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Results tagged with ag.algebraic-geometry
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user 1384
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
7
votes
When two k-varieties with the same underlying topological spaces isomorphic?
Here's another type of counterexample not (as I write) ruled out by the hypotheses of the question: consider an inclusion of fields $K\to L$, with $K$ and $L$ finite extensions of $k$. Now take the sp …
- 40.3k
8
votes
The central role of varieties (a comment from Mumford's Red Book)
The kernel of an isogeny between abelian schemes is flat. The reason is that it's true over a field, and then you use the fibrewise criterion for flatness. I don't know of any other proof. So there's …
- 40.3k
13
votes
Colimits of schemes
Sounds to me like you don't want to hear the proof, but you want to hear "the point". The point is that a scheme must by definition be covered by affine schemes, and sometimes when doing exercises in …
- 40.3k
2
votes
Accepted
Dense section of sheaves of modules
If U is an open set in X, but U isn't X, then there are non-zero sheaves on X whose support lies outside U. Now add O_X to one of these to get a counterexample.
- 40.3k
1
vote
Question on determining the minimal polynomial for an algebraic quotient
Here's a suggestion. Use polcompositum(FA,FC) (with FA the min poly of A, FC the min poly of C) to find a number field K=Q(alpha) containing roots of both your polynomials, and then use lindep() to fi …
- 40.3k
13
votes
Accepted
Why is the prime spectrum not useful in non-archimedean analytic geometry?
I am surprised that Brian got to this one first without making what I thought was another obvious comment: affinoids are Jacobson rings! A function which is zero at all points of an affinoid rigid spa …
- 40.3k
11
votes
Accepted
The closure of the set of rational points in the Adeles
Here's an example where $X(\mathbf{Q})$ is Zariski-dense but the first inequality is not an equality.
Let $X$ be an elliptic curve over the rationals, such that the group $X(\mathbf{Q})$ is isomorph …
- 40.3k
9
votes
Accepted
Z_p flatness and irreducible components.
Your proof seems wrong to me. I might be misunderstanding some things you wrote, but surely $\mathbf{Q}{}_p=\mathbf{Z}_p[X]/(pX-1)$ is finite type over $\mathbf{Z}_p$, and contains many elements which …
- 40.3k
10
votes
How do you explicitly compute the p-torsion points on a general elliptic curve in Weierstras...
http://en.wikipedia.org/wiki/Division_polynomials
That's not the best wikipedia page. "The division polynomials form an elliptic divisibility sequence." is mentioned well before the far more importan …
- 40.3k
3
votes
Is there a schemetical construction for modular curves over the rationals?
Passing comment: Mumford's GIT constructs modular curves as quotients---not of the upper half plane, but of some parameter space of subspaces of projective space, by an algebraic group. As for the las …
- 40.3k
133
votes
6
answers
21k
views
what mistakes did the Italian algebraic geometers actually make?
It's "well-known" that the 19th century Italian school of algebraic geometry made great progress but also started to flounder due to lack of rigour, possibly in part due to the fact that foundations ( …
- 40.3k
5
votes
When is a coarse moduli space also a fine moduli space?
IIRC there's an example due to Gabber in the book by Katz and Mazur of a representable moduli problem where objects have automorphisms (I forget the trick---perhaps he rigged it so that every object h …
- 40.3k
3
votes
Accepted
Regularity of schemes under base change
I don't think this will be true in general.
Say $K=\mathbf{Q}$ and $K'=\mathbf{Q}(\sqrt{2})$, and let $X_0$ be $Spec(R')$. Then $X_0$ is regular of dimension 1 and the map down to $S$ is projective a …
- 40.3k
12
votes
2
answers
605
views
Image of projective 1-space contained in projective 1-space over a smaller field?
This is inspired by
Does "all points rational" imply "constant" for this "cubic" curve over an arbitrary field? .
Say $K/F$ is a finite separable extension of fields. Assume $F$ is infinite (or el …
- 40.3k
17
votes
Uniform Faltings
On the contrary, some conjectures suggest that the answer is NO! It follows from the Bombieri-Lang conjecture (sometimes known as Lang's conjectures) that a uniform bound should exist.
More precisel …
- 40.3k
21
votes
1
answer
2k
views
Naive question about constructing constructible sheaves.
In algebraic geometry, an etale sheaf on a Noetherian scheme is called constructible if the scheme has a finite stratification by locally closed subschemes such that the pullback of the sheaf to each …
- 40.3k
4
votes
Elementary proof that projective space is a quotient
Look at the subspace of $\mathbf{A}^{n+1}$ cut out by your polynomials. This set is invariant under the diagonal action of $k^\times$. So the functions that vanish on it will be an ideal $I$ (the radi …
- 40.3k
15
votes
What is etale descent?
As I write the question looks like a muddle of two distinct notions:
1) Restriction of scalars. Given $L/K$ finite and a variety $V/L$ there's a variety $W/K$ of dimension $(dim V)[L:K]$ with $W(K)=V …
- 40.3k
68
votes
Accepted
Smooth proper scheme over Z
Hey Bjorn. Let me try for a counterexample. Consider a hypersurface in projective $N$-space, defined by one degree 2 equation with integral coefficients. When is such a gadget smooth? Well the partial …
- 40.3k
10
votes
Accepted
Is there such a thing as a non-injective flasque abelian sheaf?
Just take a non-injective abelian group and a point in your space, and form the skyscraper sheaf at that point with stalk the abelian group.
- 40.3k
13
votes
0
answers
366
views
History of use of "=" symbol to mean "is canonically isomorphic to"
Let $A$ be a commutative ring, and let $f$ and $g$ denote elements of $A$ such that the prime ideals of $A$ containing $f$ are precisely the prime ideals containing $g$ (a not completely trivial examp …
- 40.3k
3
votes
Accepted
Abelian varieties of CM type?
IIRC I learnt a lot from Katz' papers from the 1970s. Of course the basic construction is the same as the elliptic curve case: you take C^g, quotient out by the lattice coming from E via its g embeddi …
- 40.3k
6
votes
Accepted
What is known about finite morphisms from X to the projective line
No because $f$ can be ramified pretty much anywhere. Just think of a random rational function $f=p(x)/q(x)$ with $p,q$ coprime polynomials, $p$ non-constant and $q$ non-zero. That gives a finite morph …
- 40.3k
3
votes
Accepted
Does "all points rational" imply "constant" for this "cubic" curve over an arbitrary field?
This isn't an answer but I think it's progress. It started off by thinking of restriction of scalars but I've translated it down to a rather more mundane point of view.
Let me call the fields $K$ an …
- 40.3k
0
votes
Is tensoring with a module representable iff it is locally free of finite rank?
Contrary to what I guessed initially, I now think the question has a great answer: the functor is representable if and only if $M$ is locally free, and the proof is EGA I, 9.4.10.
Edit: this is an an …
- 40.3k
0
votes
Is tensoring with a module representable iff it is locally free of finite rank?
Here is an example where representability fails. If $R$ is an $A$-algebra representating $\otimes_AM$ on $A$-algebras, and if $B\to C$ is an injective map of $A$-algebras, then $R(B)\to R(C)$ will be …
- 40.3k
6
votes
Accepted
Can different modules have the same symmetric algebra? (answered: no)
I now believe a-fortiori's argument: translations are a problem, but, as a-fortiori observed, they are the only problem. Let me spell it out.
Say $f:Sym(M)\to Sym(N)$ is an isomorphism. For $m\in M$ …
- 40.3k
0
votes
Is (relatively) algebraically closed stable under finite field extensions?
Let me have a punt at this. $F$ alg closed inside $F'$ iff $\overline{F}\otimes_FF'$ is a field, right? So now it's easy because $\overline{L}$ is an algebraic closure of $F$, and I don't think I even …
- 40.3k
8
votes
Proving that a map is a morphism
You have an answer to the 'example' version of your question already, but let me offer an answer to the "actual" question: if one is faced with two schemes $A$ and $B$, and for each $a\in A$ you have …
- 40.3k
3
votes
Accepted
Serre tensor construction on finite flat group schemes
Thinking about this more, it might be easier than you think. You don't seem to even care about the actions of $\mathcal{O}_E$ or $\mathcal{O}_{E'}$ in your question so you should consider what happens …
- 40.3k
29
votes
Accepted
Why are topological ideas so important in arithmetic?
Why are topological ideas so important in arithmetic? In some sense KConrad is of course spot on, but let me offer a completely different kind of answer.
Why are complex functions of one variable so …
- 40.3k
6
votes
Accepted
Zograf's bound on the index of a modular curve for Shimura curves
Here are the answers to some of your questions. If $P$ is a (non-zero) prime ideal of the integers of $F$ then $B$ will either be split or ramified at $P$, depending on whether $B\otimes_F F_P$ is iso …
- 40.3k
24
votes
Accepted
L-functions and higher-dimensional Eichler-Shimura relation
Surprisingly, the case of modular curves is misleading! General theory of correspondences, plus the theory of the mod $p$ reduction of curves like $X_0(Np)$ ($p$ doesn't divide $N$) give a relationshi …
- 40.3k
11
votes
Accepted
Consequences of the geometric properties of the eigencurve
The eigencurve is an honest moduli space---it parametrises families of finite slope overconvergent modular eigenforms (or more precisely, of systems of overconvergent finite slope Hecke eigenvalues)-- …
- 40.3k
13
votes
Can we count isogeny classes of abelian varieties?
Here's a related question on which there has been much work. If you actually need an answer to your question for some other reason and you follow this up, I'd be interested to see where it goes.
Let …
- 40.3k
15
votes
1
answer
790
views
components of E[p], E universal in char p.
I have just realised that a group scheme I've known and loved for years is probably a bit wackier than I'd realised.
In this question, in Charles Rezk's answer, I erroneously claim that his construct …
- 40.3k
120
votes
Accepted
Etale cohomology -- Why study it?
$\DeclareMathOperator{\gal}{Gal}$
Here's a comment which one can make to differential geometers which at least explains what etale cohomology "does". Given an algebraic variety over the reals, say a s …
- 40.3k
5
votes
When is a homogeneous space a variety?
Part 2: Baily-Borel! At least if $G$ is the real points of, say, a reductive algebraic group, and $H$ is a maximal compact subgroup, and EDIT furthermore if $G/H$ admits a complex structure (I think t …
- 40.3k
3
votes
Examples and intuition for arithmetic schemes
One example I always found useful was that if you consider an elliptic curve like (the projective model of) y^2=x^3+1, then this equation gives an elliptic curve not only over the complex numbers but …
5
votes
Quotients of Tate modules
Although this question isn't really well-defined (you'd surely need to be more precise about the word "canonical" in the comment under the question) let me make two comments which hopefully put this t …
- 40.3k
32
votes
Accepted
Why is one interested in the mod p reduction of modular curves and Shimura varieties?
The Eichler-Shimura relation doesn't just prove the Hasse-Weil conjecture for modular curves. It e.g. attaches Galois representations to modular forms of weight 2. More delicate arguments (using etale …
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