Undergraduate Programme and Module Handbook 2013-2014 (archived)
Module MATH3321: ALGEBRAIC GEOMETRY III
Department: Mathematical Sciences
MATH3321:
ALGEBRAIC GEOMETRY III
Type |
Open |
Level |
3 |
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 IV (MATH4011).
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 and 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.
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.
- 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 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.
- Assignments for self-study develop problem-solving skills and
enable students to test and develop their knowledge and
understanding.
- In addition, formatively assessed assignments provide feedback
for students and the lecturer on student progress and opportunities for
the lecturer to test and enhance development of modelling and
computation skills.
- Summative examination assesses acquired knowledge,
problem-solving skills and arrange of modelling and computational
skills.
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. |
3 hours |
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