All Questions
22,770 questions
12
votes
4
answers
2k
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For which hypersurfaces in projective space does the complement admit an algebraic group structure?
For example, if $H$ is a hyperplane, then $\mathbb{P}^n - H = \mathbb{A}^n$, which is a vector space.
If $n = m^2 - 1$, then we can regard $\mathbb{A}^{n+1}$ as the space of $m \times m$ matrices and ...
- 3,918
14
votes
5
answers
2k
views
Rational maps with all critical points fixed
What can be said about rational self-maps of $\mathbb P^1$
for which all critical points are also fixed points ?
If all but one of the fixed points are critical, there is
a characterization in http://...
- 8,828
5
votes
1
answer
586
views
a general theory of configurations?
Once I found by accident an article by MacPherson: "Classical projective geometry and modular varieties", in "Algebraic analysis, geometry, and number theory" (Baltimore, MD, 1988), whose introduction ...
- 10.8k
9
votes
3
answers
3k
views
Why is the Euler characteristic of powers of a line bundle a polynomial in the power?
Mumford's book Abelian Varieties asserts that for a line bundle L on a projective variety (if necessary, you can assume it is as nice as possible), the Euler characteristic $\chi(L^k)$ of tensor ...
- 31.2k
6
votes
2
answers
1k
views
Higher vanishing cycles
The generalisation of the vanishing cycle formalism in SGA 7 is apparently since the 1970's an issue, Morava mentioned a connection with Bousfield localization. I find the Morava's remarks un-...
- 10.8k
27
votes
7
answers
6k
views
Etale covers of the affine line
In characteristic p there are nontrivial etale covers of the affine line, such as those obtained by adjoining solutions to x^2 + x + f(t) = 0 for f(t) in k[t]. Using an etale cohomology computation ...
- 52.6k
43
votes
1
answer
19k
views
What is inter-universal geometry?
I wonder what Mochizuki's inter-universal geometry and his generalisation of anabelian geometry is, e.g. why the ABC-conjecture involves nested inclusions of sets as hinted in the slides, or why such ...
- 10.8k
6
votes
2
answers
468
views
Algorithms for semistable reduction of families of curves
This is a somewhat vague question which came up MSRI a few days ago: Suppose I have a family of curves over a one dimensional base, given in a computationally explicit way. For example, maybe I have a ...
- 156k
11
votes
3
answers
1k
views
Is there a version of the valuative criteria for separateness/properness for varieties?
What I had in mind was something like the following:
X is separated/proper iff for all curves C and all maps f : C \ c -> X, f extends to C in at most/exactly one way.
Is there a good reason why ...
- 4,612
16
votes
2
answers
2k
views
Bad Categorical Quotients
Let $G$ be an algebraic group acting on a scheme $X$. Then $f: X \to Y$ is called a categorical quotient if it is constant on $G$-orbits and every $X \to Z$ constant on $G$-orbits factors through it ...
- 2,806
15
votes
7
answers
2k
views
Examples of rational families of abelian varieties.
I'd like to know examples of non-trivial families of abelian varieties over rational bases (e.g. open subschemes of the projective line P^1).
One can generate many examples as Jacobians of rational ...
- 10.5k
14
votes
4
answers
3k
views
References for syntomic cohomology
Could anyone point to good readable references for learning about syntomic cohomology?
- 5,637
6
votes
4
answers
1k
views
When is a map given by a word surjective?
Let $w(x,y)$ be a group word in $x$ and $y$.
Let $x$ and $y$ now vary in $\operatorname{SL}_n(K)$, where $K$ is a field. (Assume, if you wish, that $K$ is an algebraically complete field of ...
- 20.2k
38
votes
6
answers
6k
views
"Points" in algebraic geometry: Why shift from m-Spec to Spec?
Why were algebraic geometers in the 19th Century thinking of
m-Spec as the set of points of an affine variety associated to the
ring whereas, sometime in the middle of the 20 Century, people started ...
- 2,967
25
votes
4
answers
2k
views
algebraic group G vs. algebraic stack BG
I've gathered that it's "common knowledge" (at least among people who think about such things) that studying a (smooth) algebraic group G, as an algebraic group, is in some sense the same as studying ...
22
votes
6
answers
3k
views
Formal consequences of Riemann-Roch (multiple answers welcome)
This question aims to pin down what Riemann-Roch can tell us about a divisor on a curve, without any "geometric thinking". It can be annoying to wonder if there is some clever trick you're missing ...
- 11.2k
8
votes
1
answer
2k
views
Mirror symmetry for noncompact Calabi-Yau manifolds
In analogy with the Hodge diagram for ordinary de Rham cohomology, we should have some kind of diagram for Alexander-Spanier cohomology. Doing all the relevant duality stuff and assuming that now our ...
- 287
11
votes
2
answers
2k
views
Finiteness conditions on simplicial sheaves/presheaves
Could someone give an overview, or just some examples, of "finiteness conditions" for simplicial sheaves/presheaves and/or simplicial schemes? Any answer or comment about this would be interesting, ...
- 5,637
11
votes
8
answers
3k
views
Are good introductory/pedagogical problems in algebraic geometry rare?
I have just started reading Elementary Algebraic Geometry by Hulek. It is a nice book but I find that it doesn't give many problems (about 10 to 15 per chapter), and that the exercises present are a ...
16
votes
3
answers
5k
views
Stalks of sheaf-hom
Let $F$ and $G$ be sheaves on $X$. Under what conditions is the natural map from the stalk at $p$ of $\mathcal{H}\kern{-1pt}\mathit{om}(F,G)$ to $\mathrm{Hom}(F_p, G_p)$ an isomorphism?
- 161
14
votes
2
answers
882
views
A complex manifold which is quasiprojective in two different ways
Does there exist a complex manifold M which is a quasiprojective variety in two "essentially" different ways? Let me be more specific. I'm looking for a complex manifold M together with two ...
- 44.8k
16
votes
2
answers
2k
views
"synthetic" reasoning applied to algebraic geometry
A hyperlinked and more detailed version of this question is at
nLab:synthetic differential geometry applied to algebraic geometry.
Repliers are kindly encouraged to copy-and-paste relevant bits of ...
- 19.8k
10
votes
2
answers
943
views
Logarithmic structures on moduli of elliptic curves over Z
I've heard it stated that if you take the moduli of elliptic curves with some level structure imposed (as a moduli scheme over Spec(Z)), there is a logarithmic structure that you can impose at the ...
- 52.6k
30
votes
5
answers
4k
views
Deformation theory of representations of an algebraic group
For an algebraic group G and a representation V, I think it's a standard result (but I don't have a reference) that
the obstruction to deforming V as a representation of G is an element of H2(G,V&...
8
votes
2
answers
823
views
What is the affinization of M_g?
This question is inspired by What is an example of a function on M_g? . Consider Mg, the moduli space of genus g curves, NOT compactified. When g is 3 or greater, this is not affine. Does anyone know ...
- 156k
9
votes
1
answer
840
views
Limit Linear Series
A linear series on a curve C is a line bundle L together with a subspace V of the global sections of L. Eisenbud and Harris develeoped a theory of limit linear series which explans how (L, V) ...
- 611
8
votes
1
answer
1k
views
what is the connection between D-modules and coordinate bundles?
Fix $n$ and a field $k$ of characteristic zero. Let $G$ be the pro-algebraic group of automorphims of $k[[x_1,...x_n]]$. Let $G_0$ be the subgroup of automorphisms preserving the closed point (note ...
- 1,038
8
votes
3
answers
921
views
Generic Noether normalisation
Suppose that $M$ is a finitely generated module over $A=k[X_1,\ldots,X_n]$ of Krull dimension $m$ with $k$ an infinite field. Then one version of Noether normalisation says there is an $m$-dimensional ...
- 3,545
2
votes
1
answer
925
views
Theta Functions and Cousins
So I am (barely) familiar with the construction of the theta function of an integral lattice $L$. The theta function, as I understand it, is defined as the function which takes a variable $z$ and ...
- 27.5k
5
votes
3
answers
1k
views
Can the valuative criteria be checked "on a dense open"?
The valuative criterion for separatedness (resp. properness) says that a noetherian scheme X is separated (resp. proper) if and only if
for any DVR R, with fraction field K,
any map Spec(K)→...
7
votes
8
answers
746
views
What is an example of a function on M_g?
It feels bad talking about a space without knowing a single function on it, hah?
So what is a function on the moduli space of curves, from the geometric point of view?
From the functorial point of ...
- 5,052
5
votes
4
answers
826
views
$E_\infty$ spectrum corresponding to $\Bbb Z_p$
First of the questions about derived algebraic geometry from a noobie.
The way I understand it, every discrete ring $R$ corresponds to some ring spectrum whose $\pi_0$ is $R$. Now consider $p$-adic ...
- 15.1k
9
votes
2
answers
2k
views
What is an example of a smooth variety over a finite field F_p which does not lift to Z_p?
Somebody answered this question instead of the question here, so I am asking this with the hope that they will cut and paste their solution.
- 10.5k
15
votes
5
answers
3k
views
Existence of (smooth) models
Hi everyone,
let X be a variety over a field k, S an integral scheme such that the function field K of S is contained in k. An S-scheme X is called model of X/k if X x_S k = X, i.e. if the generic ...
- 4,450
15
votes
2
answers
2k
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What is an example of a smooth variety over a finite field F_p which does not embed into a smooth scheme over Z_p?
Such an example of course could not be projective and would not itself lift to Z_p. The context is that one can compute p-adic cohomology of a variety X over a finite field F_p via the cohomology of ...
- 10.5k
39
votes
4
answers
3k
views
Does a scheme have a "separification"?
Background:
(1) If C and D are categories and there is a forgetful functor U:C→D, then a C-ification functor F:D→C is an adjoint to U. For example, the (left) adjoint to the forgetful ...
2
votes
3
answers
1k
views
What is the base change in number theory?
I'm somewhat familiar with base change in scheme theory: sometimes a property of a morphism X \to Y survives a base change ...
- 15.1k
10
votes
1
answer
1k
views
Why are torsion points dense in an abelian variety?
Hi everyone,
let $A$ be an abelian variety of dimension $g$ over an algebraically closed field $k$ of characteristic $p\geqslant 0$. I'm trying to prove that the subgroup $A'$ which is the union of ...
- 4,450
32
votes
4
answers
3k
views
Spectrum of the Grothendieck ring of varieties
Here's a problem that may ultimately require just simple algebraic-geometry skills to be solved, or perhaps it's very deep and will never be solved at all. From the comments, some literature and my ...
- 15.1k
7
votes
1
answer
2k
views
Dualizing sheaf on singular curves
I am trying to understand the stabilization map, which takes a prestable curve (a curve with some marked points, and at worst nodal singularities) and returns a stable curve (a curve with some marked ...
- 21k
13
votes
3
answers
3k
views
What is the Theorem of the Cube?
What is the "theorem of the cube" for abelian varieties? What is the statement and how should I think about it?
- 27.5k
7
votes
2
answers
1k
views
Non-zero sheaf cohomology
Let $\mathbb{R}$ denote the real line with its usual topology. Does there exist a sheaf $F$ of abelian groups on $\mathbb{R}$ whose second cohomology group $H^{2}\left(\mathbb{R},F\right)$ is non-zero?...
- 47.5k
20
votes
2
answers
10k
views
does a line bundle always have a degree
For curves there is a very simple notion of degree of a line bundle or equivalently of a Weil or Cartier divisor. Even in any projective space $\mathbb P(V)$ divisors are cut out by hypersurfaces ...
- 3,968
14
votes
5
answers
4k
views
When are Hilbert schemes smooth?
I know that Hilbert schemes can be very singular. But are there any interesting and nontrivial Hilbert schemes that are smooth? Are there any necessary conditions or sufficient conditions for a ...
- 21k
10
votes
5
answers
1k
views
Compact Kaehler manifolds that are isomorphic as symplectic manifolds but not as complex manifolds (and vice-versa)
What are some examples of compact Kaehler manifolds (or smooth complex projective varieties) that are not isomorphic as complex manifolds (or as varieties), but are isomorphic as symplectic manifolds (...
- 21k
19
votes
6
answers
4k
views
Why and how are moduli spaces of (semi)stable vector bundles well-behaved?
The slope of a vector bundle $E$ is defined as $\mu(E) = \deg(E)/\mathrm{rank}(E)$. Then a vector bundle $E$ is called semistable if $\mu(E') \leqslant \mu(E)$ for all proper sub-bundles $E'$. It is ...
- 21k
20
votes
5
answers
4k
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Equivalent statements of the Riemann hypothesis in the Weil conjectures
In the cohomological incarnation, the Riemann hypothesis part of the Weil conjectures for a smooth proper scheme of finite type over a finite field with $q$ elements says that: the eigenvalues of ...
- 671
13
votes
3
answers
1k
views
How to topologize X(R) when R is a topological ring?
Given a topological ring $R$, under what conditions and in what way, can one induce a topology on the $R$-points of a scheme $X$? For example, if $X$ is $P^n$ or $A^n$, one has natural topology on ...
- 671
28
votes
5
answers
9k
views
Can a quotient ring R/J ever be flat over R?
If $R$ is a ring and $J\subset R$ is an ideal, can $R/J$ ever be a flat $R$-module? For algebraic geometers, the question is "can a closed immersion ever be flat?"
The answer is yes: take $J=...
18
votes
3
answers
2k
views
Is there an example of a variety over the complex numbers with no embedding into a smooth variety?
Is there an example of a variety over the complex numbers with no embedding into a smooth variety?
- 10.5k