Algebraic geometry

The course can be included in the following programs: Mathematical Sciences, Master's program (N2MAT), or taken as a stand-alone course.

Knowledge equivalent to 90 credits in mathematics, including the course MMA330 Commutative Algebra.

Algebraic curves in the plane and their local and global geometric properties. Affine and projective varieties. Morphisms and rational maps. Function fields and birational geometry. Dimension. Tangent space and regularity. Singularities and resolution of singularities. Intersection theory and intersection multiplicities. Bézout's theorem. Classical examples from projective geometry. Divisors and linear systems.

After passing the course, the student should be able to:

• explain basic concepts and examples in algebraic geometry, with a focus on algebraic curves and varieties,
• reason about geometric structures that arise on algebraic curves, such as addition laws on cubics, tangent lines and singularities,
• use affine and projective coordinates to describe and analyze algebraic varieties,
• describe regular and rational mappings between varieties and their function fields,
• reason about local and global geometric properties of varieties, such as dimension, regularity and singularities,
• use tangent spaces and intersection-theoretic ideas to analyze intersections and local behaviors,
• explain how algebraic and geometric methods interact in the study of algebraic curves, with a view to more advanced global results in algebraic geometry.

The course is assessed by an oral or written exam at the end. During the course, there may be optional assignments that give bonus points on the exam. Examples of such assignments are small written tests, labs, and oral or written presentations. Information about this is found on the course home page.

If a student who has been failed twice for the same examination element wishes to change examiner before the next examination session, such a request is to be granted unless there are specific reasons to the contrary (Chapter 6 Section 22 HF).

If a student has received a certificate of disability study support from the University of Gothenburg with a recommendation of adapted examination and/or adapted forms of assessment, an examiner may decide, if this is consistent with the course’s intended learning outcomes and provided that no unreasonable resources would be needed, to grant the student adapted examination and/or adapted forms of assessment.

If a course has been discontinued or undergone major changes, the student must be offered at least two examination sessions in addition to ordinary examination sessions. These sessions are to be spread over a period of at least one year but no more than two years after the course has been discontinued/changed. The same applies to placement and internship (VFU) except that this is restricted to only one further examination session.

If a student has been notified that they fulfil the requirements for being a student at Riksidrottsuniversitetet (RIU student), to combine elite sports activities with studies, the examiner is entitled to decide on adaptation of examinations if this is done in accordance with the Local rules regarding RIU students at the University of Gothenburg.

The grading scale comprises: Pass with Distinction (VG), Pass (G), and Fail (U).

The course is evaluated with an anonymous questionnaire and/or a discussion with the student representatives. The results of and possible changes to the course will be shared with students who participated in the evaluation and students who are starting the course.

The course MMA321 Algebraic Geometry has a large overlap in content with the course MMA320 Introduction to Algebraic Geometry. It is not permitted to register and/or be examined in more than one of these courses.