5/8/18: I have reworded Problem 5 to make it simpler. Reload HW 7 to
see the latest version.
5/8/18: I have reworded Problem 5 to make it simpler. Reload HW 7 to
see the latest version.
5/2/18: My office hours before the homework is due next Wednesday will
be as follows: Thursday, May 3: 11:15-12:05, Tuesday, May 8:
1:30-2:30, Wednesday, May 9: 2:00-3:00.
5/1/18: The last homework has been finalized.
4/22/18: I have posted the first five problems for the last homework,
as I would like to add problems during the last two weeks, depending
on what we will be able to cover in class. Thus, I am putting off the
due date for Homework 7 to May 9.
4/21/18: I have to move my office hours on Monday, April 23 to 1:25 -
2:15 from the standard 2:30-3:20.
4/16/18: There was some mess in the computation of Čech
cohomology of ℤ on the circle S1 today in class. The
problem is that what I called the constant sheaf ℤ: ℱ(U) =
ℤ, is only a presheaf, called the constant presheaf with
value ℤ. Its sheafification is what I called the locally constant
sheaf ℤ, the sheaf of locally constant functions with values in
ℤ. It is this sheaf which is called the constant sheaf
ℤ. Thus, our first computation, when H0(S1,
ℱ) = ℤ and H1(S1, ℱ) = 0 was the
computation of Čech cohomology of the constant presheaf,
whereas the second computation, when H0(S1,
ℤ) = ℤ and H1(S1, ℤ) = ℤ was
the computation of Čech cohomology of the constant
sheaf. Both computations were associated with the open cover of
the circle by two intervals. This last computation is actually done in
Hartshorne's Example III.4.0.4.
I have also messed up the computation of global sections of 𝒪(m) on ℙ0A, having put A in degree -m erroneously. The correct answer is Γ(ℙ0A, 𝒪(m)) = A, being an ungraded A-module. I will comment on this on Wednesday.
4/16/18: Expanded the hint to Problem 4 and reduced reading on
Čech cohomology from HW 6.
4/16/18: Removed the last problem from HW 6.
4/8/18: A couple of changes have been made to Homework 6: a few
clarifications; Problem 4 originally included something that was done
on Homework 4, etc.
4/8/18, 3 a.m.: Homework 6 is now posted.
4/7/18: I have procrastinated again with posting the homework. :-( I
will post it later tonight with a due date of Wednesday, April 18.
4/4/18: I have to move my office hours on Thursday, April 5, even
further, to 12:45 - 11:35 from the standard 11:15-12:05. Sorry for the
late notice and frequent changes.
4/4/18: I have to move my office hours on Thursday, April 5, to 12:20
- 11:10 from the standard 11:15-12:05. Sorry.
4/3/18: I have to move my office hours on Wednesday, April 4, to 11:15
- 12:05 from the standard 1:25-2:15.
3/29/18: I have to put off my office hours today, Thursday, March 29,
to 11:50 - 12:40. Sorry for the late notice.
3/24/18: I have a committee meeting this coming Wednesday, March 28,
during my regular office hours 1:25-2:15 and am moving them to
2:30-3:20. It is going to be a one-time change.
3/23/18: Homework 5 is now posted, due on Monday, April 2. Do not
forget to reload the Homework page, if you do not see the latest
homework.
3/4/18: Homework 4 is now posted, due on Monday, right after the
Spring Break. I have decided to leave some essential material for
your own reading during this week, while the class meetings are
canceled.
2/22/18, 5:10 p.m.: I have added the missing definition of d to
Problem 1. Thus, if you do not see d = dim R there, it is not
the latest copy you are looking at.
2/22/18, 12:05 p.m.: I have made a few minor, but important correction
to Homework 3, which I posted last night. Please make sure to use the
latest copy, the one that has a reference to Homework 2 in Problem 8.
2/22/18, 12:54 a.m.: Homework 3 is posted. Enjoy! Reload the Homework
page, if you do not see the latest homework. Do not hesitate to ask
for a hint. Work in study groups also helps.
2/21/18: Correction: I said in class on Monday, in relation to
explaining why every affine morphism X→S was separated, that
Ui ×ViUi covered X
×SX. Lucy corrected me: this way you actually get an
open covering of the diagonal ΔX(X), but this is all
we care about.
2/14/18: I learned from Alex Ma today just a few minutes before 3:35
p.m. that one condition in Problem 2 on HW 2 was missing. The
condition is that of quasi-compactness of the open embedding given by
an open subscheme of an S-scheme of finite type. I corrected the
problem only after the deadline for the homework. I apologize and will
ask the grader to take this into account. This is all nicely
discussed by Vakil, see Exercises 7.3.Q (a,b).
2/14/18: I forgot that I had a doctor's appointment in the middle of
the day today and had to reschedule my office hours. You may come and
see me between noon and 1:00 and then between 3 and 3:30 today. Or
briefly after the Algebraic Geometry Seminar (in lieu of or class
meeting). Or communicate by email.
2/9/18: Professor Daniel Litt, who will be giving a talk in lieu of
our class on Wednesday, will be giving a colloquium talk one day
earlier, on Tuesday, February 13, intended for a general mathematical
audience. This is what colloquium talks are intended to be. The
Tuesday talk is not a prerequisite his Wednesday talk, but
nevertheless may be helpful. The Tuesday talk is on Fundamental
groups in arithmetic and geometry. Abstract: Let X be an
algebraic variety -- that is, the solution set to a system of
polynomial equations. Then the *fundamental group* of X has several
incarnations, reflecting the geometry, topology, and arithmetic of
X. This talk will discuss some of these incarnations and the subtle
relationships between them, and will describe an ongoing program which
aims to apply the study of the fundamental group to classical problems
in algebraic geometry and number theory. Room: Vincent Hall
16. Time: 3:30-4:30 p.m. I encourage you to attend both talks
(in the right order ;-)).
2/8/18: Added some hints to Problem 7 on Homework 2. Some easy
material of the upcoming Monday class can also be used to solve this
problem.
2/7/18: As I mentioned on Monday in class, there will be an Algebraic
Geometry Seminar talk by a Daniel Litt (Columbia University) on
Wednesday, February 14, in lieu of our class meeting, 3:35-4:35 in our
standard classroom, VinH 313. You are encouraged to attend the talk,
but do not have to. You may bring the homework due that day either to
the talk or put it in my mailbox. Here is some more information on the
talk. Title: Canonical Paths on Algebraic
Varieties. (BTW, an algebraic variety is just a reduced
algebraic scheme over a field, i.e., a separated scheme of finite type
over a field such that sections of the structure sheaf over open
subsets never have nilpotent elements.) Abstract: Given a
path-connected topological space X and two points x and y, there is
typically no distinguished homotopy classes of paths between x and y.
If X is a normal algebraic variety over the complex numbers, however,
there is a distinguished linear combination of paths between x and y;
there is an analogous statement for a variety over any local field.
I'll make this precise and describe many applications to arithmetic
and geometry: for example, to explicit descriptions of Galois actions
on fundamental groups, and to the study of the geometry of Selmer
varieties. Some of the work described is joint with Alexander Betts.
2/7/18, 12:30 a.m.: Homework 2 is finally posted and looks exciting.
2/5/18: I still had a lot of trouble connecting to the departmental
computer from home. I will finish the homework tomorrow night. The due
date will be Wednesday, February 14.
2/4/18: I have to cancel my office hours on Monday, February 5, I am
sorry. If you have any questions, please use email. My work on the
homework has been jeopardized by the fact that my home computer's hard
drive failing. I practically lost remote connection from home.
2/1/18: I have got a cold a decided to stay at home and cancel my
today's (Thursday) office hours to prevent spreading the virus. If you
have any questions, please use email. A Skype appointment is also
possible. I will also be working on the homework from home.
1/24/18: I have added an explanation of what k(s) means to the
Hartshorne problem on HW 1.
1/22/18: To confirm what I said in class, I am moving the due date of
the first homework from this Wednesday to next Monday, January 29.
1/18/18: I will be at a conference in Philadelphia the week of March
5-9, which means there will be no classes, and you will have an
extended Spring Break. I will make sure to provide you with homework
to keep you busy, though...
1/9/18: I wish I posted the first homework earlier, but at least, I
have posted it today.
12/27/17: Stephanie has managed to change the time and classroom for
this course to MW 3:35-4:50, 313 Vincent Hall. No more trips to
another building in the harsh Minnesota winter!
12/25/17: I will post the first homework on the class web site
sometime soon in the holiday period.
12/25/17: You might have noticed that the class is scheduled to meet
MWF 3:35-4:25 in Bruininks Hall. I have requested with Stephanie to
change the time to meet on Mondays and Wednesdays, just as it was in
the Fall. I have requested a room in Vincent or at least closer, too.
12/25/17: I recommend the following way to study for this
class. Attend each class, take notes, participate in class
actively. After each class review your notes and study the
corresponding part of the text. You can find out which part of the
text at the Class Outlines page. Then do the assigned homework
problems pertinent to that material. Some students find it helpful to
read the material before it is covered in class, some prefer to do the
reading after class. And I also encourage you to work in study groups
on the homework, as you probably did in the first term.
12/25/17: If you need to register for this course, please, send a
message to Stephanie Lawson at ugrad@math.umn.edu and ask for a permission number. If
permission is granted, go to OneStop and register.
Last modified: (2018-05-08 17:09:15 CDT)