All Questions
22,546 questions
16
votes
3
answers
5k
views
What is an Oper?
Given a curve C, and a reductive group G, there is a moduli stack Loc_G(C), the stack of G-local systems. I keep reading that there's a substack of "opers" but am having trouble locating a definition....
- 16k
4
votes
1
answer
310
views
Solvable subgroups of groups of polynomial automorphisms
Does every finitely generated free solvable group embed into the group of polynomial automorphisms of some C^n?
- 41
2
votes
2
answers
550
views
Algebraic Geometry in an applied setting?
I just saw this paper recently which mentioned that the optimization on a Grassmanian Manifold can be used to get an achieve an best approximation of a multilinear rank of a tensor (in the sense of a ...
- 1,801
6
votes
1
answer
1k
views
What is Drinfeld's manuscript "Best Dream" (in Russian!) about?
I would like to know what Drinfeld's scanned manuscript "Best Dream" is about: the title makes me curious.
It's in Russian.
- 10.8k
1
vote
1
answer
162
views
Does automatic decomposition of varieties into irreducibles exist?
Varieties decompose uniquely into finitely many irreducibles, and each variety is generated by only finitely polynomials. These two finiteness properties make varieties seemingly "manageable" objects, ...
- 2,967
252
votes
37
answers
179k
views
Best algebraic geometry textbook? (other than Hartshorne)
I think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best.
Then what might be the 2nd best?
It can be a book, preprint, online lecture note, webpage, etc.
One suggestion ...
31
votes
3
answers
4k
views
Sheaf description of $G$-bundles
Now, among algebraic geometers, at least, it is well known that there is an equivalence between locally free $\mathcal{O}_X$-modules of rank $n$ and vector bundles of rank $n$. So, equivalently, ...
- 16k
10
votes
3
answers
1k
views
Hamiltonian $S^1$ actions with isolated fixed points
I have in mind the following question for some time. Is there an example of a compact symplectic manifold with a Hamiltonian $S^1$-action with isolated fixed points, that does not admit a compatible $...
44
votes
5
answers
5k
views
Several Topos theory questions
Hey. I have a few off the wall questions about topos theory and algebraic geometry.
Do the following few sentences make sense?
Every scheme X is pinned down by its Hom functor Hom(-,X) by the ...
- 12.1k
6
votes
3
answers
2k
views
What are Log Stacks
So, I've been running in both stacky circles and logarithmic circles and I've been wondering: is there a definition of log stack that is "useful"? I can imagine two such definitions:
1) A log stack ...
- 16k
108
votes
7
answers
21k
views
What is the field with one element?
I've heard of this many times, but I don't know anything about it.
What I do know is that it is supposed to solve the problem of the fact that the final object in the category of schemes is one-...
- 5,062
12
votes
4
answers
3k
views
Elliptic Curves, Lattices, Lie Algebras
I've recently started to look at elliptic curves and have three basic questions:
Is it correct to say that elliptic curves $E$ in the projective plane are in bijective correspondence with lattices $...
- 1,462
12
votes
3
answers
3k
views
ubiquitous quantum cohomology
Manin stressed that every projective scheme should have a quantum-cohomology structure. I'd like to know more about that. And since the varieties considered in texts about monodromy resp. vanishing ...
- 10.8k
13
votes
3
answers
1k
views
A comprehensive overview of finite fields
I've read numerous introductions to finite fields, but I feel like my intuition about them is fairly lacking. Considering that finite fields are the the most "inert" objects in algebraic geometry, I ...
- 11.3k
22
votes
2
answers
3k
views
What is the relationship between integrable systems and toric degenerations?
Given an integrable system on a Kahler manifold X, is there a way to associate a toric degeneration of X whose Milnor fibers are related to the fibers of the integrable system?
An integrable system ...
- 4,949
26
votes
2
answers
2k
views
Manifolds distinguished by Gromov-Witten invariants?
What is a simplest example of a manifold $M^{2n}$ that admits two symplectic structures with isotopic almost complex structures, and such that Gromov-Witten invariants of these symplectic structures ...
- 28.9k
15
votes
6
answers
3k
views
Curves with negative self intersection in the product of two curves
I wonder if the following is known: Are there two compact curves C1 and C2 of genus>1 defined over complex numbers, such that their product contains infinite number of irreducible curves of negative ...
- 28.9k
20
votes
6
answers
4k
views
What are the most important instances of the "yoga of generic points"?
In algebraic geometry, an irreducible scheme has a point called "the generic point." The justification for this terminology is that under reasonable finiteness hypotheses, a property that is true at ...
- 24.1k
75
votes
4
answers
16k
views
What's the "Yoga of Motives"?
There are some things about geometry that show why a motivic viewpoint is deep and important. A good indication is that Grothendieck and others had to invent some important and new algebraico-...
- 15.1k
6
votes
3
answers
601
views
Solving "a, b, a+b have given divisors" problem
I've read an interesting article, math.NT/0409456 where you're just trying to solve a simple problem:
For a given (finite) set of primes S find all solutions to an equation ...
- 15.1k
8
votes
1
answer
596
views
Cartographic group and flat stringy connection
There's a literature about dessins d'enfants (including my previous question here), and one amazing thing about them is that absolute Galois group Gal Q acts on ...
- 15.1k
8
votes
1
answer
1k
views
Learning about Galois representations
My goal was to learn about l-adic representations on some example — I'm a newbie in these topics.
Thus take pt = Spec F_q, ...
- 15.1k
65
votes
17
answers
17k
views
Good introductory references on algebraic stacks?
Are there any good introductory texts on algebraic stacks?
I have found some readable half-finsished texts on the net, but the authors always seem to give up before they are finished. I have also ...
- 1,538
52
votes
7
answers
5k
views
What does a projective resolution mean geometrically?
For R a commutative ring and M an R-module, we can always find a projective resolution of M which replaces M by a sequence of projective R-modules. But as R is commutative, we can consider the affine ...
- 1,056
4
votes
1
answer
416
views
Question about a family of semistable curves
Let $B$ be a curve (integral but not necessarily smooth) and let $\pi: C --> B$ be a family of curves such that each fiber is a rational curve with $g$ many elliptic tails attached.
Let $\omega$ ...
- 10.5k
27
votes
1
answer
3k
views
Stein Manifolds and Affine Varieties
When is a Stein manifold a complex affine variety? I had thought that there was a theorem saying that a variety which is Stein and has finitely generated ring of regular functions implies affine, but ...
- 16k
11
votes
1
answer
2k
views
Non-finitely generated ring of regular functions
It is remarked in Shafarevich's Basic Algebraic Geometry 1 that Rees and Nagata constructed examples of quasiprojective varieties such that the ring of regular functions is not finitely generated, but ...
user709
4
votes
1
answer
765
views
Does the fiber product of two regular varieties over perfect field remain regular?
k is a perfect field. X and Y are two regular varieties over k. Does their fiber product over k remain to be regular?
Note: When k is algebraically closed it's true by Jacobian criterion. When k is ...
- 1,091
8
votes
2
answers
4k
views
Does the fiber product of two normal varieties remain normal?
Suppose $k$ is an algebraically closed field, and $X$, $Y$ are two normal varieties over $k$. Is the product $X \times Y$ necessarily still normal?
- 1,091
50
votes
5
answers
10k
views
Definition and meaning of the conductor of an elliptic curve
I never really understood the definition of the conductor of an elliptic curve.
What I understand is that for an elliptic curve E over ℚ, End(E) is going to be (isomorphic to) ℤ or an ...
- 5,530
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
18
votes
7
answers
6k
views
Langlands Dual Groups
Can someone explain, explicitly, how to, given a reductive complex algebraic group construct the Langlands dual group? I know it is a group with the cocharacters of G as its characters, but how does ...
- 16k
22
votes
4
answers
5k
views
Examples for Decomposition Theorem
There's an important piece of geometric knowledge usually quoted as Beilinson-Bernstein-Deligne.
Here's a refresher: by $IC$ one means the intersection complex, which is just $\mathbb Q$ for a smooth ...
- 15.1k
8
votes
3
answers
832
views
Why is the Hodge class of \bar{M_g} big and nef?
Let pi: \bar{Mg,1} \to \bar{M_g} be natural projection of compactified moduli stacks of curves and let omega be the relative dualizing sheaf. Then the Hodge class \lambda of \bar{M_g} is the first ...
- 10.5k
26
votes
8
answers
3k
views
Bimodules in geometry
Grothendieck's approach to algebraic geometry in particular tells us to treat all rings as rings of functions
on some sort of space. This can also be applied outside of scheme theory (e.g., Gelfand-...
- 37.8k
8
votes
2
answers
2k
views
Properties of monodromy of a fibration?
Sorry for a loaded question.
I'm not an expert on those things, but I do know that a fibration gives rise to the representations of pointed fundamental group of the base on the cohomology of the ...
- 15.1k
58
votes
10
answers
11k
views
What are dessins d'enfants?
There was an observation that any algebraic curve over Q can be rationally mapped to P^1 without three points and this led ...
- 15.1k
5
votes
1
answer
332
views
Extending Functions on Closed Submanifolds of C^n
Functions on an algebraic subvariety X of A^n are the same as functions on A^n restricted to X. So the statement that functions on X extend to all of A^n follows by the definition. My question is: ...
- 5,548
4
votes
2
answers
759
views
What is the homology of the real coordinate ring of SO(n,R)? Other compact matrix groups?
As someone whose knowledge of cohomology is patchy and picked up on a need-to-know basis, and whose algebraic geometry is even worse, I wondered if someone could help with this question. (I ran into ...
- 25.8k
11
votes
6
answers
4k
views
What are some examples of coarse moduli spaces?
It took me some effort to work out Gerashenko's nice simple example Can a singular Deligne-Mumford stack have a smooth coarse space? of a DM stack non-equisingular with its coarse moduli space, which ...
14
votes
6
answers
2k
views
Does every morphism BG-->BH come from a homomorphism G-->H?
Given a homomorphism f:G→H between smooth algebraic groups, we get an induced homomorphism of algebraic stacks Bf:BG→BH, given by sending a G-torsor P over a scheme X to the H-torsor PxGH, ...
- 24.1k
9
votes
1
answer
2k
views
Kodaira-Spencer Theory and moduli of curves
I was looking at a paper of Farkas and the following confusing point came up.
Let $\mathscr{M}_g$ be the moduli stack of smooth genus $g$ curves and let $\pi: \mathscr{C} \to \mathscr{M}_g$ be the ...
- 10.5k
32
votes
3
answers
2k
views
Can algebraic varieties be rigidified by finite sets of points?
For an algebraic variety X over an algebraically closed field, does there always exist a finite set of (closed) points on X such that the only automorphism of X fixing each of the points is the ...
- 8,296
12
votes
2
answers
1k
views
Graded or stacky Serre duality
I am considering the following situation. $A$ is a finitely generated ring over a field $K$ with non-negative grading and $A_0=K$ of Krull dimension n+1, but I don't necessarily assume A is generated ...
- 121
13
votes
4
answers
3k
views
How to do Computations Using the Decomposition Theorem for Perverse Sheaves
This is a follow-up to this post on the Decomposition Theorem. Hopefully, this will also invite some discussion about the theorem and perverse sheaves in general.
My question is how does one use the ...
- 5,548
23
votes
3
answers
8k
views
Finite type/finite morphism
I am not too certain what these two properties mean geometrically. It sounds very vaguely to me that finite type corresponds to some sort of "finite dimensionality", while finite corresponds to "...
user709
15
votes
4
answers
2k
views
When is a scheme a zero-set of a section of a vector bundle?
Are there any general results on when a closed subscheme X of a quasi-projective smooth scheme M can be written as the zero-set of a section of a vector bundle E on M?
To put it in a diagram: When is ...
- 3,917
66
votes
4
answers
11k
views
Is there a good way to think of vanishing cycles and nearby cycles?
Once in a while I run into literature that invokes vanishing cycle machinery with a cryptic sentence like, "this follows from a standard vanishing cycle argument." Is there a good way to look at ...
- 45.7k
6
votes
1
answer
320
views
Is there a canonical notion of principal divisor on a discrete dynamical system?
I hope this question is well-posed.
Let (X, f) be a discrete dynamical system such that every x in X has finite period, i.e. there is some n such that f^n(x) = x. Let Div(X) be the free abelian ...
- 118k
14
votes
2
answers
2k
views
Can a singular Deligne-Mumford stack have a smooth coarse space?
Let XX be a Deligne-Mumford stack and let XX \to X be a coarse moduli space. Suppose that X is smooth. Is XX smooth? If not, what is an example? What if XX is of finite type over C (the complex ...
- 10.5k