Mathematics

MTHM029 - Algebraic Curves (2019)

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MODULE TITLEAlgebraic Curves CREDIT VALUE15
MODULE CODEMTHM029 MODULE CONVENERProf Mohamed Saidi (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11 0 0
Number of Students Taking Module (anticipated) 11
DESCRIPTION - summary of the module content

This module introduces you to the basic concepts of algebraic geometry and algebraic curves. This includes, affine and projective varieties, affine and projective curves, intersection theory in projective space and Bezout's Theorem. It also includes desingularisation of algebraic curves, curves and function fields in one variable, and the Riemann-Roch Theorem.

Pre-requisite Module: MTH2002 Algebra, or equivalent

AIMS - intentions of the module

The module aims to introduce you to some of the central concepts of modern algebraic geometry in an accessible form. The treatment will be in the language of varieties, and will cover the standard properties of affine and projective curves over an algebraically closed field.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:

1 Demonstrate a good understanding of the basic concepts of algebraic geometry in the context of affine and projective curves;

Discipline Specific Skills and Knowledge:

2 Reveal an enhanced understanding of the role of algebraic techniques in the formulation and solution of problems in geometry;

Personal and Key Transferable/ Employment Skills and  Knowledge:

3 Show enhanced problem-solving skills and ability to apply rigorous mathematical argument to the systematic study of geometric questions.

SYLLABUS PLAN - summary of the structure and academic content of the module

- Affine varieties: The Coordinate Ring; Hilbert's Nullstellensatz; irreducible components; multiple points and tangents;

- Projective varieties: projective space; projective plane curves; Bezout’s Theorem; morphisms and rational maps;

- Resolution of singularities;

- Riemann-Roch Theorem and applications.

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
Scheduled Learning & Teaching Activities 50.00 Guided Independent Study 100.00 Placement / Study Abroad 0.00
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
Category Hours of study time Description
Scheduled Learning and Teaching Activities 50 Lectures/example classes
Guided Independent Study 100 Private study

 

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Coursework – Problem sheets 1, 2   All Written comments on script and model solutions available

 

SUMMATIVE ASSESSMENT (% of credit)
Coursework 0 Written Exams 100 Practical Exams 0
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Written Exam – Closed Book 100 2 hours All Written/verbal on request
         
         
         
         

 

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-reassessment
All Above Written exam (100%) All August Ref/Def Period
       
       

 

RE-ASSESSMENT NOTES

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 50% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

ELE – http://vle.exeter.ac.uk

Reading list for this module:

Type Author Title Edition Publisher Year ISBN Search
Set Gibson, C.G. Elementary Geometry of Algebraic Curves: An Undergraduate Introduction Cambridge University Press 2001 978-0521646413 [Library]
Extended Walker, R.J. Algebraic Curves Springer-Verlag 1978 978-3540903611 [Library]
Extended Fulton, W. Algebraic Curves: An Introduction to Algebraic Geometry Addison-Wesley 1989 978-0201510102 [Library]
CREDIT VALUE 15 ECTS VALUE 7.5
PRE-REQUISITE MODULES MTH2002
CO-REQUISITE MODULES
NQF LEVEL (FHEQ) 7 AVAILABLE AS DISTANCE LEARNING No
ORIGIN DATE Tuesday 10 July 2018 LAST REVISION DATE Wednesday 07 August 2019
KEY WORDS SEARCH Affine Space; Algebraic Sets; Hilbert's Nullstellensatz; Coordinate Ring; Local Ring at a Point; Projective Space; Projective Varieties; Plane Projective Curves; Intersection Numbers; Bezout Theorem