TÜRKÇE
ENGLISH
Information
| Code | MT522 |
| Name | Introduction to Algebraic Geometry |
| Term | 2024-2025 Academic Year |
| Term | Spring |
| 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 Instructor |
1 |
Course Goal / Objective
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 | ||