All Questions
2,036 questions
30
votes
6
answers
8k
views
Algebraic stacks from scratch [closed]
I have a pretty good understanding of stacks, sheaves, descent, Grothendieck topologies, and I have a decent understanding of commutative algebra (I know enough about smooth, unramified, étale, and ...
30
votes
2
answers
2k
views
Diffeomorphic Kähler manifolds with different Hodge numbers
This question made me wonder about the following:
Are there orientedly diffeomorphic Kähler manifolds with different Hodge numbers?
It seems that this would require that those manifolds are not ...
- 42.9k
28
votes
4
answers
7k
views
Extending vector bundles on a given open subscheme
Let $U$ be a dense open subscheme of an integral noetherian scheme $X$ and let $E$ be a vector bundle on $U$. Suppose that the complement $Y$ of $U$ has codimension $\textrm{codim}(Y,X) \geq 2$. Let $...
- 9,497
28
votes
5
answers
3k
views
MaxSpec, Spec, ... "RadSpec"? Or, why not look at all radical ideals?
I was reading this question on why algebraic geometry looks at prime ideals instead of only maximal ideals, and I understand Anton's answer, but I'm a little confused as to how this fits with Hilbert'...
- 6,792
27
votes
3
answers
6k
views
learning Deligne-Lusztig theory
Can someone give me a roadmap for learning Deligne-Lusztig theory? (Except for the original article by Deligne and Lusztig)
Edit: You may assume knowledge of representation theory of finite groups (...
user19475
26
votes
7
answers
6k
views
When is a product of elliptic curves isogenous to the Jacobian of a hyperelliptic curve?
David's question Families of genus 2 curves with positive rank jacobians reminded me of a question that once very much interested me: when is a product of elliptic curves isogenous to the jacobian of ...
- 4,593
26
votes
1
answer
5k
views
What are the epimorphisms in the category of schemes?
Is there a known characterization of epimorphisms in the category of schemes?
It is easy to see that a morphism $f : X \to Y$ such that the underlying map $\lvert f\rvert$ is surjective and the ...
- 63.1k
24
votes
3
answers
4k
views
Are there "motivic" proofs of Weil conjectures in special cases?
This is a question meant as a first step to get into reading more on Weil conjectures and standard conjectures. It is known that the standard conjectures on vanishing of cycles would imply the Weil ...
- 7,442
24
votes
1
answer
3k
views
What is the precise relationship between Langlands and Tannakian formalism?
As anyone who's been reading the forums closely can see, I've been averaging a question a day about Tannakian formalism for the past few days. It's quite an interesting concept!
In any case, I wish ...
- 6,348
23
votes
1
answer
3k
views
What is the status of the Friedlander-Milnor conjecture today?
For the purposes of this question, the Friedlander-Milnor (FM) conjecture asserts an equality of the group homology for algebraic groups, and their discretizations in the following sense:
Conjecture ...
- 231
22
votes
1
answer
884
views
q-Catalan numbers from Grassmannians
In this question by $q$-Catalan numbers I mean the $q$-analog given by the formula $\frac{1}{[n+1]_q}\left[{2n\atop n}\right]_q$. The polynomial $\left[{2n\atop n}\right]_q$ represents the class of ...
- 85.6k
22
votes
1
answer
7k
views
What are the monomorphisms in the category of schemes?
Someone recently asked what the epimorphisms in the category of schemes are; the other day I had been wondering about the similar question: what are the monomorphisms in the category of schemes? I am ...
- 1,329
21
votes
3
answers
4k
views
How many flat connections has a line bundle in algebraic geometry?
Suppose $X$ is a projective variety over $\mathbb C$. I am happy to entertain more or different adjectives — I'm not looking for the most general statement, but rather to understand when and how ...
- 54.6k
21
votes
3
answers
4k
views
What is a twisted D-Module intuitively?
When I think about $\mathcal{D}$-Modules, I find it very often useful to envison them as vectorbundles endowed with a rule that decides whether a given section is flat. Or alternatively a notion of ...
- 13.2k
20
votes
3
answers
2k
views
what is the maximum number of rational points of a curve of genus 2 over the rationals
Conjecturally, there exists an integer $n$ such that the number of rational points of a genus $2$ curve over $\mathbf{Q}$ is at most $n$. (This follows from the Bombieri-Lang conjecture.)
We are ...
- 209
20
votes
3
answers
3k
views
What is Kirillov's method good for?
I am planing to study Kirillov's orbit method. I have seen Kirillov's method in several branch of mathematics, for instance, functional analysis, geometry, .... Why is this theory important for ...
user21574
19
votes
2
answers
2k
views
Dual versions of "folding" symmetric ADE Dynkin diagrams?
Start with the Dynkin diagram of an irreducible root system, typically associated with a simple
Lie algebra over $\mathbb{C}$ or a simple algebraic group. Most of the simply-laced ADE
diagrams ...
- 52.9k
19
votes
2
answers
5k
views
Relatively ample line bundles
Is it true that a line bundle is relatively ample iff its restsriction to fibers is? If so, what would be the reference?
- 1,590
18
votes
1
answer
2k
views
Conjecture: The number of points modulo $p$ of certain elliptic curve is $p$ or $p+2$ for $p$ of form $p=27a^2+27a+7$
Numerical evidence suggests a conjecture that the number of
points of certain elliptic curve over $\mathbb{F}_p$ is
either $p$ or $p+2$ for $p$ of certain form.
Let $p$ be prime of the form $p=27a^2+...
- 25.4k
18
votes
3
answers
5k
views
Why are local systems on a complex analytic space equivalent to vector bundles with flat connection?
Let $X$ be a complex analytic space. It is a 'well known fact' that the categories of local systems on $X$ (i.e. locally constant sheaves with stalk $C^n$), and of (holomorphic) vector bundles on $X$ ...
- 1,488
18
votes
1
answer
1k
views
Smith Normal Form of powers of a matrix
What invariants of a matrix determine the Smith Normal Form (SNF) of all the powers of a matrix?
The question makes sense over any PID $R$. If we let $M = M_n(R)$ and $G=Gl_n(R)$, then SNF is a ...
- 3,526
18
votes
1
answer
1k
views
A natural refinement of the $A_n$ arrangement is to consider all $2^n-1$ hyperplanes given by the sums of the coordinate functions. Have you seen this arrangement? Is it completely intractable?
The short version
Here is an extremely natural hyperplane arrangement in $\mathbb{R}^n$, which I will call $R_n$ for resonance arrangement.
Let $x_i$ be the standard coordinates on $\mathbb{R}^n$. For ...
- 2,372
18
votes
1
answer
5k
views
Best strategy for small resolutions
I would like to know if there is a standard technique to check if a singular variety admits a small resolution. What are the main references for these types of questions?
I am mostly interested in ...
- 3,022
17
votes
4
answers
4k
views
Completion of a local ring of a curve
Let $X$ be a smooth projective irreducible curve defined over an algebraically closed field $\mathbb{K}$ (of arbitrary characteristic), and let $p\in X$ be a closed point. Denote by $\mathcal{O}_p(X)$ ...
- 7,722
17
votes
4
answers
2k
views
Comparing fundamental groups of a complex orbifolds and their resolutions.
Let $X$ be a complex manifold with quotient singularities, and let $\tilde X$ be its resolution (that exists, for example, by Hironaka). Then I am pretty sure that $\pi_1(X)\cong \pi_1(\tilde X)$.
...
- 28.9k
16
votes
2
answers
3k
views
Is the Torelli map an immersion?
The Torelli map $\tau\colon M_g \to A_g$ sends a curve C to its Jacobian (along with the canonical principal polarization associated to C); see this question for a description which works for families....
- 10.5k
16
votes
3
answers
2k
views
Are there nonisotrivial elliptic curves over $\mathbb{G}_m$?
Is there an elliptic curve over $\mathbb{C}[t, t^{-1}]$ that has a nonconstant $j$-invariant? What is an equation for such a curve, if it exists?
- 2,663
16
votes
1
answer
2k
views
Comparison of the classical Fourier transform and the Fourier-Mukai transform [closed]
This question has been revised. Skip to the question in bold.
Two MO questions, "Heuristic behind the Fourier-Mukai transform" and "Explaining Mukai-Fourier transforms physically," compel me to ask ...
15
votes
3
answers
3k
views
algebraic de Rham cohomology of singular varieties
Hi,
Is there a simple example of an (affine) algebraic variety $X$ over $\mathbb C$ where
the $H^*_{dR}(X/\mathbb C) = H^*(\Omega^\bullet_{A/\mathbb C})$ differs from the singular cohomology $H^*_{...
- 2,842
15
votes
3
answers
3k
views
Existence of fine moduli space for curves and elliptic curves
For the moduli problem of a curve of genus $g$ with $n$ marked points, how large an $n$ is needed to ensure the existence of a fine moduli space? For this question, terminology is that of Mumford's ...
- 7,442
14
votes
2
answers
882
views
A complex manifold which is quasiprojective in two different ways
Does there exist a complex manifold M which is a quasiprojective variety in two "essentially" different ways? Let me be more specific. I'm looking for a complex manifold M together with two ...
- 44.8k
14
votes
0
answers
958
views
What is the state of art in Groebner bases
How big polynomial systems can we deal with? How do you know when you don't even have to try?
Motivation:
Recently I tried to solve a problem posed in another MO question and ultimately I got stuck ...
- 8,597
14
votes
1
answer
497
views
3-fold of general type homeomorphic to rational 3-fold
Is there a smooth (complex projective) 3-fold of general type which is homeomorphic (in the complex topology) to a rational $3$-fold?
I am aware of such examples in complex dimension $2$, for ...
- 6,995
13
votes
4
answers
2k
views
Fundamental groups of compact Kähler manifolds
This is a sort of a follow-up to this question, and especially to Sean Lawton's answer: The book Fundamental Groups of compact Kähler manifolds (which, in my opinion, is one of the best mathematics ...
- 96.4k
12
votes
1
answer
5k
views
Closest 3D rotation matrix in the Frobenius norm sense
Given a 3 by 3 matrix $M$ I would like to find the rotation matrix $R$ minimizing the Frobenius norm:
\begin{equation}
\|R-M\|_F
\end{equation}
Is there a closed form solution for $R$, or is it ...
- 561
12
votes
4
answers
2k
views
Motivation for definition of Quotient stack
I am reading "Some notes on Differentiable stacks" by J. Heinloth. In that paper, the notion of quotient stack is defined as follows.
Let $G$ be a Lie group action on a manifold $X$ (left ...
- 6,023
12
votes
5
answers
5k
views
reduced ⊗ reduced = reduced; what about connected?
Several questions actually.
All rings and algebras are supposed to be commutative and with $1$ here.
(1) Let $k$ be a field, and let $A$ and $B$ be two $k$-algebras. I need a proof that if $A$ and $...
- 33.8k
12
votes
7
answers
2k
views
Can the Category of Schemes be Concretized?
If not, are there any interesting subcategories that can be concertized? If I am not mistaken, the category of reduced finite type varieties over the complex numbers would be an example, where the ...
- 5,548
11
votes
0
answers
630
views
Inversion, Koszul duality, combinatorics and geometry
According to this MO answer Koszul duality is related to operations on generating series;
1) multiplicative inversion for quadratic algebras,
2) compositional inversion for quadratic operads,
3) ...
11
votes
3
answers
2k
views
What is the difference between Grothendieck groups K_0(X) vs K^0(X) on schemes?
More specifically, I was wondering if there are well-known conditions to put on $X$ in order to make $K_0(X)\simeq K^0(X)$. Wikipedia says they are the same if $X$ is smooth. It seems to me that you ...
- 970
11
votes
1
answer
2k
views
Descent theorems for fundamental groups and groupoids?
Grothendieck in his 1984 "Esquisse d'un programme" (Section 2) wrote (English translation):
" ..,people still obstinately persist, when calculating with fundamental groups, in fixing a single base ...
- 12.3k
10
votes
3
answers
1k
views
About the classification of commutative and of cocommutative, fin. dim. Hopf algebras
I want to prove that the cocommutative finite dimensional Hopf algebras over an algebraically closed field of characteristic zero are group algebras (for some finite group) and that the commutative f....
- 7,746
10
votes
5
answers
3k
views
Multipolynomial resultants
We know that the resultant of two polynomials can be computed as the determinant of their Sylvester matrix ( http://en.wikipedia.org/wiki/Sylvester_matrix ). How do we compute the resultant of more ...
- 103
9
votes
1
answer
1k
views
Nonabelian $H^2$ and Galois descent
I would like to know whether the following metatheorem on nonabelian $H^2$ has been ever stated and/or proved.
Let $k$ be a perfect field and $k^s$ its fixed separable closure.
Let $X^s$ be a variety ...
- 14.2k
8
votes
2
answers
764
views
Is $G/T$ a projective variety?
Let $G$ be a semisimple Lie group and $T$ be its maximal torus. Can we say that $G/T$ is a projective variety?. Is there any proof or counterexample for it?
user21574
7
votes
4
answers
3k
views
Measure on real Grassmannians
OK, so I'm reading about this nice measure you can define on a (real) Grassmannian on Wikipedia. Basically, and to save you the trip through the link, consider the Haar measure $\theta$ on $O(n)$, fix ...
- 4,586
7
votes
3
answers
1k
views
Simplest complex curves with isomorphic Jacobian
Hi.
What are the simplest examples to have in mind of non-isomorphic smooth (complex) algebraic curves with isomorphic jacobian variety?
- 73
7
votes
1
answer
436
views
Can one divide algebraic manifolds ? Make sense: $Gr(2,n)/ Gr(2,n+m) = P^{n-1}/P^{n+m-1} P^{n-2}/P^{n+m-2}$
Let's start from a little bit far.
Basic probability theory - chain rule reads:
$$ P(AB) = P(A)P(B|A)$$
Example: consider n+m balls, where n - white balls, m - black balls,
consider A - first ...
- 24.9k
7
votes
2
answers
4k
views
Number of Plücker relations for a Grassmannian
Is it true that the number of Plücker relations for a Grassmannian $Gr(k,n)$ is equal to the dimension $k(n-k)$ of said Grassmannian? So far, for $Gr(2,5)$, I get exactly five Plücker relations: $$p_{...
- 349
6
votes
1
answer
283
views
If number of points on a manifold is $q^n ( [n+1]_q )$ does it imply a geometric relation to $A^n (P^n)$?
Consider an algebraic manifold whose number of points is $q^n ([n+1]_q)$. Is there a geometric relation to $A^n (P^n)$? In particular, is there an equivalence in the Grothendieck ring of varieties ...
- 24.9k