# DEGREE REGULATIONS & PROGRAMMES OF STUDY 2015/2016

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# Undergraduate Course: Algebraic Geometry (MATH11120)

 School School of Mathematics College College of Science and Engineering Credit level (Normal year taken) SCQF Level 11 (Year 4 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 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 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.
 Pre-requisites Students MUST have passed: Honours Algebra (MATH10069) Co-requisites Prohibited Combinations Other requirements None
 Pre-requisites None High Demand Course? Yes
 Academic year 2015/16, 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
 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
 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.
 Graduate Attributes and Skills Not entered Keywords AG
 Course organiser Dr Ivan Cheltsov Tel: (0131 6)50 5060 Email: I.Cheltsov@ed.ac.uk Course secretary Mrs Alison Fairgrieve Tel: (0131 6)50 5045 Email: Alison.Fairgrieve@ed.ac.uk
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