2017 Advanced topics in Algebra G1

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Academic unit or major
Graduate major in Mathematics
Suzuki Masatoshi  Kawachi Takeshi 
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Course description and aims

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.

Student learning outcomes

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.

Competencies that will be developed

Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

Ordinary lectures. Assignments will be given during class sessions.

Course schedule/Required learning

  Course schedule Required learning
Class 1 Algebraic sets and rational maps Details will be provided during each class session.
Class 2 Sheaves
Class 3 Algebraic varieties and their maps
Class 4 Divisors and locally free sheaves
Class 5 The canonical sheaf
Class 6 Intersection numbers
Class 7 Singularities
Class 8 Algebraization theorem


None in particular.

Reference books, course materials, etc.

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

Assessment criteria and methods

Assessments on reports.

Related courses

  • MTH.A201 : Introduction to Algebra I
  • MTH.A202 : Introduction to Algebra II
  • MTH.A301 : Algebra I
  • MTH.A302 : Algebra II

Prerequisites (i.e., required knowledge, skills, courses, etc.)

No prerequisites are necessary, but enrollment in the related courses is desirable.

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