# Mathematics

## ECMM729 - Algebraic Curves (2018)

MODULE TITLE CREDIT VALUE Algebraic Curves 15 ECMM729 Prof Mohamed Saidi (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11 weeks
 Number of Students Taking Module (anticipated) 10
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 theorem. It also includes desingularisation of algebraic curves, curves and function fields in one variable, and the Riemann-Roch theorem.

Prerequisite module: ECM2711 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; 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
Not applicable

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams 20 80
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
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