SMA 6207 - Algebraic Geometry
Graduate-level introduction to Algebraic Geometry. This course equips students with the computational tools which are useful in Commutative Algebra, Algebraic Geometry, Deformation Theory, Singularity Theory as well as Algebraic Statistics, bridging abstract algebraic foundations with concrete geometric and algorithmic problem-solving techniques.
Course learning outcomes
By the end of this course, you should be able to:
- Analyse the relationship between algebraic sets and ideals in affine and projective spaces using the Zariski topology and Hilbert’s Nullstellensatz.
- Evaluate the geometric properties and birational equivalence of quasi-projective varieties through regular functions and morphisms.
- Apply dimension theory and the Jacobian criterion to compute dimensions and classify singular and non-singular points.
- Synthesise the theory of divisors and discrete valuation rings to analyse the geometry of algebraic and elliptic curves.
Prerequisites
SMA 6108 - Commutative Algebra
Course information | SMA 6207 - Algebraic Geometry | SOMAS
Below is the course content. You can click on any section here and it will take you through to this section of the course. If you are signed in and enrolled on this course we can track your progress.
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Below is the course content. You can click on any section here and it will take you through to this section of the course.
- 1. Foundations of Affine Algebraic Sets
- 1.1. Review of polynomial rings and ideals over algebraically closed fields
- 1.2. Affine algebraic sets and the Zariski topology
- 1.3. The ideal of an algebraic set and basic properties
- 1.4. Summary of Week 1 and introductory exercises
- 2. The Hilbert Basis Theorem and Nullstellensatz
- 2.1. Noetherian rings and the Hilbert Basis Theorem
- 2.2. Irreducible components of algebraic sets
- 2.3. The Weak Hilbert Nullstellensatz
- 2.4. The Strong Hilbert Nullstellensatz and the algebraic-geometric dictionary
- 3. Projective Space and Projective Nullstellensatz
- 3.1. Motivation for projective space and homogeneous coordinates
- 3.2. Projective algebraic sets and homogeneous ideals
- 3.3. The Zariski topology on projective space
- 3.4. The Projective Nullstellensatz and comparisons with the affine case
- 4. Functions, Morphisms, and Geometric Properties
- 1. Regular Functions and Morphisms
- 1.1. Regular functions on affine varieties and coordinate rings
- 1.2. Quasiprojective varieties and local rings
- 1.3. Definition and characterization of morphisms of affine varieties
- 1.4. Morphisms of projective varieties and explicit examples
- 2. Products and Rational Maps
- 2.1. The Segre embedding and products of projective varieties
- 2.2. Quasiprojective products and properties
- 2.3. Dominant morphisms and rational maps
- 2.4. Birational equivalence and examples
- 3. Separatedness and Completeness
- 3.1. The concept of separated morphisms and Hausdorff analogues
- 3.2. Definition and intuition of complete varieties
- 3.3. Proof that projective varieties are complete
- 3.4. Applications of completeness and global properties
- 1. Finite Morphisms and Dimension
- 1.1. Integral extensions and the Going-Up theorem
- 1.2. Geometric consequences of finite morphisms
- 1.3. Definition of dimension via chains of prime ideals
- 1.4. Dimension of a hypersurface and affine-projective relations
- 2. Intersection Theory and Transcendence Degree
- 2.1. Dimension of the intersection of two varieties
- 2.2. Krull's Hauptidealsatz (Principal Ideal Theorem)
- 2.3. Relation of dimension to transcendence degree of function fields
- 2.4. Computational applications and examples
- 3. Tangent Spaces and Nonsingularity
- 3.1. Zariski tangent spaces and the Jacobian criterion
- 3.2. Singular and non-singular points of a variety
- 3.3. Regular local rings and smoothness
- 3.4. Sard's theorem and the set of singular points
- 1. Curves and Discrete Valuation Rings
- 1.1. Definition and basic properties of algebraic curves
- 1.2. Non-singular points of curves and local rings
- 1.3. Connection to discrete valuation rings (DVRs)
- 1.4. Extension of morphisms from curves
- 2. Divisors and Bézout's Theorem
- 2.1. Divisors on curves and linear equivalence
- 2.2. The Picard group of a curve
- 2.3. Degree of morphisms of curves
- 2.4. Linear systems and Bézout's theorem in the projective plane
- 3. Canonical Divisors and Elliptic Curves
- 3.1. Canonical divisors and differentials on curves
- 3.2. The genus of a curve and the Riemann-Roch framework preview
- 3.3. Group law on smooth cubic curves
- 3.4. Introduction to elliptic curves and summary of the course
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