Introduction
Introduction to algebraic geometry, emphasizing basic properties and examples of varieties and maps between varieties.
Content
Affine and projective varieties, the Zariski topology, regular and rational maps. A selection of examples, such as Grassmannians, blowups, lines on cubic surfaces, or the Bézout Theorem.
Learning outcome
After completing this course, the student should be able to:
- Reproduce and exemplify the definitions of affine and projective varieties, the Zariski topology, and regular and rational maps.
- Analyse the geometry of manageable examples of varieties, such as determining the dimension, the irreducible components, and other central properties.
- Explain relations between geometric questions for varieties and algebraic questions for commutative rings.
- Carry out and convey reasoning about varieties and about regular and rational maps.
Required prerequisite knowledge
None
Recommended prerequisites
Abstract Algebra (MAT250), Manifolds (MAT510)
Exam
Oral exam
Weight 1/1
Duration 45 Minutes
Marks Letter grades
Aid None permitted
Individual oral exam
Method of work
4 hours lectures per week.
Using AI to support learning and study activities
AI tools may be used in this course as a learning aid, but must never substitute for the understanding, reasoning and skills the course is designed to build — you must be able to produce and explain your own work. Productive uses include asking AI to explain concepts you find difficult, to give feedback on or help debug and structure work you have written yourself, and to generate practice questions for self-testing before assessments. AI should not produce the answers, code, calculations or text that you submit as your own. Any use of AI in assessed work must follow the course's guidelines and, where you submit work, be disclosed as specified in the syllabus — and you remain responsible for critically evaluating whatever the AI produces.
Open for
Admission to Single Courses at Master Level at the Faculty of Science and Technology
Mathematics and Physics - Master
Mathematics and Physics - Master
Admission requirements
Must meet the admission requirements of one of the study programmes the course is open for.
Course assessment
The faculty decides whether early dialogue will be held in all courses or in selected groups of courses. The aim is to collect student feedback for improvements during the semester. In addition, a digital course evaluation must be conducted at least every three years to gather students’ experiences.
The course description is retrieved from FS (Felles studentsystem). Version 1