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DRPS : Course Catalogue : School of Mathematics : Mathematics

Undergraduate Course: Algebraic Geometry (MATH11120)

Course Outline
SchoolSchool of Mathematics CollegeCollege of Science and Engineering
Credit level (Normal year taken)SCQF Level 11 (Year 5 Undergraduate) AvailabilityAvailable to all students
SCQF Credits10 ECTS Credits5
SummaryAlgebraic 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.
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 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
Week 11. Revision.
Entry Requirements (not applicable to Visiting Students)
Pre-requisites Students MUST have passed: Honours Algebra (MATH10069)
Prohibited Combinations Other requirements None
Information for Visiting Students
High Demand Course? Yes
Course Delivery Information
Academic year 2018/19, 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:
  1. Demonstrate knowledge of the basic affine and projective geometries.
  2. Demonstrate familiarity with explicit examples including plane curves, quadrics, cubic surfaces, Segre and Veronese embeddings.
  3. Demonstrate increased knowledge of finitely generated commutative rings and their fields of fractions.
  4. 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
Course organiserDr Ivan Cheltsov
Tel: (0131 6)50 5060
Course secretaryMrs Alison Fairgrieve
Tel: (0131 6)50 5045
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