All Questions
486 questions with no upvoted or accepted answers
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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 \...
- 6,006
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0
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91
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Factorizations of closed embeddings of smooth schemes
All schemes will be of finite type over a field $k$. Say I have a closed embedding $\iota: X \hookrightarrow Y$ of smooth $S$-schemes for some scheme $S$ (in particular it is a regular embedding). ...
- 595
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0
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155
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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 ...
- 6,006
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222
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Extend a Morphism of Schemes
I have a question about following argument used in Szamuely's "Galois Groups and Fundamental Groups" in the excerpt below (or look up at page 159):
Let $X,Y$ schemes which are finite and locally free ...
- 6,006
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96
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depth and extension of sections
Let $S$ be an affine scheme, $X$ smooth affine over $S$ and $U$ an open subset of $X$, fiberwise of codimension at least two.
Suppose that we have a function on $U$, can we extend it to $X$?
- 3,472
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890
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Cohomology of the pullback of a coherent sheaf (considered as an abelian sheaf)
Let $X$ be an affine Noetherian scheme, $Y$ a separated Noetherian scheme, $f:X\rightarrow Y$ a morphism of schemes inducing a homeomorphism on the underlying topological spaces.
Let $F$ be a ...
user137767
1
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0
answers
61
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Non-constructible conditions on the fibers that lift from the generic point to a non-empty open
Let $f:X\rightarrow Y$ be a flat morphism of schemes, with an irreducible locally Noetherian target. Call a condition on the fibers of $f$ "good" if the condition holds at the generic point of $Y$ iff ...
user137767
1
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170
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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?
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298
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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
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0
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126
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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
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55
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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
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111
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Quasi-compactness of irreducible separated scheme locally of finite type
Is an irreducible separated scheme locally of finite type necessarily quasi-compact?
- 723
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127
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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
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130
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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
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180
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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
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126
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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
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168
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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
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0
answers
348
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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
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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
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81
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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
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149
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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
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208
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(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
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0
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177
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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
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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
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0
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385
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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
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0
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189
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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
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108
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(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
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89
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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
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120
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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
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205
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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
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128
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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
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134
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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
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154
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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
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301
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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
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220
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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
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0
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192
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"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
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0
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238
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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
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0
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190
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About Chern classes via Atiyah class
I am trying to understand a construction of the Chern classes of a vector bundle $\mathcal{E}$ via the Atiyah class, like is done in this text and here in section 1.4. I am interested in the case ...
- 348
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112
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Irregularity of surfaces for dominant maps
I have a question about an argument in the proof of Lemma 1.2.(1) in Quotients of K3 surfaces modulo involutions by D. Q. Zhang:
Let Let $(X, \sigma)$ be X be a smooth projective K3 surface with an ...
- 6,006
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0
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99
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Quotients of K3 surfaces vs cyclic covers
Let $X$ be an algebraic K3 surface (for sake of simplicity, with base field of char $\neq 2$) and $f: X \to X$ a non-symplectic morphism (i.e. non-symplectic in sense of that that the induced action ...
- 6,006
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111
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Lefschetz Theorem in Dolgachev's On automorphisms of Enriques Surfaces
Let $F$ be a Enriques surface over $\Bbb C$. I have a question about a detail in the proof of Proposition 2.1. from Dolgachev's On automorphisms of Enriques surfaces.
This 2.1. Proposition. states ...
- 6,006
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127
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Relative minimal models of pencils of surfaces
I would like to ask for recomendation for literature on theory relative minimal models of surfaces, where "relative" in sense of that the study objects are not surfaces alone (="absolue ...
- 6,006
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0
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124
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Counit map surjective
Let $X \to Y$ a (set theoretically) surjective morphism of schemes, $L$ a line bundle/invertible sheaf on $X$ (maybe more generally a locally free coh sheaf, but let's stick firstly on invertible ...
- 6,006
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118
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Induced action on infinitesimal thickenings by an algebraic group
Let $X$ be an irreducible locally noetherian $k$-scheme (for $k$ any field), $G$ an algebraic group acting on $X$ via $a:G \times X \to X$ and $x \in X$ a closed point, which is by Zariski's lemma ...
- 6,006
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0
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109
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Affine scheme over ring of meromorphic functions with finite poles on unit circle
I am looking into the set $S$ of meromorphic functions with a finite number of poles on the unit circle (i.e., rational functions with poles on the unit circle). I assume that any $h\in S$ has the ...
- 91
0
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0
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331
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Smooth morphisms under base change, Qing Liu's proposition 4.3.38
I have a concern about the first assertion in the proof of proposition 4.3.38 of Qing liu's "Algebraic Geometry and Arithmetic Curves". Referring to smooth morphisms, he says "The ...
- 149
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145
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Stalk at isolated point of proper map with $f_*\mathcal{O}_X=\mathcal{O}_Y$
Let $f:X \to Y$ a proper surjective map between schemes $X,Y$ with additional assumption $f_*\mathcal{O}_X=\mathcal{O}_Y$. Let $y \in Y$ with a 'isolated point fiber', i.e., $f^{-1}(y)=\{x\}$ as set. ...
- 6,006
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0
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267
views
completion and tensor product
Let $A$ be a commutative ring, consider the map $Spec(A[[t]])\rightarrow Spec(A)$, does it have geometrically connected fibers?
If $A$ is noetherian, it is clear because one has for $k$ a residue ...
- 3,472
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0
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190
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How to compute the exceptional divisor of this blow-up
Suppose that $k$ is a field and $R$ is the ring $k[x,xy,xy^2,xy^3]$.Let $I$ be the maximal ideal of $R$ generated by $x,xy,xy^2,xy^3$.Let $E$ be the exceptional divisor of the blow-up of Spec$R$ along ...
- 639
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0
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130
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Is a closed subsecheme contained in a Cartier divisor?
Let $X$ be a variety over a field $k$. For a closed subscheme $Z\hookrightarrow X$ and a closed point $x\in X$ such that $\text{codim}_XZ \geq 1$ and $x\notin |Z|$, is there an effective Cartier ...
- 349