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Online course materials for MATH32062

Algebraic Geometry


Unit code: MATH32062
Credit Rating: 10
Unit level: Level 3
Teaching period(s): Semester 2
Offered by School of Mathematics
Available as a free choice unit?: N

Requisites

Prerequisite

Additional Requirements

MATH32062 pre-requsites

Aims

To introduce students to the basic notions of affine and projective algebraic geometry.

Overview

Algebraic geometry studies objects called varieties defined by polynomial equations. A very simple example is the hyperbola defined by the equation xy = 1 in the plane. There is a way of associating rings to varieties, and then the geometric properties can be studied using algebra, for example points correspond to maximal ideals, or the geometry of the variety can give information about certain algebraic properties of the ring. Algebraic geometry originated in nineteenth century Italy, but it is still a very active area of research. It has close connections with algebra, number theory, topology, differential geometry and complex analysis.

Learning outcomes

Successful students will

  • understand the correspondences between algebraic varieties, ideals and co-ordinate rings both in the affine and projective cases,
  • be able to calculate the singular points and the dimension of algebraic varieties,
  • be able to carry out calculations on elliptic curves.

Assessment methods

  • Other - 20%
  • Written exam - 80%

Assessment Further Information

  • Coursework: 20% (in-class test in week 5, weighting within unit 8%, take-home test in week 8 or 9, weighting within unit 12%)
  • End of semester examination (2 hours): 80%.

Syllabus

1.Affine varieties, Hilbert's Nullstellensatz

2.Co-ordinate rings, function fields, morphisms and rational maps between affine varieties.

3. Tangent spaces and dimension.

4.  Projective spaces and varieties.

5. Geometry in the plane.

6. Elliptic curves.

 

Recommended reading

M. Reid, Undergraduate Algebraic Geometry, CUP, 1988,

K. Hulek, Elementary algebraic geometry AMS, 2003.

Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

Study hours

  • Lectures - 22 hours
  • Tutorials - 11 hours
  • Independent study hours - 67 hours

Teaching staff

Gabor Megyesi - Unit coordinator

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