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ФакультетыУчебные курсы

Научно-исследовательский семинар "Алгебраическая геометрия"

Учебный год: 2018/2019
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").

Textbooks:
  1. Algebraic Geometry by R. Hartshorn
  2. Algebraic varieties by G. Kempf
  3. Foundations of Algebraic Geometry, by R. Vakil; available electronically at http://math.stanford.edu/~vakil/216blog/FOAGaug2610public.pdf
  Prerequisites:
    • 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.
Weakly plan:
  • 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 (khoroshkin@gmail.com), V. Vologodsky (vologod@gmail.com) or to our course assistants Dmitry Krekov (dmkrekov@gmail.com) and Vasily Rogov (vasirog@gmail.com). Here is how we will weight these items:
    Homework: 50% total
    Final Exam: 50%.
Статус: Общефакультетский факультатив
Преподаватели: Вологодский Вадим Александрович (ведет семинары и принимает экзамены/зачеты), Хорошкин Антон Сергеевич (ведет семинары и принимает экзамены/зачеты)
Язык: русский
Когда читается: 1-й курс, 1, 2 модуль
Кредитов: 5
Прогр. уч. дисц.:  program-2428186371-58Hzoual37.pdf
Материалы по курсу: