Undergraduate Course: Algebraic Geometry (MATH11120)
Course Outline
| School | School of Mathematics |
College | College of Science and Engineering |
| Course type | Standard |
Availability | Available to all students |
| Credit level (Normal year taken) | SCQF Level 11 (Year 4 Undergraduate) |
Credits | 10 |
| Home subject area | Mathematics |
Other subject area | None |
| Course website |
None |
Taught in Gaelic? | No |
| Course description | Algebraic geometry studies geometric objects defined algebraically.
It is a classical subject with a modern face that studies geometric
spaces defined by polynomial equations in several variables.
Besides providing crucial techniques and examples to many other
areas of geometry and topology, recent decades have seen remarkable
applications to representation theory, physics and to the construction of algebraic codes.
The goal of the course is to give a basic flavour of the subject as
motivation for further study through the introduction of minimal
background material supplemented by a vast collection of examples.
This course will introduce the basic objects
in algebraic geometry: affine and projective varieties, and the maps
between them. The focus will be on explicit concrete examples.
We plan to cover Sections 1-5 and 7 from Reid's book (see Reading List below), which include :
- basics of commutative algebra,
- Hilbert Basis Theorem and the Nullstellensatz,
- affine and projective varieties,
- morphisms and rational maps between varieties,
- conics, plane curves, quadric surfaces,
- blow ups, birational equivalence,
- Segre and Veronese embeddings,
- cubic surfaces. |
Entry Requirements (not applicable to Visiting Students)
| Pre-requisites |
Students MUST have passed: (
Algebra (MATH10021) AND
Numbers & Rings (MATH10023))
|
Co-requisites | |
| Prohibited Combinations | |
Other requirements | None |
| Additional Costs | None |
Information for Visiting Students
| Pre-requisites | None |
| Displayed in Visiting Students Prospectus? | No |
Course Delivery Information
|
| Delivery period: 2013/14 Semester 2, Available to all students (SV1)
|
Learn enabled: Yes |
Quota: None |
Web Timetable |
Web Timetable |
| Course Start Date |
13/01/2014 |
| Breakdown of Learning and Teaching activities (Further Info) |
Total Hours:
100
(
Lecture Hours 22,
Seminar/Tutorial Hours 5,
Summative Assessment Hours 2,
Programme Level Learning and Teaching Hours 2,
Directed Learning and Independent Learning Hours
69 )
|
| Additional Notes |
|
| Breakdown of Assessment Methods (Further Info) |
Written Exam
95 %,
Coursework
5 %,
Practical Exam
0 %
|
| No Exam Information |
Summary of Intended Learning Outcomes
A first course in algebraic geometry is a basic requirement for study in geometry, algebraic number theory or algebra at the MSc or PhD level. Students who successfully complete this module will :
- have knowledge of the basic affine and projective geometries,
- be familiar with explicit examples including plane curves, quadrics, cubic surfaces, Segre and Veronese embeddings,
- have increased their knowledge of finitely generated commutative rings and their fields of fractions,
- have learned how to formulate and prove basic statements about algebraic varieties in precise abstract algebraic language |
Assessment Information
| See 'Breakdown of Assessment Methods' and 'Additional Notes', above. |
Special Arrangements
| None |
Additional Information
| Academic description |
Not entered |
| Syllabus |
This syllabus is for guidance purposes only :
Weeks 1-2. Projective plane, conics, plane curves.
Weeks 3-4. Cubic curves (elliptic curves). Bezout's theorem (without proof) and its applications (Cayley-Bacharach theorem).
Weeks 5-6. Affine varieties and their rings of functions. Hilbert Basis Theorem and the Nullstellensatz. Projective varieties.
Weeks 7-8. Quadric surfaces, blow ups, rational and birational maps.
Weeks 9-10. Basics of cubic surfaces. 27 lines on a smooth cubic
surface.
Week 11. Revision. |
| Transferable skills |
Not entered |
| Reading list |
The main book for this course will be the book by Miles Reid,
'Undergraduate algebraic geometry'.
Miles Reid, Undergraduate algebraic geometry, CUP.
Frances Kirwan, Complex algebraic curves, CUP.
Miles Reid, Undergraduate commutative algebra, CUP. |
| Study Abroad |
Not entered |
| Study Pattern |
Not entered |
| Keywords | AG |
Contacts
| Course organiser | Dr Susan Sierra
Tel: (0131 6)50 5070
Email: S.Sierra@ed.ac.uk |
Course secretary | Mrs Alison Fairgrieve
Tel: (0131 6)50 5045
Email: Alison.Fairgrieve@ed.ac.uk |
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