Undergraduate Programme and Module Handbook 2013-2014 (archived)
Module MATH4011: ALGEBRAIC GEOMETRY IV
Department: Mathematical Sciences
MATH4011:
ALGEBRAIC GEOMETRY IV
Type |
Open |
Level |
4 |
Credits |
20 |
Availability |
Available in 2013/14 |
Module Cap |
None. |
Location |
Durham
|
Prerequisites
- Complex Analysis II (MATH 2011) AND Algebra II
(MATH2581)
Corequisites
Excluded Combination of Modules
- Algebraic Geometry III (MATH3321).
Aims
- To introduce the basic theory of algebraic varieties and birational
geometry, with particular emphasis on plane curves.
Content
- Affine and projective plane curves over a field
k.
- Conics, Pappus' Theorem.
- Unique factorisation in polynomial rings.
- Study's lemma, irreducibility.
- Singular points, tangents, points of
inflection.
- Dual plane, linear systems of curves.
- Bezout's theorem: Resultants, weak form of Bezout,
applications of Pascal's theorem, Cayley-Bacharach theorem, group law on
a cubic. Intersection multiplicity, strong form of
Bezout.
- Bezout's theorem: applications, flexes, Hessian,
configuration of flexes of a cubic.
- Elliptic curves, Weierstrass normal form.
- Complex curves as real surfaces.
- Basic topology and manifolds.
- Degree-genus formula.
- Resolution of singularities and non-singular
models.
- Reading material on a topic in the following
area:parametrizing the branches of a curve by Puiseux
Series.
Learning Outcomes
- By the end of the module students will: be able to solve
complex, unpredictable and specialised problems in Algebraic
Geometry.
- have an understanding of specialised and complex theoretical
mathematics in the field of Algebraic Geometry.
- have mastered a coherent body of knowledge of these subjects
demonstrated through one or more of the following topic areas: Affine
and projective plane curves.
- Unique factorisation in polynomial rings.
- Linear systems.
- Bezout's theorem and applications.
- Topology of curves over the complex numbers.
- Knowledge and understanding of a topic in parametrisation of
branches of a curve.
- Students will have highly specialised and advanced
mathematical skills which will be used with minimal guidance in the
following areas: spatial awareness, abstract reasoning.
- Students will have the ability to read independently to
acquire knowledge and understanding in the area of branch
parametrisation.
- Students will have enhanced problem solving skills.
Modes of Teaching, Learning and Assessment and how these contribute to
the learning outcomes of the module
- Lectures demonstrate what is required to be learned and the
application of the theory to practical examples.
- Subject material assigned for independent study develops the
ability to acquire knowledge and understanding without dependence on
lectures.
- Assignments for self-study develop problem-solving skills and
enable students to test and develop their knowledge and
understanding.
- Formatively assessed assignments provide practice in the
application of logic and high level of rigour as well as feedback for
the students and the lecturer on students' progress.
- The end-of-year examination assesses the knowledge acquired
and the ability to solve complex and specialised problems.
Teaching Methods and Learning Hours
Activity |
Number |
Frequency |
Duration |
Total/Hours |
|
Lectures |
40 |
2 per week for 19 weeks and 2 in term 3 |
1 Hour |
40 |
|
Preparation and Reading |
|
|
|
160 |
|
Total |
|
|
|
200 |
|
Summative Assessment
Component: Examination |
Component Weighting: 100% |
Element |
Length / duration |
Element Weighting |
Resit Opportunity |
three hour written examination |
|
100% |
|
Four written assignments to be assessed and
returned. Other assignments are set for self-study and complete solutions
are made available to students.
■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University