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Regular version of the site
2019/2020

Research Seminar "Algebraic Geometry: Language of Schemes II"

Category 'Best Course for Career Development'
Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'
Category 'Best Course for New Knowledge and Skills'
Type: Optional course (faculty)
When: 3, 4 module
Language: English
ECTS credits: 6

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

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