All Questions
22,770 questions
4
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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
764
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
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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
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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
331
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
758
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, ...
9
votes
1
answer
2k
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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
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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
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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
31
votes
7
answers
10k
views
Quotients of Schemes by Free Group Actions
I've often seen people in seminars justify the existence of a quotient of a scheme by an algebraic group by remarking that the group action is free. However, I'm pretty sure they are also invoking ...
- 5,548
10
votes
3
answers
2k
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Unstable Vector Bundles
As a follow up to me other question, what can be said about unstable vector bundles? I know this is rather open ended, but what sorts of horrible things does having a subbundle of strictly greater ...
- 16k
4
votes
2
answers
2k
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Pushforwards of Line Bundles and Stability
I recently finished reading this paper, and was wondering about a couple of things relating to theorem 1, which says that for any curve X there is a curve Y and f:Y->X such that pushforward is a ...
- 16k
8
votes
1
answer
688
views
Universal covers of domains in complex projective space
The Uniformization Theorem states that the universal cover of a Riemann surface is biholomorphic to the extended complex plane, the complex plane or the open unit disk. Each of these three is a domain ...
- 4,485
34
votes
2
answers
7k
views
What is the geometric meaning of integral closure?
More precisely, how does one characterize integrally closed finitely generated domains (say, over C) based on geometric properties of their varieties? Given a finitely generated domain A and its ...
- 118k
6
votes
2
answers
1k
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Does projectiveness descend along field extensions?
Background: Properness is a much more robust notion than projectiveness. For example, properness descends along arbitrary fpqc covers (see, for example, Vistoli's Notes on Grothendieck topologies, ...
11
votes
4
answers
2k
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Moduli spaces of complex curves as algebraic varieties
Using a minimum of technical vocabulary, give a summary of why it is that the moduli space of genus g complex curves with n marked points has a natural compactification that is isomorphic (as a ...
- 7,062
17
votes
2
answers
3k
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Are curves with `fractional points' uniquely determined by their residual gerbes?
One makes precise the vague notion of "curve with a fractional point removed" (see for instance these slides) using stacks -- one should really consider Deligne-Mumford stacks whose coarse spaces are ...
- 10.5k
13
votes
5
answers
5k
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Examples and intuition for arithmetic schemes
How should a beginner learn about arithmetic schemes (interpret this as you wish, or as a regular scheme, proper and flat over Spec(Z))? What are the most important examples of such schemes? Good ...
6
votes
1
answer
777
views
Existence of proper regular models for varieties over Q and other global fields
What is known about regular proper models for smooth projective varieties over Q? Results for other global fields would also be interesting, as well as general comments and suggested references for ...
- 5,637
15
votes
7
answers
5k
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Morphisms of (quasi-)projective varieties
This is another "homework help" question, which is still hopefully of at least pedagogical interest to working mathematicians.
So, I'm currently taking an intro algebraic geometry class, and one ...
- 12.6k
6
votes
3
answers
1k
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Is there a software package that does Schubert Calculus computations?
Is there a good software package for doing computations in the cohomology ring of Grassmannians? Things like, I can write down a polynomial in, in fact, special Schubert classes, but it's one where ...
- 16k
17
votes
3
answers
1k
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R2 and S3 for rings.
For a noetherian ring R, Serre's criterion for normality states that R is normal if and only if R satisfies conditions R1 and S2, where R1 is regularity in codimension one, and S2 is Serre's condition ...
- 273
20
votes
1
answer
2k
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Functorial characterization of open subschemes?
Given a morphism of schemes f: U → X, can one determine when f is an isomorphism of U onto an open subscheme of X in terms of some induced functors between the categories of quasicoherent modules ...
- 5,407
13
votes
6
answers
3k
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Gromov-Witten theory and compactifications of the moduli of curves
Why, from a string theory perspective, is it natural to consider the Deligne-Mumford (resp. Kontsevich) compactification of the moduli of curves (resp. maps [from curves to a target space X]) rather ...
- 21k
128
votes
15
answers
51k
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A learning roadmap for algebraic geometry
Unfortunately this question is relatively general, and also has a lot of sub-questions and branches associated with it; however, I suspect that other students wonder about it and thus hope it may be ...
3
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1
answer
514
views
4
votes
1
answer
1k
views
The existence of primitive and sufficiently ample line bundles on K3 surfaces?
Let S be a surface and L be a line bundle on S. For any zero-dimensional closed subschemes x of S, there is natural map from global sections of L to the global sections of L restricting to x (which is ...
- 41
62
votes
8
answers
14k
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Sheaf cohomology and injective resolutions
In defining sheaf cohomology (say in Hartshorne), a common approach seems to be defining the cohomology functors as derived functors. Is there any conceptual reason for injective resolution to come ...
user709
6
votes
5
answers
3k
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Is very ampleness of a divisor on a curve determined entirely by degree and genus?
Edit: Apparently the answer is "no", so what is an example of two curves of genus g, and a divisor of degree d on each, such that one is very ample and the other is not?
Question as originally stated:...
- 11.2k
13
votes
1
answer
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What are the higher $\mathrm{Ext}^i(A,\mathbf{G}_m)$'s, where $A$ is an abelian scheme?
Let $S$ be a base scheme, let $A/S$ be an abelian scheme, and let $\mathbf{G}_m/S$ be the multiplicative group; consider $A$ and $\mathbf{G}_m$ as objects in the abelian category of sheaves of abelian ...
- 2,270
6
votes
2
answers
1k
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Explicit Direct Summands in the Decomposition Theorem
Let f:X→Y be a semismall resolution of singularities. Then the pushforward of the constant sheaf on X is a semisimple perverse sheaf on Y. Under these conditions, I know how to calculate the ...
- 8,866
20
votes
2
answers
4k
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"Fermat's last theorem" and anabelian geometry?
Do I remember a remark in "Sketch of a program" or "Letter to Faltings" correctly, that acc. to Grothendieck anabelian geometry should not only enable finiteness proofs, but a proof of FLT too? If yes,...
- 10.8k
9
votes
1
answer
1k
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Example where you *need* non-DVRs in the valuative criteria
The valuative criterion for separatedness (resp. properness) says that a morphism of schemes (resp. a quasi-compact morphism of schemes) f:X→Y is separated (resp. proper) if and only if it ...