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Regular version of the site
2018/2019

Research Seminar "Toric Varieties"

Type: Optional course (faculty)
When: 3, 4 module
Instructors: Karine Kuyumzhiyan
Language: English
ECTS credits: 3

Course Syllabus

Abstract

This is an introduction to the theory of toric varieties, which are algebraic manifolds obtained from convex polytopes by means of some wonderful explicit geometric construction. For example, the standard n-simplex gives in this way the projective space of dimension и. The advantage of the pass from polytopes to toric varieties is that the crutial combinatorial and geometric properties of polytopes predetermine the key properties of the corresponding varieties and vice versa. Almost all essential algebraic, geometric, and topological characteristics of a toric variety are explicitly computable in terms of its polytope. This makes the toric varieties very suitable for testing algebro-geometric and topological conjectures, seeking examples and counterexamples, etc.
Learning Objectives

Learning Objectives

  • he seminar is intended to introduce the subject area to the students, and to offer them an opportunity to prepare and give a talk.
Expected Learning Outcomes

Expected Learning Outcomes

  • Successful participants imporve their presentation skills and prepare for participation in research projects in the subject area.
Course Contents

Course Contents

  • Affine and projective toric varieties, orbit - cone correspondence, automorphisms of affine toric varieties and locally nilpotent derivations, resolution of singularities in dimension 2 and continued fractions.
  • Divisors on toric varieties.
  • Cohomology groups of nonsingular toric varieties
Assessment Elements

Assessment Elements

  • non-blocking Cumulative grade
    cumulative grade is proportional to number of tasks solved.
  • non-blocking Final exam
Interim Assessment

Interim Assessment

  • Interim assessment (4 module)
    0.3 * Cumulative grade + 0.7 * Final exam
Bibliography

Bibliography

Recommended Core Bibliography

  • Cox, D. A., Little, J. B., & Schenck, H. K. (2011). Toric Varieties. Providence, Rhode Island: AMS. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=971087

Recommended Additional Bibliography

  • Fulton, W. (1993). Introduction to Toric Varieties. (AM-131), Volume 131. Princeton: Princeton University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1432979