All Questions
1,159 questions
1
vote
0
answers
146
views
Factoriality of schubert cells in affine flag variety
Take for simplicity $G=SL_n$ and consider the affine flag variety $Fl=G(\mathbb{C}((t)))/I$ for $I$ the Iwahori corresponding to the Borel of upper triangular matrices of determinant one.
For each $...
- 3,472
14
votes
2
answers
4k
views
explanation on a scheme which is not affine scheme
Hartshorne at the end of page 76 of his Algebraic Geometry book gives an example of a scheme which is not an affine scheme. The scheme is constructed by gluing two affine lines together along their ...
- 261
10
votes
2
answers
2k
views
Parahoric Group Scheme
I am looking for the definition of a parahoric group scheme in the sense of Bruhat and Tits? I couldn't find a reference for that? at least a "clear" reference!
thanks
- 307
2
votes
0
answers
215
views
A closed point in the closure of any point in the closure of any point of an irreducible scheme
Let $X$ be the underlying space of an irreducible scheme. In particular, $X$ is non-empty.
Note that the closure of any point is a closed irreducible subset. We say that a point has codimension $n$ ...
user137767
7
votes
3
answers
752
views
Does every linear group admit a subgroup of dimension 1?
Suppose that $G$ is a linear group of positive dimension, defined over some field $k$. Is that true, that $G$ admits a (closed) one-dimensional subgroup?
I'm pretty much sure this is true in ...
- 267
6
votes
0
answers
421
views
Where is the local structure theory of étale morphisms needed?
In Stacks 02GH it is mentioned étale morphisms are locally standard étale. This seems to be the analogue of the local form of local diffeomorphisms in the smooth category. In the latter, local ...
- 10.5k
2
votes
1
answer
206
views
Quasi-compactification of locally spectral spaces
Let $X$ be a locally spectral topological space (i.e. a space admitting an open cover by spectral spaces). Does there necessarily exist a quasi-compact locally spectral space $Y$ and an injective ...
user138661
31
votes
4
answers
5k
views
The Frobenius morphism
I found the following list on the "Frobenius Page" by David Ben-Zvi, described by the author as "an outdated collection of intuitive ways to think about raising to the p-th power".
Generates a ...
3
votes
0
answers
193
views
Motivic strong bellows conjecture
There is a theorem due to Gaifullin--Ignashchenko stating that the Dehn invariant of any flexible polyhedron in the $n$-dimensional Euclidean space ($n\geq 3$) is constant during the flexion.
Is ...
user138661
22
votes
3
answers
2k
views
One dimensional (phi,Gamma)-modules in char p
I would like to better understand the simplest case of the correspondence between Galois representations and (phi,Gamma)-modules. Namely, consider 1-dimensional Galois representations of $G_{Q_p}$ ...
- 1,680
3
votes
0
answers
197
views
Topological properties of Noetherian affine schemes that do not hold for general Noetherian spectral spaces
I used to think that the only reason why an affine scheme with a Noetherian space can fail to be Noetherian is nilpotents. It turns out that this is not true.
This leads me to the following question: ...
user139949
9
votes
1
answer
847
views
Algebro-geometric version of {vector fields} $\longleftrightarrow$ {flows} correspondence?
Main Question: What Is the correpondence between flows and vector
fields in algebraic geometry?
Here is a more precise statement could be an answer If it was true (I have no idea it is):
"...
- 7,799
2
votes
0
answers
177
views
vanishing of higher algebraic de Rham cohomology and sheaves of differentials for singular curves
I'm looking for some results or references about de Rham cohomology of curves in less-than-optimal cases. The two vanishing results I care about are:
$H^{k}_{\text{dR}}(X) = 0$ for $k>2$, since $H^...
- 1,142
5
votes
0
answers
1k
views
Base change of integral scheme of finite type over a field
Let $X$ be an integral scheme of finite type over a field $k$. If $k'\supseteq k$ is a field extension, then $X' = X\otimes_k k'$ is not necessarily integral.
Why does each irreducible component of ...
2
votes
0
answers
293
views
Direct sums of invertible sheaves commuting with global sections and the functor of points approach
I am looking at the Stacks Project's treatment of the functor of points for projective space.
Let's restrict to the case that $S$ is a graded ring, generated by $S_{1}$ as an $S_{0}$ algebra. The ...
- 453
2
votes
0
answers
103
views
Existence of an open dense of a connected scheme such that the preimage under a surjective finite etale map is connected
Let $K$ be a field, $G$ a smooth finite linear algebraic group over $K$, $X$ a proper reduced connected separated scheme of finite type over $K$, $g: Y \to X$ a connected etale $G$-torsor over $X$ (so ...
- 21
0
votes
1
answer
89
views
Morphism of schemes with non-sober image
Let $f:X\rightarrow Y$ be a morphism of schemes. Can the image of $f$ endowed with the subspace topology not be sober?
user138661
13
votes
1
answer
2k
views
Which local ringed spaces are schemes?
(This was originally asked on math.stackexchange, but didn't get any responses. I figured it might be worthwhile to move it here and try again.)
This paper gives a proof that the underlying ...
- 639
4
votes
0
answers
133
views
Non-cyclic Galois groups over the field of formal Laurent series in positive characteristic
This should be an easy question, but I am unfortunately not able to give an answer, so I am sorry if this is not the appropriate level for the site.
Let $C$ be an algebraically closed field of ...
- 41
11
votes
1
answer
857
views
Sheaf associated to presheaf Aut
Let $S$ be a scheme and let $C$ be the category of schemes flat and locally of finite presentation over $S$. Endow $C$ with the fppf topology (or perhaps any subcanonical topology). Let $\mathcal P$ ...
1
vote
0
answers
113
views
Question about Immersion
Let $X,Y$ Noetherian integral schemes and assume we have an immersion
$$i:X \to Y$$
An immersion is according to https://stacks.math.columbia.edu/tag/01IO a composition of a closed immersion $c: X \...
- 6,016
10
votes
5
answers
5k
views
pushforward of locally free sheaf is locally free?
Hi,
Is there an example of a proper smooth map of schemes $f:X\to Y$ and a vector bundle $E$ on $X$
such that $f_*E$ is not locally free on $Y$?
Thanks
- 2,842
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
0
votes
0
answers
153
views
Locus of trivialization of an extension of a vector bundle
Let $X$ be a normal noetherian affine scheme. Let $j:U\rightarrow X$ be a codimension two open of $X$ and $\mathcal{E}$ a vector bundle on $U$.
We assume that $j_*\mathcal{E}$ is a vector bundle. In ...
- 3,472
1
vote
0
answers
155
views
Sheaf of Kähler Differentials is Invertible in Dense Open Subset
Let $f:S→B$ be an elliptic fibration from an integral surface $S$ to integral curve $B$
.
Here I use following definitions:
A surface (resp. curve) is a $2$
-dim (resp. $1$-dim) proper k scheme ...
- 6,016
1
vote
0
answers
95
views
Units in the coordinate ring on a reductive group
Let $K$ be a field and $G$ a connected reductive group over $K$.
Can we describe $K[G]^{*}$?
- 3,472
1
vote
0
answers
92
views
Topological space modeled by special topological structures
Let $X$ be a topological space. Suppose it is "modeled by" topological spaces of the form $\text{Spec}(A)$ for some commutative ring $A$, then, (along with some other conditions/structure), we call $...
- 6,023
1
vote
0
answers
56
views
local acyclicity when restricting to an hypersurface
Let $X$ be a smooth scheme over $\mathbb{C}$ and a constructible sheaf $K$ of complex vector spaces on $X\times\mathbb{A}^1$ and a function $g:X\rightarrow \mathbb{A}^1$.
Suppose that $K$ is locally ...
- 3,472
11
votes
1
answer
928
views
Non-algebraic K3 surfaces in characteristic $p$
I have a very naive question.
Recall that over the field of complex numbers, there exist non-algebraic K3 surfaces. Namely, smooth non-projective simply connected compact complex surfaces with ...
- 21.4k
7
votes
1
answer
452
views
Recovering the Zariski topology from the Zariski topology over an extension
Suppose $A$ is a $k$-algebra, with $k$ a field, and let $\ell$ be a field extension of $k$. Is there an easy way to see/recover $\mathrm{Spec}(A)$ in/from $\mathrm{Spec}(A \otimes_k \ell)$, using the ...
- 4,605
0
votes
0
answers
170
views
Linear Morphism of Schemes
Let $U$ be an arbitrary scheme and we consider the schematic fiber product $\mathbb{A}^1 _U = \mathbb{A}^1 _{\mathbb{Z}} \times_{\mathbb{Z}} U$.
My question referer to Bosch's "linear morphisms" (of ...
- 6,016
4
votes
1
answer
2k
views
On Q-Cartier Divisors
I have my question on Q-Cartier Weil divisor.
People say $D$ is Q-Cartier divisor if $nD$ is Cartier for some $n \geq 1$. Especially for $n > 1$, I have never seen the `rigorous' definition of $...
- 781
1
vote
2
answers
508
views
Base change of a finite morphism
Let $X$, $Y$, $Z$ be integral schemes of finite type over a field $K$ (i.e., locally affine opens are finitely-generated algebras over $K$). Suppose we have the following condition$\colon$
$f \colon ...
- 563
8
votes
2
answers
814
views
Are there varieties with non finitely generated Picard group and vanishing irregularity?
Let $X$ be a smooth projective variety over an algebraically closed field $k$.
Can it happen that $q(X) := \dim H^1(X,\mathcal O_X) =0$ and $\textrm{Pic} \,X$ is not finitely generated?
Certainly, ...
- 83
4
votes
1
answer
322
views
Pairing on arithmetic surfaces
Let $f: X \to S$ be an arithmetic surface, where $S=\operatorname{Spec } O_K$ for a number field $K$. It is well known that if we want to introduce a reasonable intersection theory on $X$ we have to ...
- 321
0
votes
0
answers
146
views
A scheme whose underlying space is the product of the underlying spaces of schemes
We know that the product of two spectral topological spaces is spectral.
If $X$ is the underlying space of the scheme $\mathrm{Spec}\,\mathbb{Z}[x]$, what is a simple example of an affine scheme ...
user140269
0
votes
0
answers
144
views
Making a quasi-compact open into an affine open
Let $X$ be a spectral topological space, $U\subset X$ be a quasi-compact open subspace. Is there necessarily some scheme structure on $X$ (we do not require it to be affine) such that $U$ endowed with ...
user138661
2
votes
1
answer
220
views
Does cohomology with compact support contain "ample" elements
Let $X$ be a quasi-projective integral variety over $\mathbb{C}$. If $X$ is projective, then $\mathrm{H}^2(X,\mathbb{Z})$ contains "ample" classes. These "ample" classes are defined as being the image ...
- 181
4
votes
0
answers
162
views
Embed FPPF group scheme into smooth one
Let $A$ be a ring and $G$ be an affine commutative FPPF group scheme over $A$. Can we embed $G$ into a smooth group scheme over $A$?
- 3,472
5
votes
0
answers
238
views
Cohomology groups on small fppf site and small etale site are not the same
Let $F$ be a quasi-coherent sheaf on a scheme $X$. Is there an example where cohomology groups of $F$ on small fppf site of $X$ and small etale site of $X$ are not isomorphic?
- 723
1
vote
2
answers
321
views
Embeddings of fields and rational points
Let $S$ be a irreducible scheme over a field $k$ (for example a smooth projective curve over algebraically closed field). Denote by $k(S)$ its field of fractions. Let $K$ be a(n algebraically closed) ...
- 2,305
4
votes
1
answer
804
views
Pushouts of schemes along closed immersions
Is there a closed immersion $i:Z\to X$ and a finite morphism $f:Z\to Y$ of schemes such that the pushout of the span $Y\stackrel{f}{\leftarrow} Z\stackrel{i}{\rightarrow} X$ does not exist in the ...
- 349
2
votes
0
answers
132
views
Functorial subscheme structure on non-locally closed subsets
If we have a scheme and a locally closed subset of the underlying topological space, then there is a canonical way to put a scheme structure on it so that the inclusion map can be upgraded to a ...
user138661
-1
votes
1
answer
480
views
How do I find the algebra representing the projective bundle of a direct sum of line bundles over a projective space?
I am trying to learn how to compute the projective bundle $\mathbb{P}(\mathcal{O}(a_1)\oplus \cdots \mathcal{O}(a_k))$ over some projective space using relative proj. How can I find a presentation for ...
- 1,716
0
votes
1
answer
444
views
Example of projective variety that do not contain algebraic curves of genus strictly greater to $1$
Does exist a smooth, complex, projective variety $X$ of dimension $d\geq2$ such that $X$ does not contain smooth, complex, projective curves of wichever genus $g\geq2$?
Answer by Bertie: No, it does ...
- 828
15
votes
1
answer
1k
views
Number of curves over a finite field
Let $K$ be a finite field. Is there a formula for the number of isomorphism classes of genus $g$ smooth curves over $K$?
In other words does there exists a formula for the number of rational points ...
- 8,998
2
votes
0
answers
136
views
Size of the ring of functions on open subschemes
This question consists of two related sub-questions.
Let $X$ be a Noetherian integral affine scheme. Under what assumptions on $X$ does every open subscheme of $X$ have a Noetherian ring of global ...
user138661
20
votes
1
answer
2k
views
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
2
votes
1
answer
965
views
on universal homeomorphisms between schemes
We are taught since when we are young that schemes are cool because they take into account "nilpotents". This means also that we can distinguish between schemes which have the same underlying ...
- 155
5
votes
0
answers
568
views
Eisenbud-Goto conjecture in Positive Characteristic
Eisenbud-Goto conjecture predicted that the Castelnuovo-Mumford regularity
${\rm reg}(X)$ of a non-degenerate projective variety $X\subset \mathbb{P}^N$
is bounded by the $\deg(X)-{\rm codim}(X,\...
- 1,595