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MODULE TITLE	Algebraic Curves	CREDIT VALUE	15
MODULE CODE	MTHM029	MODULE CONVENER	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	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	20	Written Exams	80	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	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

ELE – http://vle.exeter.ac.uk

Reading list for this module:

Type	Author	Title	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, MTH2010, MTH2011
CO-REQUISITE MODULES

NQF LEVEL (FHEQ)	7	AVAILABLE AS DISTANCE LEARNING	No
ORIGIN DATE	Tuesday 10 July 2018	LAST REVISION DATE	Wednesday 15 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

MTHM029 - Algebraic Curves (2023)