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Algebraic Geometry: Language of Schemes

2019/2020
Учебный год
ENG
Обучение ведется на английском языке
6
Кредиты
Статус:
Дисциплина общефакультетского пула
Когда читается:
1, 2 модуль

Course Syllabus

Abstract

The course will cover most of "Algebraic Geometry" by Harshorne. Additional topics may include: general Riemann-Roch theorem, The Hilbert scheme and its application to the existence theorems, a proof of the Weil conjectures for curves over finite fields, rational curves on Fano varietis ("bend and break trick")
Learning Objectives

Learning Objectives

  • Master the language of modern Algebraic Geometry
Expected Learning Outcomes

Expected Learning Outcomes

  • Get prepared for research in Algebraic Geometry and related areas such as Geometric Representation Theory and Number Theory
Course Contents

Course Contents

  • Review of commutative algebra
  • Schemes, fiber products
  • Proper morphisms, valuation criteria
  • Coherent sheaves
  • Divisors, the Picard group
  • The case of curves
  • Differentials, smooth morphisms
  • Cohomology of coherent sheaves
  • Serre duality
  • Riemann - Roch theorem
  • Applications to counting points over finite field
  • Introduction to the deformation theory with applications to rational curves on Fano varieties
Assessment Elements

Assessment Elements

  • non-blocking homemade problem sheets
  • non-blocking written exam
Interim Assessment

Interim Assessment

  • Interim assessment (2 module)
    0.5 * homemade problem sheets + 0.5 * written exam
Bibliography

Bibliography

Recommended Core Bibliography

  • Hartshorne, R., & American Mathematical Society. (1975). Algebraic Geometry, Arcata 1974 : [proceedings]. Providence: AMS. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=772699

Recommended Additional Bibliography

  • Dolgachev, I. (2012). Classical Algebraic Geometry : A Modern View. Cambridge: Cambridge University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=473170