In this course the major subjects of algebraic geometry such as sheaves, divisors and related topics are treated. Divisors and its associated sheaves, and the canonical sheaf and the canonical divisor defined from algebraic varieties are displayed. Then properties the algebraic variety has might be determined by divisors and its intersection number.
Based on ring theory and modules, algebraic varieties and sheaves are defined. The geometric properties as manifold are expressed in algebraic terms, also the algebraic properties are characterized geometrically. Mathematical relationship between algebraic and geometric subjects will be acquired through this course.
Course goals are:
Understanding the sheaves and the divisiors, and could find the direct and inverse image of sheaves and divisors.
Calculate the intersection number of divisors.
Understanding the canonical sheaf and the canonical divisor, and could determine the ramification divisor.
Algebraic variety, sheaf, divisor, singularity.
|✔ Specialist skills||Intercultural skills||Communication skills||Critical thinking skills||Practical and/or problem-solving skills|
Ordinary lectures. Assignments will be given during class sessions.
|Course schedule||Required learning|
|Class 1||Algebraic sets and rational maps||Details will be provided during each class session.|
|Class 3||Algebraic varieties and their maps|
|Class 4||Divisors and locally free sheaves|
|Class 5||The canonical sheaf|
|Class 6||Intersection numbers|
|Class 8||Algebraization theorem|
None in particular.
Audun Holme, "A Royal Road to Algebraic Geometry", Springer, ISBN 978-3-642-42921-7
Igor R. Shafarevich, "Basic Algebraic Geometry 1", Springer, ISBN 978-3-642-42726-8
Igor R. Shafarevich, "Basic Algebraic Geometry 2", Springer, ISBN 978-3-662-51401-6
Robin Hartshorne, "Algebraic Geometry", GTM 52, Springer-Verlag, ISBN 0-387-90244-9
Assessments on reports.
No prerequisites are necessary, but enrollment in the related courses is desirable.