Master Program in Mathematics

Algebraic Geometry(MATH 513)

Content
Course Code Course Name Semester Theory Practice Lab Credit ECTS
MATH 513 Algebraic Geometry 2 3 0 0 3 7
Prerequisites
Admission Requirements
Language of Instruction English
Course Type Elective
Course Level Masters Degree
Course Instructor(s) Ayberk ZEYTİN azeytin@gsu.edu.tr (Email)
Assistant
Objective The aim of this course is to understand affine and porjective geometry and to learn the relation between algebraic notions and geometric structures
Content Ring theory and fields (summary), Polynomials and afine space, Affine algebraic sets, İdeals of algebraic sets, Hilbert’s Nullstellensatz theorem, Radical ideals and Nullstellensatz theorem;
Zariski topology and irreducible algebraic sets, Decomposition of an algebraic set, Polynomial mappings and polynomial functions, Coordinate ring of an algebraic set, Affine change of coordinates, Rational functions and local rings;
Projective space, Projective algebra-geometry dictionary, Homogeneous coordinate ring and function field, Projective change of coordinates, Dehomogenization and homogenization of polynomials, Affine-projective transfer of algebraic sets, Multiprojective space and Segre product;
Algebraic set of a monomial ideal, Hilbert function and dimension, Dimension of a projective algebraic set, Elementary properties of dimension;
Tangent spaces and singularities, blow-up, Smooth algebraic sets, Blow-up of curves and surfaces, Examples.
Course Learning Outcomes To know the definitions and the properties of algebraic sets in affine and projective space
To be able to examine the properties of geometric structures using the properties of ideals
Teaching and Learning Methods Lecture, discussion, problem solving
References A Primer of Algebraic Geometry, Huishi Li
Ideals, Varieties and Algorithms, D. Cox, J. Little, D. O’Shea
Print the course contents
Weekly Contents
Theory Topics
Week Weekly Contents
1 Ring theory and fields (summary), Polynomials and afine space
2 Affine algebraic sets, Ideals of algebraic sets
3 Hilbert’s Nullstellensatz theorem, Radical ideals and Nullstellensatz theorem
4 Zariski topology and irreducible algebraic sets, Decomposition of an algebraic set
5 Polynomial mappings and polynomial functions, Coordinate ring of an algebraic set
6 Affine change of coordinates, Rational functions and local rings
7 Projective space, Projective algebra-geometry dictionary
8 Homogeneous coordinate ring and function field, Projective change of coordinates
9 Dehomogenization and homogenization of polynomials, Affine-projective transfer of algebraic sets
10 Multiprojective space and Segre product
11 Algebraic set of a monomial ideal, Hilbert function and dimension
12 Dimension of a projective algebraic set, Elementary properties of dimension
13 Tangent spaces and singularities, Blow-up, Smooth algebraic sets
14 Blow-up of curves and surfaces, Examples
Practice Topics
Week Weekly Contents
Assessment System
Contribution to Overall Grade
  Number Contribution
Contribution of in-term studies to overall grade 5 50
Contribution of final exam to overall grade 1 50
Toplam 6 100
In-Term Studies
  Number Contribution
Assignments 4 10
Presentation 0 0
Midterm Examinations (including preparation) 1 40
Project 0 0
Laboratory 0 0
Other Applications 0 0
Quiz 0 0
Term Paper/ Project 0 0
Portfolio Study 0 0
Reports 0 0
Learning Diary 0 0
Thesis/ Project 0 0
Seminar 0 0
Other 0 0
Toplam 5 50
Relation of Proficiency
No Program Learning Outcomes Contribution
1 2 3 4 5
1 X
2 X
3 X
4 X
5 X
6 X
7
8
9 X
10
11
12 X
13 X
14 X
ECTS
Activities Number Period Total Workload
Class Hours 14 4 56
Working Hours out of Class 14 3 42
Assignments 4 2 8
Midterm Examinations (including preparation) 1 15 15
Final Examinations (including preparation) 1 25 25
Total Workload 146
Total Workload / 25 5,84
Credits ECTS 6
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