All Questions
22,546 questions
4
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3
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715
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Is a holomorphic vector bundle on a projective variety locally trivial in the Zariski topology?
By the GAGA principle we know that a holomorphic vector bundle E->X is analitically isomorphic to an algebraic one, say F->X, and by definition F is locally trivial in the Zariski topology. But since ...
- 14.7k
22
votes
2
answers
1k
views
Is Hodge theory somehow connected with a Galois group action Gal(C/R)?
I'm currently taking a course in Hodge theory ... and I wonder if all the splittings in $\{i,-i\}$ Eigenvalue pairs come from the Galois group action (of the extension $\mathbb{R}\rightarrow\mathbb{C}$...
- 1,926
56
votes
8
answers
8k
views
Questions about analogy between Spec Z and 3-manifolds
I'm not sure if the questions make sense:
Conc. primes as knots and Spec Z as 3-manifold - fits that to the Poincare conjecture? Topologists view 3-manifolds as Kirby-equivalence classes of framed ...
9
votes
3
answers
2k
views
Crepant resolutions of toric varieties
Given a toric variety, is it easy to see if a crepant resolution exists? If so, how can it be explicitly constructed?
- 611
3
votes
1
answer
1k
views
Iso-lines to 3D Surface Generation
I have a set of isolines points ( or contour points) such as this:
Each point has their own respective X, Y and Z. Since they are isolines, that means that all of the points will have a unique X-Y ...
- 381
17
votes
2
answers
1k
views
Can Hom_gp(G,H) fail to be representable for affine algebraic groups?
Let $G$ and $H$ be affine algebraic groups over a scheme $S$ of characteristic 0 and let $\textbf{Hom}_{S,gp}(G,H)$ be the functor $T \mapsto \text{Hom}\_{T,gp}(G,H)$
Theorem (SGA 3, expose XXIV, 7....
- 10.5k
16
votes
12
answers
11k
views
Are there any interesting connections between Game Theory and Algebraic Topology?
I've been learning game theory on my own and was just curious how it connected with previous things I've learned. So are there any interesting connections between Game Theory and Algebraic Topology? ...
- 809
2
votes
0
answers
526
views
How much of math could be taught without using mathematical notation? [closed]
Given that mathematics is not about number, and that it is not even about the cryptic notation used to describe mathematical problems, how much of mathematics could be taught without reference to ...
2
votes
1
answer
167
views
Triviality of the Hodge bundle for a special family of semistable curves
Let g,h be positive integers. Let E be an elliptic curve, C be a genus h curve, and D be a genus g-h-1 curve. Let c,d,e be points on (resp.) C,D, and E.
Let f:CC --> E-e be the family whose fiber ...
- 10.5k
25
votes
4
answers
4k
views
What are the automorphism groups of (principally polarized) abelian varieties?
What are the possible automorphism groups of a principally polarized abelian variety $(A,\lambda)$ of dimension $g,$ say an abelian surface ($g=2$) over the complex numbers or algebraic closure of a ...
- 4,265
0
votes
1
answer
485
views
Understanding a lemma in "Loop Spaces and Langlands Parameters" article
First, some background. I was trying to read the article Loop Spaces and Langlands Parameters but I get immediately stuck at Theorem 2.1 in the introduction.
This was actually forward-referring to ...
- 15.1k
13
votes
1
answer
1k
views
When do six operations work?
This question comes (heavily edited) from my notes, thus slightly unusual structure.
We know that algebraic maps have very strict structure, and in many settings the operations ...
- 15.1k
11
votes
4
answers
3k
views
What does ramification have to do with separability?
Does ramification have anything to do with inseparability? It feels like an extension of Q in which p ramifies should somehow correspond to an extension of F_p(t). Does totally ramified <--> purely ...
- 15.4k
11
votes
3
answers
2k
views
Nonprojective Surface
Let k be an algebraically closed field. It's well known that every complete curve, period, is projective. Also, that every smooth surface is, and that there are smooth 3-folds which are not, and ...
- 16k
15
votes
3
answers
2k
views
Polynomials that are sums of squares
Is any algorithm known for determining whether or not a multivariate polynomial with integer coefficients can be written as a sum of squares of such polynomials?
By way of background, if we one ...
- 151
13
votes
3
answers
1k
views
Decomposition of k[G]
There's a well-known decomposition of $L^2(G)$, a regular representation of compact complex group Lie $G$, called Peter-Weyl theorem.
Turns out for some reason I automatically think that there is a ...
- 15.1k
3
votes
2
answers
1k
views
Castelnuovo Positivity (Rewrite of: Weil's original proof for FP^2)
Weil's proof of the Riemann Hypothesis for projective curves relies upon the following positivity result: Let $\mathbb{F}q$ be the finite field with $q$ elements, $\overline{\mathbb{F}q}$ its closure, ...
- 1,462
15
votes
2
answers
2k
views
Total Spaces of Quasicoherent Sheaves
You can construct a total space of a quasicoherent sheaf on an scheme by taking relative spec of the symmetric algebra of the dual sheaf. For locally free sheaves, you get vector bundles, and every ...
- 5,548
16
votes
3
answers
3k
views
When does direct image with proper support have a right adjoint?
For $f: X → Y$ a morphism of schemes, does anybody know conditions for the existence of an adjunction $(f_!,f^!)$ between the module-categories (not the quasicoherent), where $f_!$ is direct image ...
- 12.3k
3
votes
2
answers
242
views
Vector spaces of singular planar cubics
What is the largest dimensional linear space of singular planar cubics? Is this known?
Think of the space of planar cubics as a PP^9 (parametrized by the coefficients). The discriminant \Delta is ...
- 2,955
2
votes
1
answer
173
views
Projective Curves which are Principal Bundles
I have a very specific question: does anyone know of a (non-trivial) example of a projective curve which is also a homogenous space (or just a principal bundle)? The trivial example being CP^1 = SU(2)/...
- 1,462
8
votes
3
answers
1k
views
Is there a stable algorithm for polynomial division (in several variables)?
Suppose you have a homogeneous ideal $I$ inside the algebra $\mathbb{C}[x_1,...,x_d]$ of complex polynomials in $d$-variables. Can one find a basis for $I$, say $\{f_1,...,f_k\}$, such that every $h \...
- 550
3
votes
1
answer
320
views
limits of algebraic varieties
I'm looking for a reference which deals with limits of families of algebraic varieties as the degree increases (or at least keywords from this subject).
For the kind of example I have in mind, ...
- 2,901
18
votes
4
answers
2k
views
What are the Benefits of Using Algebraic Spaces over Schemes?
I have heard that algebraic spaces have better formal properties than schemes. What are these benefits? Also, is there a natural way to go straight from affine schemes to algebraic spaces bypassing ...
- 5,548
13
votes
2
answers
3k
views
Is the fixed locus of a group action always a scheme?
Suppose $G$ is an algebraic group with an action $G\times X\to X$ on a scheme. Does the fixed locus (the set of points x∈X fixed by all of $G$) have a scheme structure? You can obviously define the ...
- 24.1k
10
votes
3
answers
1k
views
When does Tannakian theory work over affine schemes besides fields?
By 'work' I would like the correspondence between fiber functors (to finitely generated projective modules) and algebraic groups to be the same as in the field case.
Specifically, if $A$ is an affine ...
- 5,062
14
votes
2
answers
989
views
Do orbits and stable loci of group actions have natural scheme structures?
Suppose G is an algebraic group with an action G×X→X on a scheme. Then many of the usual constructions you make when you talk about group actions on sets can be made scheme-theoretically. ...
- 24.1k
55
votes
3
answers
5k
views
What are the higher homotopy groups of Spec Z ?
The homotopy groups of the étale topos of a scheme were defined by Artin and Mazur. Are these known for Spec Z? Certainly π1 is trivial because Spec Z has no unramified étale covers,...
- 8,004
5
votes
4
answers
666
views
Sections of a divisor on elliptic curve
I'm interested in producing explicit bases for the sections of a line bundle on an embedded genus 1 curve. Let me restrict to the first case that I don't know how to do, so that I can be as concrete ...
- 2,955
87
votes
15
answers
37k
views
The importance of EGA and SGA for "students of today"
That fact that EGA and SGA have played mayor roles is uncontroversial. But they contain many volumes/chapters and going through them would take a lot of time, especially if you do not speak French.
...
8
votes
2
answers
481
views
Division Algebras as Algebraic Groups
If I'm given a division algebra D with Z(D)=F, then how can I view Dx as an algebraic group defined over F? I'd like to see first how Dx can be given the structure of a variety defined over F, and ...
- 2,799
3
votes
1
answer
261
views
Over which schemes can there exist non-trivial G_a bundles?
The group scheme G_a here is the one-dimensional additive group.
- 8,866
20
votes
2
answers
3k
views
Derived functors vs universal delta functors
I would like to understand the relationship between the derived category definition of a right derived functor $Rf$ (which involves an initial natural transformation $n: Qf \rightarrow (Rf)Q$, where $...
- 11.3k
9
votes
0
answers
1k
views
"A theory of generalized Donaldson-Thomas invariants" by Joyce & Song
Is anyone else working through this paper: A theory of generalized Donaldson-Thomas invariants, by Dominic Joyce, Yinan Song? I am trying to verifying example 6.2 (m=2 for simplicity) using only the ...
- 2,218
9
votes
5
answers
2k
views
Analogues of the Weierstrass p function for higher genus compact Riemann surfaces
There was a previous post on the correspondence between Riemann surfaces and algebraic geometry. I want to ask a related but more detailed question.
BACKGROUND:
Engelbrekt gave an overview of how ...
- 3,968
6
votes
2
answers
1k
views
Equivalence of derived categories which is not Fourier-Mukai
D. Orlov proved that any equivalence of bounded derived categories F:Db(X) -> Db(Y) is a Fourier-Mukai transform, when X and Y are smooth projective varieties. Is there any example of such equivalence,...
- 912
11
votes
4
answers
5k
views
When does the sheaf direct image functor f_* have a right adjoint?
Say f: X → Y is a morphism of schemes. The sheaf direct image functor f★ always has a left adjoint, namely the sheaf inverse image functor f★ (with tensoring).
Under what (...
- 11.3k
12
votes
4
answers
715
views
Behaviour of Zeta-function under Finite Morphism
Let X ---> Y be a finite surjective morphism of smooth, projective, connected varieties over a finite field F_q. Can one describe the zeta function Z(X, t) in terms of the zeta-function Z(Y,t) of ...
- 671
7
votes
1
answer
449
views
How does one intersect non-transverse divisors on Mg-bar.
Let Mg-bar be the Deligne-Mumford compactification of genus g curves, and let δ1 be the divisor of degenerate curves of the form `genus 1 meeting a genus g-1 transversely".
Question 1: What ...
- 10.5k
16
votes
6
answers
3k
views
Can any topological space be the result of a scheme?
Maybe this is trivial but lets give it a try anyways..
Obviously there is a forgetful functor from schemes to topological space.. but is it surjective on objects? i.e. I ask whether any topological ...
- 2,275
6
votes
4
answers
828
views
Can isomorphisms of schemes be constructed on formal neighborhoods?
Let (A,m) be a complete local Noetherian ring and let X and Y be two schemes of finite type over A (and flat over A). Let Xn and Yn be the reductions of X and Y mod mn+1.
Question: Suppose there is ...
- 10.5k
33
votes
7
answers
13k
views
Links between Riemann surfaces and algebraic geometry
I'm taking introductory courses in both Riemann surfaces and algebraic geometry this term. I was surprised to hear that any compact Riemann surface is a projective variety. Apparently deeper links ...
- 2,967
12
votes
1
answer
1k
views
Formalism of homotopy theory of schemes
I have some vague knowledge about the philosophy that schemes should be thought of as similar to topologic spaces, and we should divide everything by homotopy, and that the space should be actually ...
- 15.1k
64
votes
5
answers
9k
views
Intuition about the cotangent complex?
Does anyone have an answer to the question "What does the cotangent complex measure?"
Algebraic intuitions (like "homology measures how far a sequence is from being exact") are as welcome as ...
- 12.3k
8
votes
2
answers
759
views
Can any countably generated k-algebra occur as the ring of global sections of some variety?
In the answer to this question we saw that there exists a nonsingular quasi-projective threefold over a field with non-finitely generated global sections.
I was talking about this previous ...
- 8,681
19
votes
4
answers
4k
views
Are there any recordings of Grothendieck online?
Illusie mentions tape recordings of Grothendieck explaining his trace formula and more. Are they or similar recordings online? I guess, even if (what I doubt) everything he thought about that is ...
- 10.8k
13
votes
4
answers
3k
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"Albanese" schemes: When does an "initial abelian scheme" exist under a given scheme?
For a variety V, its Albenese variety Alb(V) is a variety with a map V → Alb(V) which factors uniquely into any map from V to an abelian variety. Can we say something similar for an arbitrary ...
- 11.3k
15
votes
7
answers
4k
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Different definitions of the dimension of an algebra
I know of three ways to define the dimension of a finitely-generated commutative algebra A over a field F:
The Gelfand-Kirillov (GK) dimension, based on the growth of the Hilbert function.
The Krull ...
- 118k
4
votes
1
answer
321
views
Reverse Langlands transform
What os the meaning of a reverse Langlands transform to which Drinfeld seems to refer?
- 15.1k
22
votes
3
answers
3k
views
Homotopy theory of schemes examples
Is it possible give an example of (or explain) how the Voevodsky et al.'s homotopy theory of schemes computes higher Chow groups?
- 15.1k