TÜRKÇE
ENGLISH
Information
| Unit | FACULTY OF SCIENCE AND LETTERS |
| MATHEMATICS PR. | |
| Code | MT490 |
| Name | Elementary Algebraic Geometry |
| Term | 2018-2019 Academic Year |
| Semester | 8. Semester |
| Duration (T+A) | 3-0 (T-A) (17 Week) |
| ECTS | 5 ECTS |
| National Credit | 3 National Credit |
| Teaching Language | Türkçe |
| Level | Belirsiz |
| Type | Normal |
| Label | E Elective |
| Mode of study | Yüz Yüze Öğretim |
| Catalog Information Coordinator | Prof. Dr. YILMAZ DURĞUN |
| Course Instructor |
Prof. Dr. YILMAZ DURĞUN
(Bahar)
(A Group)
(Ins. in Charge)
|
Course Goal / Objective
This course will introduce the basic objects in algebraic geometry: afine and projective varieties, and the maps between them. The focus will be on explicit concrete examples.
Course Content
Affine and Projective spaces, Unit circle as a motivation for rational curves, rational curves, Conics and easy cases of Bezout s theorem; Cubics and the group law, Pascal s mystic hexagon, Curves and their genus; Noetherian rings, Hilbert basis theorem ; Noether normalization, Zariski topology, Nullstelensatz ; Regular functions and regular maps, product of affine varieties; Irreducible algebraic subsets, Rational functions, Rational maps, Birational maps; Projective variety, Projective Nullstellinsatz, Morphisms of projective varieties; Quadratics, Veronese varieties, Quasiprojective varieties and morphisms; Grassmannian variety, Product of varieties, Segre embedding; Nonsingular points of a hypersurface, Tangent space, Dimension ; Dimension of intersection with a hypersurface, Blow-up in afine space, Blow-up in; projective space; Resolution of singularities; Lines on surfaces
Course Precondition
Resources
Notes
Course Learning Outcomes
| Order | Course Learning Outcomes |
|---|---|
| LO01 | will be able to give the definitions of an affine and a projective varieties |
| LO02 | Will be able to express the relation between algebraic varieties, ideals and coordinate rings in both affine and projective cases |
| LO03 | will be able to calculate the singular points and the dimension of algebraic varieties |
| LO04 | will be able to calculate the genus of a curve |
| LO05 | will be able to resolve simple singularities via blow-ups |
Relation with Program Learning Outcome
| Order | Type | Program Learning Outcomes | Level |
|---|---|---|---|
| PLO01 | - | Comprehend the ability to prove the mathematical knowledge gained in secondary education on the basis of theoretical basis. | |
| PLO02 | - | Understands importance of basic consepts of Algebra, Analaysis and Topology. | |
| PLO03 | - | Mathematical reasoning demonstrates the ability to develop and write mathematical proofs by gaining maturity. | |
| PLO04 | - | Demonstrate the ability to express the basic theories of mathematics both correctly. | |
| PLO05 | - | Understands the relationship between the different fields of mathematics and its relation to other disciplines. | |
| PLO06 | - | Comprehends the ability to understand the relationships between the objects in the most understandable way while creating a model for any problem. | |
| PLO07 | - | Comprehend and explain mathematical models such as formulas, graphs, tables and schema. | |
| PLO08 | - | Demonstrate the ability to mathematically rearrange, analyze, and model the problems they encounter. | |
| PLO09 | - | Comprehends at least one of the computer programming languages. | |
| PLO10 | - | Demonstrate the ability to use scientific methods and appropriate technologies effectively in problem solving. | |
| PLO11 | - | Understands sufficient knowledge of foreign language to be able to understand Mathematical concepts and communicate with other mathematicians | |
| PLO12 | - | In addition to their professional development, they demonstrate their ability to continuously improve themselves by identifying their educational needs in scientific, cultural, artistic and social areas in line with their interests and abilities. | |
| PLO13 | - | Understands the programming techniques and shows the ability to do programming. | |
| PLO14 | - | Demonstrates the ability to study mathematics both independently and as a group. | |
| PLO15 | - | Demonstrate an awareness of the universal and social impacts and legal consequences of mathematical applications in the field of study. | |
| PLO16 | - | Demonstrate the ability to select, use and develop effectively for contemporary mathematical applications. | |
| PLO17 | - | It has ability of lifelong learning awareness, access to information, monitoring developments in science and technology and self-renewal ability. | |
| PLO18 | - | Gains the ability to use information technologies effectively for contemporary mathematical applications. | |
| PLO19 | - | Gains the ability to design, conduct experiments, field work, data collection, analysis, archiving, text solving and / or interpretation according to mathematics fields. | |
| PLO20 | - | Gains the consciousness of prefesional ethics and responsibility. |
Week Plan
| Week | Topic | Preparation | Methods |
|---|---|---|---|
| 1 | Affine and Projective spaces, Unit circle as a motivation for rational curves, rational curves, Conics and easy cases of Bezout s theorem | Review of the relevant pages from sources | |
| 2 | Cubics and the group law, Pascal s mystic hexagon, Curves and their genus | Review of the relevant pages from sources | |
| 3 | Noetherian rings, Hilbert basis theorem | Review of the relevant pages from sources | |
| 4 | Noether normalization, Zariski topology, Nullstelensatz | Review of the relevant pages from sources | |
| 5 | Regular functions and regular maps, product of affine varieties | Review of the relevant pages from sources | |
| 6 | Irreducible algebraic subsets, Rational functions, Rational maps, Birational maps | Review of the relevant pages from sources | |
| 7 | Projective variety, Projective Nullstellinsatz, Morphisms of projective varieties | Review of the relevant pages from sources | |
| 8 | Mid-Term Exam | Review of the relevant pages from sources | |
| 9 | Quadratics, Veronese varieties, Quasiprojective varieties and morphisms | Review of the relevant pages from sources | |
| 10 | Grassmannian variety, Product of varieties, Segre embedding | Review of the relevant pages from sources | |
| 11 | Nonsingular points of a hypersurface, Tangent space, Dimension | Review of the relevant pages from sources | |
| 12 | Dimension of intersection with a hypersurface, | Review of the relevant pages from sources | |
| 13 | Blow-up in afine space, Blow-up in projective space | Review of the relevant pages from sources | |
| 14 | Resolution of singularities | Review of the relevant pages from sources | |
| 15 | Lines on surfaces | Review of the relevant pages from sources | |
| 16 | Term Exams | Review of the relevant pages from sources | |
| 17 | Term Exams | Review of the relevant pages from sources |
Assessment (Exam) Methods and Criteria
| Assessment Type | Midterm / Year Impact | End of Term / End of Year Impact |
|---|---|---|
| 1. Midterm Exam | 100 | 40 |
| General Assessment | ||
| Midterm / Year Total | 100 | 40 |
| 1. Final Exam | - | 60 |
| Grand Total | - | 100 |