All Questions
2,036 questions
24
votes
2
answers
4k
views
Why is the degree:rank ratio of a vector bundle called its "slope"?
Whenever one studies moduli spaces of vector bundles on curves, one of the first things to be introduced is the "slope" of a vector bundle, i.e., its degree:rank ratio. Is there a nice (preferably ...
- 7,338
24
votes
3
answers
4k
views
Example of a variety with $K_X$ $\mathbb Q$-Cartier but not Cartier
I know the definition of $K_X$ on a normal, singular variety, but I don't have a good set of examples in my mind. What's an example of a variety where $K_X$ is $\mathbb Q$-Cartier but not Cartier? ...
- 255
23
votes
2
answers
3k
views
Criteria for irreducibility of polynomial
If $f, g\in \mathbb C[a,b]$ are polynomials in two variables, are there easy criteria that allow to see if $f(x,y)-g(t,z)\in \mathbb C[x,y,t,z]$ is irreducible?
Thank you very much,
best
- 669
22
votes
1
answer
883
views
Is being of general type stable under generization
This question is about how varieties of general type over an algebraically closed field of characteristic zero $k$ behave under generization in families.
Definition. An integral projective ...
- 9,497
22
votes
1
answer
2k
views
Do line bundles descend to coarse moduli spaces of Artin stacks with finite inertia?
According to the Keel-Mori theorem if $\mathcal{X}$ is a separated Artin stack of finite type over a scheme S with finite inertia, then it admits a coarse moduli space $X$ (which is an algebraic space)...
- 1,060
22
votes
2
answers
8k
views
Geometric vs Arithmetic Frobenius
If an algebraic variety $X$ over a field characteristic p is given by equations $f_i(x_1,...,x_k) = 0$, we can consider the variety $X^{(p)}$ obtained by applying p-th powers to all the coefficients ...
- 1,783
22
votes
2
answers
2k
views
Applications of derived categories to "Traditional Algebraic Geometry"
I would like to know how derived categories (in particular, derived categories of coherent sheaves) can give results about "Traditional Algebraic Geometry". I am mostly interested in classical ...
21
votes
5
answers
12k
views
Nonsingular/Normal Schemes
I always had trouble remembering this. Is it true that a curve over a non-algebraically-closed field is normal implies that it's non-singular? How about a 1 dimensional scheme? How about dimension 2? ...
- 1,386
21
votes
1
answer
3k
views
Motivation and potential applications of spectral algebraic geometry
Nowadays there is a lot of talk about derived algebraic geometry, but not so much about the related subject of spectral algebraic geometry.
Now I'm curious what future is there for spectral algebraic ...
- 433
21
votes
3
answers
3k
views
Are schemes that "have enough locally frees" necessarily separated
Let me motivate my question a bit.
Thm. Let $X$ be a locally noetherian finite-dimensional regular scheme. If $X$ has enough locally frees, then the natural homomorphism $K^0(X)\longrightarrow K_0(X)...
- 9,497
21
votes
10
answers
6k
views
Not especially famous, long-open problems which higher mathematics beginners can understand
This is a pair to
Not especially famous, long-open problems which anyone can understand
So this time I'm asking for open questions so easy to state for students of subjects such as undergraduate ...
21
votes
7
answers
3k
views
What should be taught in a 1st course on Riemann Surfaces?
I am teaching a topics course on Riemann Surfaces/Algebraic Curves next term. The course is aimed at 1st and 2nd year US graduate students who have have taken basic coursework in algebra and manifold ...
- 3,284
21
votes
2
answers
2k
views
Topologically contractible algebraic varieties
From a post to The Jouanolou trick:
Are all topologically trivial (contractible) complex algebraic varieties necessarily affine? Are there examples of those not birationally equivalent to an affine ...
- 15.1k
20
votes
2
answers
3k
views
D-modules and Algebraic Solutions of PDEs
I am not certain if this is a complete question and I fear it might be shot down. Anyway, I try to pose it. My question is in connection to using D-modules to study PDEs (and systems of PDEs). When ...
- 403
20
votes
1
answer
2k
views
Geometric generic fibre
This is a pretty elementary question about schemes, but it came up in the course of research, so let's try it here rather than MSE.
Question 1: Are the fibres of a family of complex varieties ...
20
votes
2
answers
9k
views
Does module Hom commute with tensor product in the second variable?
Let $A$ be a commutative ring, and $L, M, N$ be $A$-modules. Then is it true that
$$\text{Hom}_A (L, M)\otimes_A N \cong \text{Hom}_A (L, M\otimes_A N)$$
as $A$-modules?
(Note that there is a ...
- 1,906
20
votes
2
answers
1k
views
Why do flag manifolds, in the P(V_rho) embedding, look like products of P^1s?
Bert Kostant mentioned an odd fact to me some time ago. As usual (with such statements), fix a
complex, connected, reductive) Lie group $G$, with maximal torus $T$, and Weyl vector $\rho$ equal to ...
- 27.9k
20
votes
7
answers
9k
views
Elementary reference for algebraic groups
I'm looking for a reference on algebraic groups which requires only knowledge of basic material on the theory of varieties which you could find in, for example, Basic Algebraic Geometry 1 by ...
- 15.4k
20
votes
1
answer
4k
views
Hochschild (co)homology of Fukaya categories and (quantum) (co)homology
There is a conjecture of Kontsevich which states that Hochschild (co)homology of the Fukaya category of a compact symplectic manifold $X$ is the (co)homology of the manifold. (See page 18 of ...
- 21k
20
votes
1
answer
2k
views
Are flat morphisms of analytic spaces open?
Let $f:X\to Y$ be a morphism of complex analytic spaces. Assume $f$ is flat (or, more generally, that there is a coherent sheaf on $X$ with support $X$ which is $f$-flat). Is $f$ an open map?
The ...
- 19.6k
20
votes
2
answers
2k
views
Integral points on varieties
I recently came across an interesting phenomenon which confused me slightly, concerning integral points on varieties.
For example, consider $X = \mathbb{A}_{\mathbb{Z}}^{n+1} \setminus \{0\}$, affine ...
- 21.4k
20
votes
1
answer
2k
views
Categorical interpretation of quasi-compact quasi-separated schemes
Let $X$ be a scheme. Consider the global section functor $\Gamma : \mathrm{Qcoh}(X) \to \mathrm{Ab}$. It is well-known that $\Gamma$ preserves filtered colimits if $X$ is assumed to be quasi-compact ...
- 63.1k
19
votes
3
answers
2k
views
Can a coequalizer of schemes fail to be surjective?
Suppose $g,h:Z\to X$ are two morphisms of schemes. Then we say that $f:X\to Y$ is the coequalizer of $g$ and $h$ if the following condition holds: any morphism $t:X\to T$ such that $t\circ g=t\circ h$ ...
- 24.1k
19
votes
3
answers
2k
views
Elkies' supersingularity theorem in higher dimension
The following is a theorem of Elkies:
Let $X$ be an elliptic curve over $\mathbb{Q}$. Then there are infinitely many primes $p$ such that the action of Frobenius on $H^1(\mathcal{O}, X)$ is zero.
...
- 156k
19
votes
1
answer
6k
views
dualizing sheaf of a nodal curve
I'm trying to understand the dualizing sheaf $\omega_C$ on a nodal curve $C$, in particular why is $H^1(C,\omega_C)=k$, where $k$ is the algebraically closed ground field. I know this sheaf is ...
- 2,108
19
votes
1
answer
825
views
Is the regularity of finitely generated rings decidable?
Q: Is there an algorithm to decide whether a given finitely generated (over $\mathbb{Z}$) commutative ring is regular?
I mean by regular that the localization at every prime ideal is a regular local ...
- 343
19
votes
3
answers
7k
views
Evidences on Hartshorne's conjecture? References?
Hartshorne's famous conjecture on vector bundles say that any rank $2$ vector bundle over a projective space $\mathbb{P}^n$ with $n\geq 7$ splits into the direct sum of two line bundles.
So my ...
- 2,444
19
votes
1
answer
3k
views
When does the Torelli Theorem hold?
The Torelli theorem states that the map $\mathcal{M}_g(\mathbb{C})\to \mathcal{A}_g(\mathbb{C})$ taking a curve to its Jacobian is injective. I've seen a couple of proofs, but all seem to rely on the ...
- 16k
19
votes
4
answers
2k
views
What is the geometric object corresponding to a subalgebra in a polynomial ring
Many introductory texts on algebraic geometry set up some sort of algebra-geometry dictionary in which radical ideals correspond to varieties, and so on. I am wondering if there is a geometric way to ...
- 1,961
19
votes
14
answers
4k
views
Excellent uses of induction and recursion
Can you make an example of a great proof by induction or construction by recursion?
Given that you already have your own idea of what "great" means, here it can also be taken to mean that the chosen ...
19
votes
3
answers
3k
views
When are (finite) simplicial complexes (smooth) manifolds?
Hi,
is there an algorithm that determines if a given simplicial complex is
a.) a manifold
b.) a smooth manifold
c.) homotopy equivalent to a manifold
d.) a real algebraic variety
?
- 949
19
votes
2
answers
5k
views
New Geometric Methods in Number Theory and Automorphic Forms
The MSRI is organising a programme with the above title from Aug 11, 2014 to Dec 12, 2014. Here is a short description from their website :
The branches of number theory most
directly related to ...
- 18.7k
19
votes
1
answer
956
views
"Real algebraic varieties" vs finite type separated reduced $\mathbb{R}$-schemes with dense $\mathbb{R}$-points
This question is partly motivated by a few comments here. Let me denote by $R$ the (real-closed) field of real numbers $\mathbb{R}$; everything is probably the same over an arbitrary real-closed field....
- 23.4k
19
votes
1
answer
3k
views
Questions about the "universal elliptic curve" over the affine $j$-line punctured at 0 and 1728
So my question refers to families of elliptic curves over the $\mathbb{A}^1_\mathbb{C}\setminus\{0,1728\}$ whose fiber above a point $j$ has $j$-invariant equal to $j$ (I understand it's not universal)...
- 10.7k
19
votes
2
answers
4k
views
Whitney stratifications
Many results on characteristic classes of singular varieties (as well as other singularity-theoretic constructions) make use of a so-called "Whitney stratification" of the variety under ...
- 673
18
votes
1
answer
802
views
Relative Picard functor for the Zariski topology
I'm trying to understand better the relative Picard functor, as defined, for example, in Kleiman's article.
Let $X \to S$ be a smooth projective morphism of schemes whose geometric fibres are ...
- 21.4k
18
votes
3
answers
703
views
Existence of a ring with specified residue fields
Given a finite set of fields $k_1, \ldots, k_n$, is there a (commutative with $1$) ring $R$ with (maximal) ideals $m_i$ such that $R/m_i \cong k_i$?
To prevent things from being too easy, I require ...
- 706
18
votes
0
answers
496
views
What is the logical complexity of the Hodge conjecture?
The Hodge conjecture seems to me the most mysterious among the Millennium problems
(and many others). In particular, I am not sure about its logical complexity. It is not difficult to see that the ...
- 6,891
18
votes
2
answers
2k
views
Homotopy types of schemes
Let $X$ be a scheme over $\mathbb{C}$.
When does the topological space $X\left(\mathbb{C}\right)$ of $\mathbb{C}$-points have the homotopy type of a finite CW-complex?
When does the topological ...
- 15.5k
18
votes
3
answers
3k
views
Algebraic varieties which are topological manifolds
Inspired by this thread, which concludes that a non-singular variety over the complex numbers is naturally a smooth manifold, does anyone know conditions that imply that the topological space ...
- 2,218
17
votes
2
answers
4k
views
Path connectedness of varieties
Let $X$ be a variety. Then, is $X$ path connected? And by path connected, I mean any two closed points $P, Q$ on the variety can be connected by the image of a finite number of non-singular curves.
- 1,510
17
votes
1
answer
1k
views
Any real algebraic variety is diffeomorphic to a real algebraic variety defined over $\mathbb{Q}$
Given a smooth proper real algebraic variety can you find a smooth proper real algebraic variety defined over $\mathbb{Q}$ that is diffeomorphic to it?
user145520
17
votes
2
answers
3k
views
Infinite products exist *only* for affine schemes?
Infinite products don't exist in the category of schemes (see Jonathan Wise's answer here). However, all limits of affine schemes exist in the category of schemes (and are affine). I would like to ...
- 63.1k
17
votes
1
answer
2k
views
What is the precise relationship between o-minimal theory and Grothendieck's "Esquisse d'un programme"?
I have seen various references in the literature to such a connection but they tend to assume that the reader is familiar with the connection, and limit themselves to providing additional detail. So ...
- 16.6k
17
votes
4
answers
5k
views
Why do automorphism groups of algebraic varieties have natural algebraic group structure?
I am not sure that all automorphism groups of algebraic varieties have natrual algebraic group structure.
But if the automorphism group of a variety has algebraic group structure, how do I know the ...
- 2,444
17
votes
0
answers
1k
views
What is the expected dimension of the Zariski closure of the rational points on the moduli space of curves?
For each genus $g$, there are many curves of genus $g$ defined over $\mathbb Q$. How many? We might study this question by considering the rational points of the Deligne-Mumford moduli space of curves ...
- 149k
16
votes
2
answers
3k
views
If the direct image of f preserves coherent sheaves on noetherian schemes, how to show f is proper?
The other direction is well known.
I think it is true and I was told by several other guys doing algebraic geometry that it is indeed true but they did not know how to prove. I am also wondering ...
- 1,982
16
votes
3
answers
5k
views
Do we have non-abelian sheaf cohomology?
Lets $X$ be a complex manifold (algebraic variety), $N$ an integer, and consider the sheaf $F$ defined by:
$F(U)$ ={ holomorphic maps $f: U\rightarrow GL(N,\mathbb{C})$ } with multiplicative ...
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
16
votes
4
answers
2k
views
Coboundaries and Gluing in Cech Cohomology - Intuition?
I'm trying to develop an intuition for Cech cohomology geometrically, but am currently failing. A lot of people seem to say that the groups $H^n$ measure obstructions to gluing local sections to get ...
- 827