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 15 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)
Prerequisites 
Students MUST have passed: (
Algebra (MATH10021) AND
Numbers & Rings (MATH10023))

Corequisites  
Prohibited Combinations  
Other requirements  None 
Additional Costs  None 
Information for Visiting Students
Prerequisites  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 12. Projective plane, conics, plane curves.
Weeks 34. Cubic curves (elliptic curves). Bezout's theorem (without proof) and its applications (CayleyBacharach theorem).
Weeks 56. Affine varieties and their rings of functions. Hilbert Basis Theorem and the Nullstellensatz. Projective varieties.
Weeks 78. Quadric surfaces, blow ups, rational and birational maps.
Weeks 910. 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 

