BSc Mathematics with Placement Year

Algebraic geometry studies objects called varieties defined by polynomial equations. A very simple example is the hyperbola defined by the equation xy = 1 in the plane. There is a way of associating rings to varieties, and then the geometric properties can be studied using algebra, for example points correspond to maximal ideals, or the geometry of the variety can give information about certain algebraic properties of the ring. Algebraic geometry originated in nineteenth century Italy, but it is still a very active area of research. It has close connections with algebra, number theory, topology, differential geometry and complex analysis.

To introduce students to the basic notions of affine and projective algebraic geometry.

Successful students will be able to:

• define the basic concepts of algebraic geometry,
• prove the main theorems about the properties of algebraic varieties, morphisms and rational maps between them, and the correspondence between algebraic varieties and ideals, rings and fields,
• define the tangent space to an affine algebraic variety at a point, prove its properties and calculate the dimension and the singular locus of affine algebraic varieties by using tangent spaces,
• apply the concepts and theorems of algebraic geometry in concrete examples,
• prove properties of the automorphism groups of projective spaces and calculate explicit automorphisms of P¹ by using the cross ratio,
• define elliptic curves and the addition operation on them, and prove properties of elliptic curves and the addition operation,
• carry out calculations involving elliptic curves.

1. Affine varieties, Hilbert's Nullstellensatz

2. Co-ordinate rings, function fields, morphisms and rational maps between affine varieties.

3. Tangent spaces and dimension.

4.  Projective spaces and varieties.

5. Geometry in the plane.

6. Elliptic curves.

• Coursework: 20% (one take home test worth 20%)
• End of semester examination: 80%.

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

M. Reid, Undergraduate Algebraic Geometry, CUP, 1988,

K. Hulek, Elementary algebraic geometry AMS, 2003.

D. A. Cox, D. O'Shea and J. Little, Ideals, Varieties and Algorithms, Springer, 2015. (Ebook is available via the library website and earlier editions are also suitable.)

The independent study hours will normally comprise the following. During each week of the taught part of the semester:
 
•         You will normally have approximately 60-75 minutes of video content. Normally you would spend approximately 2-2.5 hrs per week studying this content independently
•         You will normally have exercise or problem sheets, on which you might spend approximately 1.5hrs per week
•         There may be other tasks assigned to you on Blackboard, for example short quizzes or short-answer formative exercises
•         In some weeks you may be preparing coursework or revising for mid-semester tests
 
Together with the timetabled classes, you should be spending approximately 6 hours per week on this course unit.
The remaining independent study time comprises revision for and taking the end-of-semester assessment.
 
The above times are indicative only and may vary depending on the week and the course unit. More information can be found on the course unit’s Blackboard page.