## 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

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:

• define the basic concepts of algebraic geometry,
• prove the main theorems about the properties of algebraic varieties, morphisms and rational maps between them, and the correspondence between algebraic varieties and ideals, rings and fields,
• define the tangent space to an affine algebraic variety at a point, prove its properties and calculate the dimension and the singular locus of affine algebraic varieties by using tangent spaces,
• apply the concepts and theorems of algebraic geometry in concrete examples,
• prove properties of the automorphism groups of projective spaces and calculate explicit automorphisms of P1 by using the cross ratio,
• define elliptic curves and the addition operation on them, and prove that the addition operation defines a group structure,
• carry out calculations involving elliptic curves.

#### Assessment methods

• Other - 20%
• Written exam - 80%

#### Assessment Further Information

• Coursework: 20% (one take home test worth 20%)
• 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.

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