Math 627 Course Webpage
Algebraic Geometry
Fall 2025, Duke University
General information |
Course description |
AI and academic honesty |
Homework assignments |
Grading policies |
Other texts |
Links
Lecture: Tuesday and Thursday, 11:45 – 13:00, Physics Building 259
Contact information for the Instructor
Name: Prof. Ezra Miller (you should call me "Ezra")
Address: Mathematics Department,
Duke University, Box 90320,
Durham, NC 27708-0320
Office: Physics 209
Phone: (919) 660-2846
Email: ezra
math.duke.edu
Webpage:
https://math.duke.edu/people/ezra-miller
Course webpage: you're already looking at it...
but it's http://math.duke.edu/~ezra/627/627.html
Office hours: Tuesday, 14:15 – 15:15 in Physics 209 or outside
Thursday, 13:00 – 14:00 in Physics 209 or outside
Course content: selected from Chapters 2 – 17 of [Vakil]
=
The Rising Sea: Foundations of Algebraic Geometry, by Ravi Vakil,
dated September 8, 2024
- sheaves
- spectrum of a ring
- schemes (affine, projective, neither)
- types of schemes (reduced, normal, factorial, regular)
- morphisms of schemes: immersions, finiteness conditions
- quasicoherent sheaves
- vector bundles
- fiber products
- affine and projective morphisms
- separated and proper morphisms
Prerequisite: Solid courses on algebra and commutative algebra
covering fields and field extensions, rings, modules, ideals,
localization, primary decomposition, normality, dimension, and unique
factorization, as well as basic notions from category theory:
functors, universal properties, exact sequences, Hom, tensor product,
abelian categories, direct and inverse limits; see [Vakil, Chapter 1]
for a guide to what you might need along these lines. If you are
lacking in some of these, that's okay if you're willing to learn them
on the go by keeping up with the reading.
- Consulting generative AI is prohibited for all aspects of
this course.
- The purpose of this course is to train your mind to think in
certain ways. Past experience has shown that this goal is
impaired or circumvented entirely by AI consultation. Instead
of asking a computer, ask a classmate or the instructor. There
is a place for AI in mathematics, but this is not it.
- It is beyond disrespectful to ask me or a grader to waste our
valuable time providing feedback on machine-generated
output. You wouldn't ask a woodshop instructor to evaluate
a store-bought birdhouse. You wouldn't ask an athletic coach to
evaluate artificially generated video of you taking foul shots.
And in those cases you wouldn't learn anything about carpentry
or basketball. So why would you submit anything AI generated
for homework?
- You may at any time be required to defend your written work
orally, with no advance warning. Don't submit it in writing
if you can't explain it orally.
- If a student is responsible for academic dishonesty on a graded
item in this course, then the student will have an opportunity
to admit the infraction and, if approved by the Office of
Student Conduct, resolve it directly through a faculty-student
resolution agreement; the terms of that agreement would then
dictate the grading response to the assignment at issue. If the
student is found responsible through the Office of Student
Conduct and the infraction is not resolved by a faculty-student
resolution agreement, then the sanctions can be severe and,
depending on the nature of the infraction, might be out of the
hands of the instructor, being instead determined by the Office
of Student Conduct or another University body.
- Students are expected to adhere to the
Duke Community Standard more generally. Students
affirm their commitment to uphold the values of the Duke
University community by signing a pledge that states:
- I will not lie, cheat, or steal in my academic endeavors;
- I will conduct myself honorably in all my endeavors;
- I will act if the Standard is compromised.
- The instructor has a long record of detecting and convicting
violations, resulting in a wide range of sanctions, from
grade changes to removal from the University. The instructor
has prosecuted a lot of AI usage violations. Every single
incident tagged by the instructor—concerning AI or any
other infraction—has resulted in sanctions: I have never
lost a case. Don't test me.
- Tentative due dates for the homework assignments this semester
are listed in the table below. You will have at least two
weeks to do each assignment.
- All solutions you turn in must be typewritten using the
LaTeX templates provided for each of the homework assignments.
Communicating your ideas is an integral part of being a
mathematician. It is essential that you learn this skill in
graduate school. Simply fill your responses into the provided
LaTeX files; the formatting and macros in the preamble are
designed to enable grading, and altering those would disable
their functionality. I will be happy to answer any questions
you might have about LaTeX, although you should ask your
classmates first.
- Submit your homework and midterm solutions using Canvas
"Assignments":
Math 627 Canvas –>
Assignments
Submit both your .tex file (the grader or I may comment on your
LaTeX usage) and your .pdf file, which can serve as
verification that your system produces the same output as ours
do. (There have been cases where .tex files upload improperly;
the .pdf file then also serves as proof of on-time submission.)
- Do not print your solutions. This will save paper.
Nobody will read or otherwise use the printed version anyway.
- I encourage collaboration on homework, but
- Each student must write their own solutions using their own
words, and
- Indicate—on the homework page—who your
collaborators were.
- You must cite sources in your solutions. If you rely on
so-and-so's theorem, then you must state the theorem and tell
me where you found it. Be specific: "the local-ringed spaces
proposition" is not precise; in contrast, "[Vakil, Proposition
7.3.1]" is. (Even better would be, "a key result on morphisms
of affine schemes as local-ringed spaces [Vakil, Proposition
7.3.1]".) Theorems can be known by many names or designations,
so I'm likely not to recognize many theorems by names you might
attach.
- Late homework will not be accepted. Early homework is
fine.
- Homework solutions should be thoroughly explained: there will
be no credit for unsupported answers.
Check here two weeks before each homework is due for the
specifics of the assignments. If an assignment hasn't been posted,
and you think it should have been, then please do email me. Sometimes
I encounter problems (such as, for example, my own internet or the
department's servers going down) while posting assignments. Other
times, I might simply have forgotten to copy the assignment into the
appropriate directory, or to set the permissions properly.