MT522 Introduction to Algebraic Geometry

6 ECTS - 3-0 Duration (T+A)- . Semester- 3 National Credit

Information

Code MT522
Name Introduction to Algebraic Geometry
Semester . Semester
Duration (T+A) 3-0 (T-A) (17 Week)
ECTS 6 ECTS
National Credit 3 National Credit
Teaching Language Türkçe
Level Yüksek Lisans Dersi
Type Normal
Mode of study Yüz Yüze Öğretim
Catalog Information Coordinator Prof. Dr. DOĞAN DÖNMEZ


Course Goal

Investigation of properties of algebraic sets in affine and projective spaces. Their relation to geometry and algebra.

Course Content

Properties of algebraic sets in affine and projective spaces. Proof of Bezout's Theorem.

Course Precondition

None.

Resources

Algebraic Curves: An Introduction to Algebraic Geometry, William Fulton, Addison-Wesley Publishing Company, Advanced Book Program, 1989.

Notes

None.


Course Learning Outcomes

Order Course Learning Outcomes
LO01 Grasps the basic properties of the ring of polynomials
LO02 Understands the algebraic sets and their properties in affine and projective sapce.
LO03 Understands the relationship between algebraic sets and ideals.
LO04 Understands the intersection number for plane curves
LO05 Can state Bezout s Theorem for plane curves


Relation with Program Learning Outcome

Order Type Program Learning Outcomes Level
PLO01 Bilgi - Kuramsal, Olgusal Knows in detail the relationship between the results in her area of expertise and other areas of mathematics. 5
PLO02 Bilgi - Kuramsal, Olgusal Knows in detail the relationship between the results in his area of ​​expertise and other areas of mathematics. 4
PLO03 Bilgi - Kuramsal, Olgusal Establishes new mathematical models with the help of the knowledge gained in the field of specialization. 5
PLO04 Bilgi - Kuramsal, Olgusal Has basic knowledge in all areas of mathematics. 4
PLO05 Bilgi - Kuramsal, Olgusal It presents the knowledge gained in different fields of mathematics and their relations with each other in the simplest and most understandable way.
PLO06 Bilgi - Kuramsal, Olgusal Effectively uses the technical equipment needed to express mathematics. 5
PLO07 Bilgi - Kuramsal, Olgusal poses original problems related to field and presents different solution techniques.
PLO08 Bilgi - Kuramsal, Olgusal carries out original and qualified scientific studies on the subject related to its field. 4
PLO09 Bilgi - Kuramsal, Olgusal Analyzes existing mathematical theories and develops new theories. 3
PLO10 Beceriler - Bilişsel, Uygulamalı Knows the teaching-learning techniques in areas of mathematics that require expertise and uses these techniques effectively at every stage of education. 2
PLO11 Yetkinlikler - Bağımsız Çalışabilme ve Sorumluluk Alabilme Yetkinliği To have knowledge of a foreign language at a level to be able to follow foreign sources related to the field and to communicate verbally and in writing with foreign stakeholders. 4
PLO12 Yetkinlikler - Bağımsız Çalışabilme ve Sorumluluk Alabilme Yetkinliği presents and publishes its original works within the framework of scientific ethical rules for the benefit of its stakeholders.
PLO13 Yetkinlikler - Öğrenme Yetkinliği Adheres to the ethical rules required by its scientific title 4


Week Plan

Week Topic Preparation Methods
1 Ideals in the ring of polynomials and the Hilbert s Basis Theorem. Studying the relevant parts of the texts Öğretim Yöntemleri:
Anlatım
2 Affine algebraic sets and their properties. Studying the relevant parts of the texts Öğretim Yöntemleri:
Anlatım
3 Zariski topology. Irreducible algebraic sets and their relationship with prime ideals Studying the relevant parts of the texts Öğretim Yöntemleri:
Anlatım
4 Hilbert s Nullstellensatz. Projective sapces and their properties. Studying the relevant parts of the texts Öğretim Yöntemleri:
Anlatım
5 Homogeneous polynomials, homogeneous ideals and projective algebraic sets. Studying the relevant parts of the texts Öğretim Yöntemleri:
Anlatım
6 Homogenization and dehomogenization of polynomials. Their properties. Studying the relevant parts of the texts Öğretim Yöntemleri:
Anlatım
7 Projective Nullstellensatz. Zariski Topology in Projective space. Relationship with the affine space. Studying the relevant parts of the texts Öğretim Yöntemleri:
Anlatım
8 Mid-Term Exam Solving the homework questions Ölçme Yöntemleri:
Ödev
9 Coordinate rings and homogeneous coordinate rings. Field of rational functions. Studying the relevant parts of the texts Öğretim Yöntemleri:
Anlatım
10 Ring of regular functions. Local rings and their properties. Studying the relevant parts of the texts Öğretim Yöntemleri:
Anlatım
11 Resultant and its properties. Studying the relevant parts of the texts Öğretim Yöntemleri:
Anlatım
12 İntersection number for plane curves. Its properties. Studying the relevant parts of the texts Öğretim Yöntemleri:
Anlatım
13 An isomorphism theorem for two plane curves with finitely many common points. Studying the relevant parts of the texts Öğretim Yöntemleri:
Anlatım
14 Proof of Bezout s Theorem Studying the relevant parts of the texts Öğretim Yöntemleri:
Anlatım
15 Group law on regular plane cubic curves. Studying the relevant parts of the texts Öğretim Yöntemleri:
Anlatım
16 Term Exams Solving the homework questions Öğretim Yöntemleri:
Anlatım
17 Term Exams Solving the homework questions Ölçme Yöntemleri:
Ödev


Student Workload - ECTS

Works Number Time (Hour) Workload (Hour)
Course Related Works
Class Time (Exam weeks are excluded) 14 3 42
Out of Class Study (Preliminary Work, Practice) 14 5 70
Assesment Related Works
Homeworks, Projects, Others 0 0 0
Mid-term Exams (Written, Oral, etc.) 1 15 15
Final Exam 1 30 30
Total Workload (Hour) 157
Total Workload / 25 (h) 6,28
ECTS 6 ECTS