All Questions
22,770 questions
30
votes
2
answers
10k
views
When is fiber dimension upper semi-continuous?
Suppose $f\colon X \to Y $ is a morphism of schemes. We can define a function on the topological space $Y$ by sending $y\in Y$ to the dimension of the fiber of $f$ over $y$.
When is this function ...
11
votes
5
answers
3k
views
Ribbon graph decomposition of the moduli space of curves
What is a ribbon graph? What is the ribbon graph decomposition of the moduli space of curves? What are some good references for this material?
- 21k
20
votes
10
answers
7k
views
Resources on invariant theory
What are resources on invariant theory? Basically I've run into a need to teach myself some of the basics of invariant theory and was looking for a good place to start. I'd prefer online / freeish ...
27
votes
7
answers
4k
views
How do you see the genus of a curve, just looking at its function field?
Yuhao asked in the 20-questions seminar:
The genus of a curve is a birational invariant; the function field of a curve determines it up to birational equivelance.
How do you see the genus directly ...
- 1,059
6
votes
2
answers
673
views
are deformations of torsion modules always torsion?
Let's say I have a field $\mathbb{K}$ and a flat family of $\mathbb{K}[t]$-modules $M$ over the formal disk $Spec \mathbb{K}[[h]]$.
Now, assume that $M/hM$ is torsion as a $\mathbb{K}[t]$-module (...
- 44.7k
9
votes
1
answer
531
views
Can one check formal smoothness using only one-variable Artin rings?
Let $f:X\rightarrow Y$ be a morphism of schemes over a field $k$. Can one check that $f$ is formally smooth using only Artin rings of the form $k^{\prime}\left[t\right]/t^{n}$, where $k^{\prime}$ is ...
- 10.5k
22
votes
11
answers
13k
views
What is the exact statement of "there are 27 lines on a cubic"?
I think there was a theorem, like
every cubic hypersurface in $\mathbb P^3$ has 27 lines on it.
What is the exact statement and details?
- 15.1k
15
votes
1
answer
2k
views
Is there an example of a scheme X whose reduction X_red is affine but X is not affine?
For Noetherian schemes this follows from Serre's criterion for affineness by a filtration argument.
- 10.5k
20
votes
4
answers
4k
views
What is interesting/useful about Castelnuovo-Mumford regularity?
What is interesting/useful about Castelnuovo-Mumford regularity?
- 10.5k
10
votes
1
answer
2k
views
Can the valuative criteria for separatedness/properness be checked "formally"?
Suppose f:X→Y is a morphism of finite type between locally noetherian schemes. The valuative criterion for separatedness (resp. properness) says roughly that f is a separated (resp. proper) ...
9
votes
1
answer
1k
views
Stack with affine stabilizers but not quasi-affine diagonal
Give an example of a stack X with affine stabilizer groups and separated but not quasi-affine diagonal.
Remarks:
1) If X has finite stabilizer groups then the diagonal is quasi-finite and separated, ...
- 5,039
19
votes
3
answers
2k
views
Can a coequalizer of schemes fail to be surjective?
Suppose $g,h:Z\to X$ are two morphisms of schemes. Then we say that $f:X\to Y$ is the coequalizer of $g$ and $h$ if the following condition holds: any morphism $t:X\to T$ such that $t\circ g=t\circ h$ ...
32
votes
6
answers
9k
views
What is the universal property of normalization?
What is the universal property of normalization? I'm looking for an answer something like
If X is a scheme and Y→X is its
normalization, then the morphism
Y→X has property P and any ...
7
votes
3
answers
585
views
Weil divisors on non Noetherian schemes
Let X be an integral scheme that is separated (say over an affine scheme). Define a Weil divisor as a finite integral combination of height 1 points of X, where the height of a point of X is the ...
- 3,968
9
votes
3
answers
1k
views
If $\Omega_{X/Y}$ is locally free of rank $\mathrm{dim}\left(X\right)-\mathrm{dim}\left(Y\right)$, is $X\rightarrow Y$ smooth?
Suppose I have a morphism $f:X\rightarrow Y$ such that the relative sheaf of differentials $\Omega_{X/Y}$ is locally free. Does it follow that $f$ is smooth?
The answer is no, but for a silly reason. ...
12
votes
2
answers
2k
views
Non-quasi separated morphisms
What are some examples of morphisms of schemes which are not quasi separated?
- 611
2
votes
1
answer
406
views
Are there any criteria for a presheaf which is an etale sheaf to be a sheaf in the fppf topology?
I am happy to hear answers to variants too. For instance, my situation I actually have a sheaf in the smooth topology.
- 10.5k
3
votes
2
answers
857
views
Is there an example of an algebraic stack whose closed points have affine stabilizers but whose diagonal is not affine?
Burt Totaro has a result that for a certain class of algebraic stacks, having affine diagonal is equivalent to the stabilizers at closed points begin affine. Is there an example of this equivalence ...
- 10.5k
38
votes
18
answers
24k
views
Learning about Lie groups
Can someone suggest a good book for teaching myself about Lie groups? I study algebraic geometry and commutative algebra, and I like lots of examples. Thanks.
0
votes
0
answers
2k
views
Ignore this question [closed]
This question is a hacky way to create some tags for you to use. Move along.