All Questions
1,159 questions
1
vote
0
answers
170
views
Epimorphisms from an affine scheme?
Let $X$ be an affine scheme. Let $f:X\rightarrow Y$ be an integral morphism that is an epimorphism in the category of schemes. Is $Y$ affine?
1
vote
1
answer
317
views
Subschemes of the affine line over PID
Let $R$ be a PID with infinitely many prime ideals. Suppose we have two integral locally closed subschemes of $\mathrm{Spec}\,R[x]$ such that
both have non-empty intersection with the affine open $\...
user137767
1
vote
0
answers
298
views
Fully faithful functor from schemes to spaces
Is there a fully faithful functor from the category of schemes
to the category of topological spaces and continuous maps (or some other sufficiently topological objects, e.g. smooth manifolds and ...
user137517
1
vote
0
answers
126
views
Valuative criterion over non-locally Noetherian base
Let $X$ be an irreducible scheme and $f:X\rightarrow S$ be a morphism of finite type. Let $\eta$ be the generic point of $X$. Assume that for any (not necessarily discrete) valuation ring $A \subset K=...
- 231
1
vote
0
answers
55
views
Schemes for conditional distributions (monads)
(Note: This is a heuristic question. I'm trying to work out if this idea makes sense. I don't have much background in this area, so apologies if I'm wide of the mark.)
Suppose you have a monad ...
- 121
1
vote
0
answers
111
views
Quasi-compactness of irreducible separated scheme locally of finite type
Is an irreducible separated scheme locally of finite type necessarily quasi-compact?
- 723
1
vote
0
answers
127
views
Explicit description of the scheme obtained by relative gluing data over a base scheme
I have recently been trying to get a better understanding of the projective space bundle of a quasi-coherent sheaf of graded algebras over a scheme $X$. The key idea is the following construction ...
- 453
1
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0
answers
130
views
Exceptional Curves of a Fibration
Let $f:X \to Y$ be a dominant morphism between two integral proper surfaces (therefore $2$-dimensional, proper $k$-schemes).
Futhermore we assume
for the structure sheaf holds $\mathcal{O}_Y= f_*(\...
- 6,006
1
vote
0
answers
180
views
Weaker version of smooth base change for étale sheaves
Consider the cartesian square of schemes
$$ \require{AMScd}
\begin{CD}
X' @>{g'}>> X \\
@V{f'}VV @VV{f}V \\
S' @>>{g}> S
\end{CD}
$$
and the base change map
$$ \eta : ...
- 11
1
vote
0
answers
126
views
Which object related to families of algebraic varieties over a scheme $ S $ corresponds to the tensor product of vector bundles?
I asked yesterday on math.stackexchange.com that if the fiber product of two vector bundles seen in general as the fiber product of two families of special algebraic varieties over a scheme $ S $ ...
- 325
1
vote
0
answers
168
views
direct image and commutative diagram
Suppose we have following commutative diagram (not a square i.e not a base change) of schemes:
$X\xrightarrow{p_1} Y$, $Y\xrightarrow{\pi_2} Z$, $X\xrightarrow{\pi_1} W$, $W\xrightarrow{p_2} Z$.
...
user111251
1
vote
0
answers
348
views
rigid analytic geometry positive characteristic
I am a beginning graduate student. I have the following basic question I am very confused about:
Suppose $C$ is a smooth geometrically irreducible curve over a finite field $\mathbb{F}_q$, $q=p^m$, $...
- 147
1
vote
0
answers
187
views
Unitary dual of the Heisenberg group over non-archimedean local fields of characteristic two
What is the unitary dual of the Heisenberg group over non-archimedean local fields k of characteristic two? This is well-known for the real Heisenberg group and in fact, when local fields have ...
- 1,702
1
vote
0
answers
81
views
Geometry of componentially locally strongly separable algebras
Janelidze's categorical Galois theory yields, for nice adjunctions, a good notion of covering morphism.
The category of finitely affine schemes admits such an adjunction into the category of ...
- 10.5k
1
vote
0
answers
149
views
Smoothability of stable curves in mixed characteristic
Let $R$ be a complete DVR with residue field $k$ algebraically closed of characteristic $p$ and fraction field $K$ of characteristic zero. Let $C$ be a stable curve (in the sense of Mumford-Deligne) ...
- 2,323
1
vote
0
answers
208
views
(Ordered) Configuration space in algebraic geometry
Let $X$ be a topological space and denote by $F_n(X)$ the following subspace:
$$F_n(X):=\{(x_1,\cdots ,x_n)\in X^n: x_i\neq x_j \forall i\neq j\}.$$
Note that, we are not considering the quotient of $...
- 173
1
vote
0
answers
177
views
Is there an analytic criterion for quasi-compactness of a scheme?
Let $X$ be a locally finite type scheme over $\mathbb C$.
I'm looking for the analogue of the notion "finite type" for $X^{an}$ and an SGA 1 Exp. XII type of criterion which says that
The scheme $X$...
- 11
1
vote
0
answers
133
views
functions coming from a perverse sheaf
Let's take a scheme $X$ over a finite field $k$ and $f:X(k)\rightarrow\mathbb{Q}_{\ell}$
What kind of condition do I need on $f$ if I want that it comes from an irreducible perverse sheaf on $X$?
- 3,472
1
vote
0
answers
385
views
About complete residues on curves
Preliminaries:
Let $X$ be a projective smooth curve (scheme of finite type, integral and of dimension $1$) over a perfect field $F$. Let $K=K(X)$ be the function field of $X$ and for a closed point $...
- 1,237
1
vote
0
answers
189
views
Verdier duality on excellent schemes
Let $f:X\rightarrow Y$ be a regular morphism between $k$-schemes which are noetherian and excellent with a funcion of dimension.
In the book by Illusie-Laszlo-Orgogozo, there is a theorem (4.4.1 in ...
- 3,472
1
vote
0
answers
108
views
(Affine) Schemes and the point of view of morphisms with values in a field
Let $A$ be a commutative ring with unity. If $\varphi_1 : A \rightarrow K_1$ and $\varphi_2 : A \rightarrow K_2$ are ring morphisms from $A$ to fields, $\varphi_1$ and $\varphi_2$ are said to $1$-...
- 255
1
vote
0
answers
89
views
Holomorphic convergence conditions on $\mathbb C((z))$-valued points of a group $G$
Let $G$ be a complex, connected, simply connected, semisimple group. I'm trying to compare the following two spaces: The free loop space $LG$ of $G$, and the $\mathbb C((z))$-valued points of $G$, $G(\...
- 627
1
vote
0
answers
120
views
Vanishing theorems that work in positive characteristic
Let $X$ be a smooth projective variety over a field of characteristic $p>0$ of dimension at least $2$. I am looking for some examples when $H^2(\mathcal{O}_X)$ vanishes. Is there any standard way ...
- 1,981
1
vote
0
answers
205
views
Reducibility of fibers over closed points implies reducibility of the generic geometric fiber?
Suppose that $f\colon X\to Y$ is a proper (or even projective) morphism of (reduced) algebraic varieties over an algebraically closed field $k$. If fibers of $f$ over all closed points of $Y$ are ...
- 1,939
1
vote
0
answers
128
views
smooth morphism from a finite type source
Let $f: X\rightarrow Y$ a smooth morphism over a field $k$. We assume that $X$ is locally of finite type, does it imply that $Y$ is also locally of finite type?
- 3,472
1
vote
0
answers
134
views
ind scheme and Jacobson property
Let $G$ a semisimple group over $k$ and $k$ algebraically closed.
Let $G(k((t)))$ the corresponding ind-scheme, does it satisfies the Jacobson property, say closed points are dense in it?
- 3,472
1
vote
0
answers
154
views
closed subscheme of ind scheme
Let $X$ a ind-scheme of ind-finite type and ind-affine. (e.g, take a k- smooth, affine scheme of finte type $T$, $C$ a smooth projective curve over $k$ and $x$ a closed point, then $X=T(C-x)$ verifies ...
- 3,472
1
vote
0
answers
301
views
How to Construct a ''Nice'' Birational Model in Characteristic $p>0$?
Let $X={\rm Spec} A$ be a normal affine variety of dimension $n$ over an algebraically closed field $k$ of characteristic $p>0$. Also, assume that $S\subset X$ is a prime Weil-divisor on $X$.
Now,...
- 165
1
vote
0
answers
220
views
Dimension of faithful irreducible representations of $\mathbb{Z}_q\rtimes \mathbb{Z}_{p^2}$ in characteristic p,q
Let $p,q$ be primes s.t. $q=np+1$. Denote $m=p^2$. Then $\mathbb{Z}_p$ acts non-trivially on $\mathbb{Z}_q$, so we have a non-abelian semi-direct product $\mathbb{Z}_q\rtimes \mathbb{Z}_m$, with the ...
- 407
1
vote
0
answers
192
views
"Higher" Tangent spaces in char-p geometry - definition?
Hi, everyone!
I have some construction that requires exact definition.
I considered polynomial homomorphisms from $\Bbbk$ to $U_n(\Bbbk)$ (unitriangular matrices), where $char \Bbbk = p>0$ (more ...
- 1,422
1
vote
0
answers
238
views
Classification of fibres in pencils of curves of genus two
For the case of characteristic positive, there exist some clasification of families of curves of genus two over a elliptic curve?.
- 11
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
0
votes
1
answer
146
views
Are projective bundles corresponding to non-isomorphic vector bundles always non-isomorphic?
Suppose we are given a scheme $S$ and two vector bundles $V$ and $W$ over $S$. Is it always true that $\mathbb{P}(V)\cong \mathbb{P}(W)$ implies that $V\cong W$ as $S$-schemes?
If the statement is ...
- 187
0
votes
1
answer
384
views
Relation between canonical bundles under étale maps
Let $X$ and $Y$ be two integral separated Noetherian Gorenstein schemes over a base field $k$ of arbitrary characteristic whose local rings are unique factorization domains and $f: X\to Y$ an étale ...
- 6,006
0
votes
2
answers
621
views
Doing scheme theory with Hausdorff spaces
Suppose I have an allergy to non-Hausdorff spaces but I really want to do, say, arithmetic geometry. I wonder if there is some perverse way I could develop scheme theory that would accomodate for my ...
user137767
0
votes
1
answer
336
views
Self-intersection of zero section of line bundle over elliptic base curve
Let $C$ be an elliptic curve over $k=\mathbb{C}$ and $\mathcal{L}$ a line bundle of degree $d$. It induces naturally a $\mathbb{A}^1$-fibration $L \to C$ where $L=\underline{\operatorname{Spec}} (\...
- 6,006
0
votes
1
answer
188
views
Support of a coherent sheaf over a fiber product scheme
I'm trying to prove the following fact which I don't know if it is true since I am not able to find a counterexample:
Let $X,S$ be two $K$-scheme of finite type with $K$ an algebraically closed field....
- 395
0
votes
2
answers
386
views
Zariski closures of one parameter additive maps in positive characteristic
Suppose we are in characteristic $p$, and that the field, $K$, that we are working over is imperfect. We have a map $\Theta: K \to K^{\delta}$ where each coordinate function $\Theta_i$ is an additive ...
- 75
0
votes
1
answer
338
views
Schemes with open generic point
Let $X$ be any irreducible scheme with the property that the generic point $\eta$ of $X$ is an set open wrt underlying Zariski topology.
What can we say about the structure of such schemes? ...
- 6,006
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
0
votes
2
answers
428
views
Automorphism group of a scheme, 2
Hi,
I have the following two questions about automorphism groups of schemes.
First of all, let $S$ be a scheme, and $S^c$ its set of closed points. What
is the connection between $Aut(S)$ and $Aut(...
- 4,605
0
votes
2
answers
2k
views
non discrete valuation ring [closed]
Hi,
I am looking for examples of non-discrete valuation rings. Could you help me?
Thanks
- 141
0
votes
1
answer
175
views
An inseparable lift of a regular variety.
Let $X$ be a variety over an (imperfect) field $k$, that is regular as a scheme. Let $k'/k$ be an algebraic inseparable extension (I am interested in $k'$ being the perfection or the algebraic closure ...
- 16.9k
0
votes
1
answer
271
views
Proof of rigidity lemma
I have problems to understand a proof in this paper by Pierrick Dartois on Abelian varieties:
Theorem 1.13 (rigidity lemma). Let $ \varphi: X \times_k Y \to Z$ be a morphism of $k$-schemes. Assume ...
- 6,006
0
votes
1
answer
554
views
Completed stalks of the pushforward of the structure sheaf
Let $\pi:X\to S$ be a proper morphism of Noetherian schemes. Is it possible that $\pi_*\mathcal{O}_X\neq \mathcal{O}_S$ but the natural map $\mathcal{O}_s^\wedge\to (\pi_*\mathcal{O}_X)_s^\wedge$ is ...
user158636
0
votes
1
answer
97
views
Automorphisms of Lie algebra of type $A_5$ modulo its center in characteristic 2
Let $L$ be classical Lie algebra of type $A_5$ over field of characteristic 2; let $M$ be the quotient $L/Z(L)$ modulo its center $Z(L)$.
What about the group of automorphisms of M?
Does anybody ...
- 11
0
votes
1
answer
528
views
Artin approximation of a diagram
Let consider $f:(X,x)\to (Z,z)$ and $g:(Y,y)\to (Z,z)$ morphisms of pointed $k$-schemes of finite type ($k$ is a field). Suppose that there exists a map on the level of formal neighborhoods $\phi:X_{x}...
- 3,472
0
votes
1
answer
502
views
Closed immersion of closed fiber?
Suppose $f: X \rightarrow Y$ is a morphism of schemes, and $y \in Y$ is closed point.
We know the first projection morphism $p_1: X \ $ x$_{Y} \ k(y) \rightarrow X$ is a homeomorphism onto $f^{-1}(y)...
- 79
0
votes
1
answer
131
views
curve through a point avoiding an hypersurface
Let $H$ be a closed hypersurface in $\mathbb{A}^{n}$, $n$ big enough over $\mathbb{C}$. Let $U$ be the complementary open subset.
Let $x\in H$, Is it possible to find an curve $C\subset\mathbb{A}^{...
- 3,472
0
votes
1
answer
270
views
Confusion with the field of definition of a variety [closed]
Fix a field extension $k\subseteq K$ (assume that the fields have characteristic $0$) and consider the two following definitions:
Now let's restrict our attention to a closed subscheme $X\subseteq\...
- 1,237