Research Seminar "Algebraic Geometry: Language of Schemes II"
- Prepare students to reading specialized literature in the area of Algebraic Geometry and conducting independent research.
- 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)
- Kahler differential, differential operators, cotangent complex, smooth morphismsWe 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 curvesWe 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 applicationsSome 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.
- 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
- 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