Research Seminar "Algebraic Geometry: Language of Schemes II"

2019/2020

Учебный год

ENG

Обучение ведется на английском языке

6

Кредиты

Кто читает:

Международная лаборатория зеркальной симметрии и автоморфных форм

Статус:

Дисциплина общефакультетского пула

Когда читается:

3, 4 модуль

Преподаватель

Вологодский Вадим Александрович

Course Syllabus

Abstract

The course will cover basic results on cohomology of coherent sheaves on schemes, including the Grothendieck-Serre duality and the Riemann-Roch Theorem. Other topics to be covered include the Weil conjectures for curves over finite fields, rational curves on Fano varieties ("bend-and-break trick"), and the Suslin-Voevodsky theorem on singular cohomology of abstract algebraic varieties.

Learning Objectives

• Prepare students to reading specialized literature in the area of Algebraic Geometry and conducting independent research.

Expected Learning Outcomes

• Master basic technique in modern Algebraic Geometry and be able solve problems on the topic (such as the most of the problems from R. Vakil’s textbook)

Course Contents

• Kahler differential, differential operators, cotangent complex, smooth morphisms
  We will see how the basic concepts of differential geometry can be carried over to scheme theory. Some applications to the deformation theory will be discussed
• The Riemann-Roch Theorem and Serre's duality for curves
  We will give an ``elementary'' proof (due to J, Tate) of the two fundamental results for curves. This approach can be also used to study number fields.
• Cohomology of quasi-coherent sheaves on schemes. The Serre duality.
  We will introduce cohomology, prove basic finiteness results for cohomology of coherent sheaves on proper schemes and a general form of the Serre duality theorem
• Some applications
  Some existence results (due to Mori and Bogomolov-Mumford) for rational curves on algebraic varieties will be discussed. Also, we will prove the Weil conjectures for curves.

Assessment Elements

• Work during the classes
• Exam

Interim Assessment

• Interim assessment (4 module)
  0.5 * Exam + 0.5 * Work during the classes

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

• Voisin, C., & Schneps, L. (2003). Hodge Theory and Complex Algebraic Geometry II: Volume 2. Cambridge: Cambridge University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=120395