Research Seminar "Algebraic Geometry"
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'
Delivered at: Faculty of Mathematics
When: 1 year, 1, 2 module
ECTS credits: 5
Syllabus: The course will cover most of Hartshorn's book (see below). Additional topics: general Riemann-Roch Theorem, the Hilbert scheme and its application to the existence theorems, a proof the Weil conjectures for curves over finite fields, rational curves on Fano varieties ("bend-and-break trick").
- Algebraic Geometry by R. Hartshorn
- Algebraic varieties by G. Kempf
- Foundations of Algebraic Geometry, by R. Vakil; available electronically at http://math.stanford.edu/~vakil/216blog/FOAGaug2610public.pdf
- Commutative algebra: it is expected that students have studied the material from the book "Introduction To Commutative Algebra" by Atiyah and MacDonald, though some of the results will be reviewed and even reproved.
- Homological Algebra and sheaf theory. Cohomology of sheaves. I recommend the book "Methods of homological algebra" by Gelfand and Manin though it has more material then we will actually use. If you can reproduce a proof that the cohomology of the constant sheaf R on a smooth manifold can be computed by the de Rham complex you should not worry.
- Lecture 1: Definition of ringed spaces and schemes. Fiber products, Proj construction, closed immersions, dimension of a scheme. (Hartshorn, Chapter II, Section 2)
- Seminar 1: A commutative algebra test. (The test grade will not be counted towards the course grade).
- Lecture 2: The existence of affine schemes: spectrum of a ring. Topological properties of schemes. Fiber products. Proj construction. Closed immersions.
General Policy: There will be weakly HW assignments and and a two hour final exam. Homework will usually be assigned weekly on Tuesday and due Friday the next week. Students are expected to explain their solutions to A. Khoroshkin (email@example.com), V. Vologodsky (firstname.lastname@example.org) or to our course assistants Dmitry Krekov (email@example.com) and Vasily Rogov (firstname.lastname@example.org). Here is how we will weight these items:
Homework: 50% total
Final Exam: 50%.