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Recommended text.
Hartshorne, Algebraic geometry.
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Prerequisities.
An absolute must is
a year long sequence in graduate algebra, e.g., GRV-I and II.
In particular, you
should be comfortable working with groups, rings, ideals, and modules.
Field theory (including Galois theory) is not needed as such.
If you have not taken the
topics course in Algebraic geometry that I offered in Fall 2008, you are
strongly urged to read the online notes for the course, available at:
http://www.math.fsu.edu/~agashe/ag.html
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Course description.
This course will be a continuation of the topics course "Algebraic
geometry" that was offered in Fall 2008, although students who missed the
course above may still be able to take this course. We will start by
recalling basic concepts such as varieties, schemes, and morphisms between
them (which were treated in detail in the previous course; the material
was chosen from sections I.1 to I.3 and II.1 to II.3 of the text).
Then we will discuss some geometric objects associated to schemes,
namely divisors (invertible sheaves) and differentials. After this, we
shall briefly review some homological algebra and define cohomology groups
associated to sheaves. Then we shall discuss the Serre duality theorem
and explain how it implies the Riemann-Roch theorem, which is a very
important classical theorem in algebraic geometry.
We will cover material from sections II.3 to II.8, III.1 to III.7, and IV.1
(but not all of these sections, and not necessarily in that order).
If time permits, we
will do additional topics based on student interest.
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Grading.
The grade will be based on homeworks and class attendance.
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Honor Code. The Academic Honor System at The Florida State
University is based on the premise that each student has the
responsibility 1) to uphold the highest standards of academic
integrity in the student's own work, 2) to refuse to tolerate
violations of academic integrity in the University community, and
3) to foster a high sense of integrity and social responsibility
on the part of the University community. A copy of the University
Academic Honor Code can be found in the current Student Handbook
and you are bound by it in all your academic work.
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American Disabilities Act. Students with disabilities needing
academic accommodations should register with and provide documentation
to the Student Disability Resource Center (SDRC), and bring a letter
from the SDRC to the instructor indicating their needs.This should
be done within the first week of class.
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