2017 Advanced topics in Algebra H1

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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 classification of singularities and their properties are treated. A resolution of singularities and birational transformation to a minimal model are described.
Unlike smooth manifolds algebraic varieties usually contain singular locus. The exceptional divisor of a resolution contains information of singularities, it helps to understanding the character of singularities.

Student learning outcomes

Course goals are:
Find a resolution of singularities and determine its strict transformation and the exceptional divisor.
Classify the type of singularities.
Understanding the differences of each singularities.


Resolution of singularities, blowing-up, canonical singularity, terminal singularity, rational singularity, minimal model.

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 Blowing-ups Details will be provided during each class session.
Class 2 Resolution of singularities
Class 3 Canonical and terminal singularities
Class 4 Resolution of plane singularities
Class 5 Rational singularities
Class 6 Non rational singularities
Class 7 Minimal model
Class 8 Minimal models


None in particular.

Reference books, course materials, etc.

Shihoko Ishii, "Introduction to Singularities", Springer-Verlag, ISBN 978-4-431-56261-0
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 (100%).

Related courses

  • MTH.A201 : Introduction to Algebra I
  • MTH.A202 : Introduction to Algebra II
  • MTH.A301 : Algebra I
  • MTH.A302 : Algebra II
  • MTH.A507 : Advanced topics in Algebra G1

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

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

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