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Study information

# Algebraic Curves - 2023 entry

MODULE TITLE CREDIT VALUE Algebraic Curves 15 MTHM029 Prof Andreas Langer (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 0 11 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, or MTH2010 and MTH2011, 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 Guided Independent Study Placement / Study Abroad 50 100 0
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 Written Exams Practical Exams 20 80 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 80 2 hours - Summer Exam Period All Results released online
Coursework 1 10 - All Written comments on script and model solutions available
Coursework 2 10 - All Written comments on script and model solutions available

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
Written exam* Written exam (100%) All August Ref/Def Period
Coursework 1* Coursework 1 All August Ref/Def Period
Coursework 2* Coursework 2 All August Ref/Def Period

*Please refer to reassessment notes for details on deferral vs. Referral reassessment

RE-ASSESSMENT NOTES

Deferrals: Reassessment will be by coursework and/or written exam in the deferred element only. For deferred candidates, the module mark will be uncapped.

Referrals: Reassessment will be by a single written exam worth 100% of the module only. As it is a referral, the mark will be capped at 50%.

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

Reading list for this module:

Type Author Title Edition Publisher Year ISBN
Set Gibson, C.G. Elementary Geometry of Algebraic Curves: An Undergraduate Introduction Cambridge University Press 2001 978-0521646413
Extended Walker, R.J. Algebraic Curves Springer-Verlag 1978 978-3540903611
Extended Fulton, W. Algebraic Curves: An Introduction to Algebraic Geometry Addison-Wesley 1989 978-0201510102
CREDIT VALUE ECTS VALUE 15 7.5
PRE-REQUISITE MODULES MTH2002, MTH2010, MTH2011
NQF LEVEL (FHEQ) AVAILABLE AS DISTANCE LEARNING 7 No Tuesday 10th July 2018 Wednesday 15th March 2023
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

Please note that all modules are subject to change, please get in touch if you have any questions about this module.