
- Check effect on singularities: Determine if singularities are resolved
Common Challenges and Solutions
| Challenge | Solution Approach |
|---|---|
| Singularities | • Use blow-ups to resolve<br>• Study desingularization algorithms |
| Non-closure in affine space | • Work in projective space<br>• Add “points at infinity” |
| Dimension calculation | • Use Krull dimension<br>• Apply Hilbert polynomial techniques |
| Reducibility testing | • Factor polynomials<br>• Check primality of ideals |
| Intersection multiplicities | • Use Serre’s Tor formula<br>• Apply intersection theory |
| Global-to-local principles | • Use sheaf theory<br>• Apply localization techniques |
Important Theorems and Results
| Theorem | Statement | Significance |
|---|---|---|
| Nullstellensatz | For an algebraically closed field, maximal ideals ↔ points | Fundamental bridge between algebra and geometry |
| Riemann-Roch | dim L(D) – dim L(K-D) = deg(D) – g + 1 | Relates geometry (genus) to algebra (divisors) |
| Bezout’s Theorem | Two projective plane curves of degrees d and e intersect in d·e points (counting multiplicity) | Foundational for intersection theory |
| Hartshorne’s Connectedness | A proper variety over an algebraically closed field is connected | Important topological property |
| Serre Duality | H^i(X,F) ≅ H^{n-i}(X,F^∨⊗ω_X)^∨ | Deep symmetry in cohomology |
Best Practices for Calculations
- Start projective: Work in projective space when possible to avoid issues with “points at infinity”
- Use homogeneous coordinates: Makes projective calculations more systematic
- Localize computations: Focus on affine patches when studying local properties
- Employ elimination theory: To find projections and eliminate variables
- Apply birational transformations: To simplify varieties when possible
- Use computer algebra systems: For complex calculations (Macaulay2, Singular, Sage)
Resources for Further Learning
Foundational Textbooks
- Hartshorne, R. “Algebraic Geometry” (Standard graduate text)
- Shafarevich, I. “Basic Algebraic Geometry” (More accessible introduction)
- Eisenbud, D. & Harris, J. “The Geometry of Schemes” (Modern introduction to schemes)
- Liu, Q. “Algebraic Geometry and Arithmetic Curves” (Includes arithmetic perspective)
Advanced Topics
- Griffiths, P. & Harris, J. “Principles of Algebraic Geometry” (Complex algebraic geometry)
- Fulton, W. “Intersection Theory” (Definitive treatment of intersection theory)
- Mumford, D., Fogarty, J., & Kirwan, F. “Geometric Invariant Theory” (Moduli problems)
- Kollár, J. & Mori, S. “Birational Geometry of Algebraic Varieties” (Classification theory)
Online Resources
- The Stacks Project (stacks.math.columbia.edu): Comprehensive reference
- MathOverflow: Questions and discussions at research level
- arXiv.org (math.AG): Latest research papers
- MIT OpenCourseWare: Video lectures and course materials