Fall 2015, Course 18.725, Algebraic geometry
TuTh 1-2:30, Room E17-136
Instructor: Roman Bezrukavnikov, office E17-426.
e-mail: bezrukav@math.mit.edu
TA: Donkwan Kim, sylvaner@math.mit.edu
The goal of the course is to introduce basic objects of today’s algebraic
geometry: algebraic varieties, coherent sheaves, cohomology etc.
Prerequisites. Basic notions of commutative algebra, such as localization, Noetherian property and prime ideals. Familiarity with the following
topics is helpful though not strictly necessary: basic notions of category
theory: Yoneda Lemma, (co)limits, (co)products; calculus on manifolds, including vector fields, differential forms and de Rham cohomology; beginning
graduate topology: (co)homology, fundamental groups, topological vector
bundles; complex analysis: compact Riemann surfaces.
Textbook. The theory will be presented mostly following Kempf’s book
”Algebraic varieties”, complemented with examples from other sources, especially Shafarevich ”Basic algebraic geometry” and Harthshorne ”Algebraic
geometry”. Time permitting we will introduce schemes following Harthshorne
and/or cover some intersection theory following Shafarevich, chapter 4.
Homework. Weekly homework assignments will be due on Tuesday in
class or electronically by 1pm. You can bring your solutions to class or
my office or e-mail it to Dongkwan (cc Roman) in any reasonable electronic
format, including a readable scan of your handwritten paper.
Collaborations are allowed but you must write up your solution yourself.
Acknowledge your collaborators and sources (other than textbook) consulted
on your paper.
Course materials including homeworks will be posted at:
http : //math.mit.edu/ ∼ bezrukav/18 725.html
Final grade is based on homeworks, no final exam will be given. You
have the option to write an expository paper for extra credit, you should
contact me as early as possible but not later than November 3 to arrange a
topic.
Office hours. Instructor’s OH: Tu, Th 3-4 or by appointment. TA’s
OH: by appointment.
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