Required in stage
Optional in stage
First term
Second term
Both terms
Not organised
This year
Next year
Alternating years
External
Prerequisites
Taught by
Language of instruction
Duration
Identical coursesAims
The course offers an introduction to the classical geometry of solution sets of systems of polynomial equations in several variables (affine and projective varieties). We will explain and illustrate some of the fundamental interactions between algebra and geometry using techniques from algebra and topology.
By the end of the course, the student should have a thorough understanding of the basic objects and techniques in classical algebraic geometry. The student should be able to translate geometric problems into algebraic terms and vice versa, apply algebraic methods to analyze the local and global structure of algebraic varieties.
Previous knowledge
The student needs a good knowledge of algebraic structures as treated in Algebra I (G0N88A). Helpful would be some basic geometry as treated in Meetkunde I (G0N31B) and Meetkunde II (G0N92B).
Is included in these courses of study
Activities
5 ects. Algebraic Geometry I (B-KUL-G0A80a)
Content
- Basic Concepts from algebra and topology: finitely generated algebras, dimension theory of rings and topological spaces, Zariski topology, relation with quotients and localization, algebraic versus topological dimension, basic examples.
- Affine varieties: algebraic sets, morphisms of algebraic sets, Hilbert's Nullstellensatz, dimension of an affine variety.
- Quasi-projective varieties: raded rings, projective, quasi-projective and quasi-affine varieties, morphisms and regular functions, rational maps, blowing-ups, the local ring of a variety at a point, singular and regular points.
- Intersection theory in projective spaces: Bézout theorem and generalizations, Bertini theorem.
Course material
Course notes + Toledo
Format: more information
Lectures with assignments during the lecture.
1 ects. Algebraic Geometry I: Exercises (B-KUL-G0A81a)
Content
- Basic Concepts from algebra and topology: finitely generated algebras, dimension theory of rings and topological spaces, Zariski topology, relation with quotients and localization, algebraic versus topological dimension, basic examples.
- Affine varieties: algebraic sets, morphisms of algebraic sets, Hilbert's Nullstellensatz, dimension of an affine variety.
- Quasi-projective varieties: graded rings, projective, quasi-projective and quasi-affine varieties, morphisms and regular functions, rational maps, blowing-ups, the local ring of a variety at a point, singular and regular points.
- Intersection theory in projective spaces: Bézout theorem and generalizations, Bertini theorem.
Course material
Course notes + Toledo
Evaluation
Evaluation: Algebraic Geometry I (B-KUL-G2A80a)
Explanation
There will be one take-home exam during the semester.
The final exam is also take-home and consists either of classical exam questions or of submission of a short expository paper on a topic of own choice related to the course and agreed upon by the instructor. This paper has to contain, beside some clear introductory theory, non-trivial explicit examples, agreed upon by the instructor, worked out to illustrate the theory.
In order to pass, the student must obtain at least the score 10/20. The take-home exam during the semester will count 5 points, the final exam will count 15 points. If the student has failed to pass, for the second-chance examination no points will be carried forward from the take-home exam or the final exam. The student will be given the chance to pass the course via, again, a package consisting of a new take-home exam and a new final exam, with the same format and score share.