Undergraduate Course: Algebraic Geometry (MATH11120)
Course Outline
School  School of Mathematics 
College  College of Science and Engineering 
Credit level (Normal year taken)  SCQF Level 11 (Year 5 Undergraduate) 
Availability  Available to all students 
SCQF Credits  10 
ECTS Credits  5 
Summary  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. 
Course description 
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.
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.

Entry Requirements (not applicable to Visiting Students)
Prerequisites 
Students MUST have passed:
Honours Algebra (MATH10069)

Corequisites  
Prohibited Combinations  
Other requirements  None 
Information for Visiting Students
Prerequisites  None 
High Demand Course? 
Yes 
Course Delivery Information

Academic year 2019/20, Available to all students (SV1)

Quota: None 
Course Start 
Semester 2 
Timetable 
Timetable 
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 )

Assessment (Further Info) 
Written Exam
80 %,
Coursework
20 %,
Practical Exam
0 %

Additional Information (Assessment) 
Coursework 20%, Examination 80% 
Feedback 
Not entered 
Exam Information 
Exam Diet 
Paper Name 
Hours & Minutes 

Main Exam Diet S2 (April/May)  Algebraic Geometry (MATH11120)  2:00  
Learning Outcomes
On completion of this course, the student will be able to:
 Demonstrate knowledge of the basic affine and projective geometries.
 Demonstrate familiarity with explicit examples including plane curves, quadrics, cubic surfaces, Segre and Veronese embeddings.
 Demonstrate increased knowledge of finitely generated commutative rings and their fields of fractions.
 Formulate and prove basic statements about algebraic varieties in precise abstract algebraic language.

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. 
Additional Information
Graduate Attributes and Skills 
Not entered 
Keywords  AG 
Contacts
Course organiser  Dr Ivan Cheltsov
Tel: (0131 6)50 5060
Email: I.Cheltsov@ed.ac.uk 
Course secretary  Mr Martin Delaney
Tel: (0131 6)50 6427
Email: Martin.Delaney@ed.ac.uk 

