All Questions
22,548 questions
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inertia stratification
Let $X$ be a nice algebraic variety (say smooth, projective) over a field of characteristic 0. Let $G$ be an abelian group acting on $X$. For each subgroup $H$ of $G$, denote by $X^H$ the closed ...
- 11
3
votes
1
answer
214
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Is there a common general setup for both Weil cohomologies and generalized cohomology theories?
My question can be simply (and loosely) stated as follows:
Is there a general (but not too general) construction that captures, as specializations, both Weil cohomologies in algebraic geometry and ...
- 23.4k
3
votes
1
answer
386
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Is the quotient functor of points of a Lie group with the subfunctor of a closed subgroup a sheaf?
Let G be a Lie group and $H\subseteq G$ be a closed subgroup. It can be shown that $H$ has a unique differentiable structure such that the inclusion map $H\to G$ is an embedding of manifolds.
The ...
5
votes
0
answers
332
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Extensions of maps between graded modules
Let $R$ be a connected graded ring (like $R=\mathbb R[x_1,\dots,x_d]$ with the usual grading) and let $N \subset R^{\oplus n}$ be a graded submodule, i.e. $$N= \bigoplus_{i \in \mathbb N} (N \cap R^{\...
- 25.5k
1
vote
1
answer
296
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Computing the connected component without primary decomposition
Given an algebraically closed field $\mathbb{F}$ of characteristic $0$ and a closed subgroup $G$ of $GL_n(\mathbb{F})$. Let $\{g_1,\ldots, g_r\}$ be a Gr"obner basis for the correpsonding ideal $\...
- 53
4
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1
answer
291
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Can a branched cover of relative curves be suddenly ramified along a vertical divisor if you blow up the curve on the bottom?
If we have a $G$-Galois branched covering $Y \rightarrow \mathbb{P}^1_R$ of curves over a complete DVR, $R$ (assume $R$ is equi-characteristic $0$, and let $t$ be $R$'s parametrizing element). Assume ...
- 5,929
1
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0
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249
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On inverse images with respect to Zariski-etale topology.
For a variety $X$ I define its Zariski-etale site as follows: the category is the category of etale $X$-schemes, and the coverings are Zariski ones. Note that this topology is more coarse than the ...
- 16.9k
1
vote
0
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140
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on a decomposition lemma in adelic groups
Let X a curve over an algebraically closed k.
Fix $x$ and $y$ two distinct closed points of X.
Let G be a connected reductive group over k.
We denote Spec $\hat{\mathcal{O}}_{X,x}$ the formal ...
- 3,472
1
vote
1
answer
188
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Question related to the abelianization of simplectic groups
Let $H \subset \mathrm{Sp}(\mathbb{Z})$ be a subgroup of the simplectic group of (square of even dimension) matrices with integer entries, and let $H^{(\ell)}$ denote its pro-$\ell$ completion for ...
- 51
4
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571
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Étale cohomology of linear groups
This is in a sense a follow up question to the answer here Analytic tools in algebraic geometry
Let $k$ be an algebraically closed field of positive characteristic and let $R$ be the result of ...
- 23.5k
1
vote
1
answer
152
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cokernel of the symmetric product of an injection.
Clearly, my question can be asked more generally, but suppose for simplicity that $X$ is a smooth surface and that $0 \to E \to F \to K \to 0 $ is exact with $E ,F$ rank two bundles on $X$ and $K$ a ...
- 974
4
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0
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420
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Etale cohomology analogue for the semistable reduction theorem
Let $K$ be a field, $X/K$ a smooth projective variety, $l\neq char(K)$ a prime number and $q\ge 0$. Then we define $\overline{X}:=X_{K_{sep}}$ and denote by
$\rho_{X, l}^{(q)}$ the representation of $...
- 1,673
1
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152
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Image of linear projection
Let $X \subset \mathbb{P}^n$ be a projective variety (i.e. zariski closed), and let $\pi : X \dashrightarrow \mathbb{P}^m$ a linear projection ($\pi$ is not in every point of $X$ defined).
Under ...
- 175
4
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314
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Quasi-algebraic closure
I am interested in $C_1$ fields. First of all a field $F$ is called $C_1$ field if every non-constant homogeneous polynomial $P$ over $F$ has a non-trivial zero provided the number of its variables is ...
- 41
2
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195
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Henselization of a smooth affine scheme in closed smooth subscheme and henselization of normal bundle
Hi!
Let $Z\hookrightarrow X$ - is a closed embedding of smooth affine schemes. Let $N_{X/Z}$ be corresponding normal bundle.
I am curious if there is an isomorphism between henselizations:
$X^h_Z\...
- 21
3
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0
answers
147
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Exceptional sheaves and double dual
In his beautiful paper Moduli of bundles on $K_3$ surfaces, Mukai proves that if $F$ is a rigid (i.e. $\mathrm{Ext}^1(F,F) = 0$) torsion free sheaf on a surface $S$ with $|-K_S| \neq 0$, then $F^{**}$,...
- 7,310
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171
views
Weaker conditions for potential good reduction of Abelian varieties
We are concerned with slight weakening of a result of the Serre-Tate paper "Good reduction of abelian varieties" google. I think the title of the question conveys what we are in here for. So, I'll ...
- 1
1
vote
0
answers
332
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Singular conics on certain algebraic surfaces
Let S be an algebraic surface in 3-dimensional complex projective space. Suppose that:
The degree of S is either 5 or 6;
The generic plane section of S is a curve of genus 1.
(Equivalently, the ...
- 1,488
4
votes
1
answer
311
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Model category with formally smooth morphisms as fibrations?
Let's view the category of algebraic spaces as a full subcategory of the category of "spaces" over the opposite category of commutative rings, that is, the category of sheaves on $CRing^{op}$ in the ...
- 19.6k
2
votes
0
answers
128
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supersingular curve detector
Suppose I give you a prime $p$ and ask for a non-CM supersingular elliptic curve over $\mathbb{F}_p.$ Can this be done in polynomial time (so, polynomial in $\log p$)?
- 96.4k
1
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1
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261
views
How the complex conjugation on sheaves of modules is defined?
(Probably some basic question, but I've never worked in the real world.)
Let $X\subset\mathbb{P}^n_\mathbb{C}$ be a complex variety with the complex conjugation $\tau:X\to X$. So $\tau$ acts on $\...
- 2,232
4
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495
views
Spectral sequence for cohomology of open subset
Let $X$ be a smooth compact orientable manifold (or variety) and let $j: U \subset X$ be the complement to a union $\bigcup_{i \in I} X_i$ of smooth compact orientable manifolds. Suppose that $A$ is a ...
11
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answers
383
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What are the endomorphisms of Drinfeld's "special formal O_D-modules"?
Let $F$ be a nonarchimedean local field, and let $D/F$ be the central division algebra of invariant $1/d$. Let $k$ be the algebraic closure of the residue field of $F$ and let $\pi$ be a uniformizer.
...
- 4,487
7
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700
views
A variety always has a compactification. Is there an easy proof?
This is of course a special case of the more difficult question on compactifiable morphisms.
I would like to know if there is an easier proof of the following fact:
Every algebraic variety $X$ ...
- 23.4k
3
votes
1
answer
261
views
Over which schemes can there exist non-trivial G_a bundles?
The group scheme G_a here is the one-dimensional additive group.
- 8,866
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0
answers
85
views
$\mathcal{F}$-- twists of Lie algebras
I am trying to figure out with Drindfel's Opers. Let us consider Lie group $G$ and $G$ -- bundle $\mathcal{F}$ on the smooth algebraic curve $Y$. Can anybody help me and clarify definition of the $\...
- 181
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561
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How to get a Dehn-twist presentation of a periodic map of a Riemann surface?
Let $f:S\to S$ be a given periodic map of order $n$, and $(m_i,\lambda_i, \sigma_i)$ be the valency of the multiple point $p_i$ of $f$ ( $i=1,\cdots,r$ ).
A classical result says such $f$ is ...
- 471
2
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answers
179
views
Notation for a canonical quotient of an abelian variety in positive characteristic
This is a light question about notation, but I received no answer in Stackexchange.
Let $k$ be an algebraically closed field of characteristic $p>0$ and let $A=A_{/k}$ be an ordinary abelian ...
- 797
2
votes
0
answers
600
views
cohomological dimension of the push-forward functor
Let $\ell$ be a prime number and let $f:X \to Y$ be a morphism of schemes of finite type over the complex numbers (or a regular scheme of dimension at most 1, in which $\ell$ is invertible). How to ...
- 4,265
7
votes
0
answers
341
views
The ample cone and ranks of Frobenius on cohomology
Suppose that $X$ is a smooth projective algebraic variety over a perfect field $k$ of characteristic $p > 0$ (I'm also interested in the non-smooth case).
Suppose that $D$ is a divisor or possibly ...
- 20.5k
7
votes
0
answers
377
views
The scheme-theoretic flow-in locus
Let $R$ be a ring with an $\mathbb{N}\times\mathbb{Z}$-grading. The $\mathbb{N}$-grading allows you to construct the scheme $X = \operatorname{Proj} R$, and the $\mathbb{Z}$-grading defines an action ...
- 2,537
6
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0
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450
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Differential equation of line tangent to caustics
This problem (or rather, statement that I cannot understand) has arisen in a paper I have been reading "Geometry of Integrable Billiards and Pencils of Quadrics" by Dragovic and Radnovic. I'd be most ...
- 281
5
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232
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Coherence of the monoid algebra of a non-finitely generated monoid
Let $P$ be an integral, sharp, finitely generated commutative monoid (say even torsion-free and saturated if you like), and consider the "rational cone" $P_\mathbb{Q}\subseteq P^{gp}\otimes_\mathbb{Z}...
- 1,030
3
votes
0
answers
340
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Rank of Subgroup of Elliptic Curve
I'm currently looking at two rational points $ p, q $ on an elliptic curve $E$ over $ \mathbb{Q} $. SAGE tells me that $E$ has rank 5 and no torsion, and that $p$ and $q$ both have infinite order. ...
- 309
2
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answers
331
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Is there asymptotic expansion of heat kernel of complex laplacian?
On real Riemannian manifold , the heat kernel of the laplacian have an asymptotic expansion . But on complex manifold , i haven't seen a result like this , i.e. the heat kernel of the Kodaira ...
- 399
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0
answers
130
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morphisms in the construction of the moduli space of curves by mumford
Hi fellow mathematicians,
I'm just studying mumfords proof of the existence of a coarse modulispace for curves of genus $g$in his great GIT book. On page 102 (in the proof of proposition 5.2) he ...
- 9
2
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0
answers
140
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Contracting Fano divisors
Suppose we are given a smooth complex algebraic variety $X$ with a smooth, irreducible divisor $D$ such that $D$ is Fano and the normal bundle $L$ to $D$ is anti-ample. Then we can contract $D$ to a ...
- 21
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0
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223
views
Can a surface of the following type contain a line?
This is a follow-up question to my previous post: Sufficient conditions to tell whether a surface contains a line
Suppose that $f(x_1, x_2)$ is an irreducible binary form with integer coefficients of ...
- 26.9k
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votes
0
answers
144
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From abstract to concrete complex projective varieties
By definition, a(n abstract) complex projective variety $X$ has an expression $E$ (many actually) of the form $\bigcap_i V_i$, with each $V_i$ a hypersurface in a ${\Bbb P}^m$ ($m$ depending on $E$). ...
- 17.6k
3
votes
1
answer
424
views
Principal bundle for contractible group is weak homotopy equivalence for ind schemes
This is may be obvious, but I am not comfortable with ind-schemes.
I have an ind-scheme $X$ over $\mathbb{C}$. Every point has a neighborhood which can be written as an ascending union of regular ...
- 156k
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1
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302
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Chow group of a fiber product of grassmann bundles
Let $X$ be a smooth projective variety and $E\longrightarrow X$ a vector bundle of rank $n$. For any $0\leq k\leq n$ the associated Grassmann bundle $G_k(E)\longrightarrow X$ yields and we have the so-...
- 193
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0
answers
85
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different and discriminant for finite invariants
Let $k$ be an algebraically closed field.
Let $B$ a $k$-algebra of finite type, normal and Cohen-macaulay. Let $G$ a finite group acting on $B$. We assume that the order of $G$ is prime to the ...
- 3,472
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votes
0
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159
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Further Questions Regarding Cohomolgoy Theory Of Sheaves
In continuation to my previous post:
Question Regarding Riemann-Hurwitz Formula Proof
I'll be glad to receive some explanations regarding the following:
1) I know that when taking a sheaf $F$ , then ...
- 305
2
votes
2
answers
354
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formulate edge length problem as convex optimization problem
I want to us convex optimization to describe a problem in computational geometry.
Let $E = (E_1, E_2,\ldots , E_m) $ be a sequence of line segments in the plane, where $E_1$ and $E_m$ may be points ...
3
votes
1
answer
312
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Constructing affine hypersurfaces with one singularity
This is a followup to my previous poorly-worded question.
Consider a finite collection of points $S \subset \mathbb N^n$ lying in a hyperplane $H$. These points define exponents of a collection of ...
- 459
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votes
1
answer
294
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Relative Jacobian condition
This question is really elementary.
Where can I learn about the Jacobian condition not over a field?
I think I heard once that there's a Jacobian condition even for $f: X \rightarrow Y$. I'm not ...
- 5,929
4
votes
0
answers
255
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On (the cohomology of) Hensel pairs
I would like to study the cohomology of the Henselization $H_X(Z)$ of a closed subvariety $Z$ of a variety $X$.
I would like the following facts to be true (and to make sense!:)).
a.) The motivic ...
- 16.9k
0
votes
0
answers
152
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Kählerdifferentials and normal crossing divisors
Let $k$ be an algebraically closed field of arbitrary characteristic, $X$ a smooth surface over $k$, and $D_i \subset X$ be an regular, effective Divisor such that $D=\sum D_i$
has normal crossings ...
3
votes
1
answer
335
views
Decomposition of primes, where the residue field extensions are allowed to be inseparable
I've been dealing with the following situation:
Let $R\subseteq S$ be an extension of Dedekind rings, where $Quot(R)=:L \subseteq E:=Quot(S)$ is a $G$-Galois extension. Let $\mathfrak{p}$ be a prime ...
- 1,386
1
vote
1
answer
174
views
$Hilb_{lines}^{x}(X)$ and $Hilb_{lines}^{x}(X_{red})$
Let $X$ be a irreducible closed subscheme of $\mathbb{P}^N_{\mathbb{C}}$,
and $U$ is a nonempty open where $X$ is smooth and moreover for every $x\in U$ and for every line $l\subseteq X$ with $x\in l$ ...
- 1,159