12/13/22: A clean solution of 10.23 (b) requires a universal
hypersurface trick. I have added a hint to Homework 6. Refresh the
pdf file, if you do not see the hint.
12/13/22: A clean solution of 10.23 (b) requires a universal
hypersurface trick. I have added a hint to Homework 6. Refresh the
pdf file, if you do not see the hint.
12/3/22: Sorry that the homework posted yesterday was not
accessible. I forgot to change permissions on the file before posting
it. It should be working now.
12/2/22: Homework 6, due Wednesday, December 14, is posted, see the
course home page.
12/1/22: The slides from yesterday's Zoom lecture are posted within
the Class Outlines, see the course home page.
11/16/22: I have to move my office hours this Thursday, November 17,
from 1:25-2:15 p.m. to 11:00-11:50 a.m., because of a thesis defense.
11/15/22: Yes, Tobi was right questioning that statement that I
presented in class last Friday: the blowup X̃ is given in X
× ℙr-1 by the equations yi
fj(x) = yj fi(x). In general, we only
have an inclusion: the blowup X̃ satisfies the above equations,
but these equations may describe a larger set. An example would be the
proper (strict) transform of a variety in 𝔸2 passing
through the origin under blowing up the origin in 𝔸2,
see any of the examples in Example 9.21. The proper transform may be
identified with the blowup of the origin in the variety itself (Remark
9.12). And the set of solutions V(y1 f2(x) -
y2 f1(x)) will contain the whole exceptional set
ℙ1 of the blowup of 𝔸2, while the
proper transform will only intersect the exceptional set at finitely
many points. The argument I gave for the erroneous statement was based
on the assumption that V(yi fj(x) =
yj fi(x)) ∩ U is dense in V(yi
fj(x) = yj fi(x)), which follows from
nowhere and is not satisfied in the above examples. However, the
erroneous in general statement is true in certain cases, such as
blowing up the origin in 𝔸n, see Example 9.15.
11/11/22: Homework 5, due Friday, November 18, is posted, see the
course home page. Sorry for the delay in posting it.
11/10/22, 12:10 a.m.: I need to move my standard Zoom office hours
1:25-2:15 p.m. today, Thursday, November 10, one hour ahead, that is
to say, to 12:20-1:10 p.m. today. I will be there on Zoom and in my
office at the same time.
11/1/22, 3:30 p.m.: I will add some hints to Homework 4 soon. Go to
the homework and make sure you reload the page.
10/25/22, 1:30 a.m.: Homework 4, due November 2, is posted, see the
course home page. Sorry for the delay in posting it.
10/19/22: Hint to homework Problem 4.19(e-f): how many generators does
the maximal ideal of the stalk of stucture sheaf of the affine variety
(e) at 0 have? And are there points like that on the affine variety
(f)?
10/18/22: Reminder: please upload your homework to Gradescope, not
Canvas or Speedgrader.
10/07/22: Homework 3, due October 19, is posted, see the course home
page.
10/04/22: I have added a hint to Problem 3.12 to the pdf file of HW2.
9/27/22: Correction to what I said at the end of class Friday,
September 23: there is no evaluation map ℱa → k
for a general sheaf ℱ, only 𝒪X,a → k.
9/26/22: The second homework is posted, see the course home page.
9/20/22: Here are instructions on how to submit your homework. (The
first one is due by the beginning of class on Wednesday, September
21.) You may submit homework electronically of bring a printout to
class. Electronic submission to Gradescope at
https://www.gradescope.com/courses/445177
is encouraged. If you do not have a Gradescope account, you may sign
up for one using your umn.edu address or access it through Canvas.
9/14/22: Starting next week, September 21, I will attempt to broadcast
class meetings on Zoom, so that you are able to stay at home, should
you get sick or need to quarantine. It is the same link for all the
class meetings:
Join Lecture on Zoom
https://umn.zoom.us/j/92246209888?pwd=Y3I0d2FrMTM2aHdGdWJGcmU4eFc1QT09
Meeting ID: 922 4620 9888
Passcode: uf9vSg
I also have a Zoom link for office hours:
Join Office Hours on Zoom
https://umn.zoom.us/j/96063441796?pwd=dnN0TC9Dc3NmRFVTS0l5Nk1VZlBSZz09
Meeting ID: 960 6344 1796
Passcode: 4WGRVf
My Monday office hours will take place in my office VinH
324 and on Zoom, while my Tuesday and Thursday office hours will be
held on Zoom.
9/11/22: Sorry, I needed to change my office hours again: my schedule
is gradually stabilizing. See the course Homepage or Syllabus.
9/11/22: I have started writing Class Outlines, see the course web
page.
9/11/22: The first homework is posted, see the course home page. The
homework includes reading the end of Chapter 1 (subvarieties) and
Chapter 2 (the Zariski topology) of Gathmann's notes on your
own. Normally, I would not ask you to do that, but this week is
special, as I am out of the country.
9/11/22: 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
textbook. You can find out which part of the text at the Class
Outlines page. Then do the assigned homework problems pertinent to
that material. I also encourage you to work in study groups on
the homework, as most of you did in the first term. Here is a guide
on How
to Form a Successful Study Group.
9/11/22: If you need to register for this course and the system
prevents you from doing that, 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: (2022-12-13 01:47:13 CST)