All Questions
22,547 questions
1
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Q-Divisor and Determinant Map on a Maximal Order
Given a smooth projective surface $X$, let $A$ be a sheaf of maximal orders in a division ring.
Let us for simplicity assume $A$ ramifies in one curve $C$ with ramification index $e$. Let $A^*$ be the ...
- 1,391
3
votes
0
answers
166
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On subadditivity of multiplier ideals
If $\varphi$ and $\psi$ are two plurisubharmonic weights on an algebraic manifold $X$, then we have $\mathcal{J}(\varphi+\psi)\subseteq \mathcal{J}(\varphi)\mathcal{J}(\psi)$.
My question is, if we ...
- 320
1
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0
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157
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On closed abelian reductive subgroups of Real reductive groups
Hello everybody. I would first like to apologise for the basic question; I'm not expert on Lie Theory. Can someone please help me with the following questions
Let $\mathrm{G}=\mathrm{K} \exp(\...
- 11
0
votes
1
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238
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Geometric explanation of an orbit space: Integer action on the affine line
Let $k$ be a field of char $0$ and let $\mathbb{Z}$ act on $\mathbb{A}^1_k$ by the action induced by $G\to\mathrm{Aut}_k(k[X]), n\mapsto X+n$. It is rather easy to show that the orbit space $\mathbb{A}...
- 1,273
2
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0
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60
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lift isomorphic in a sufficiently thick fiber
Let $P$ and $P'$ two polynomials in $k[[\pi]][t]$ for an algebraically closed field $k$ and let $A=k[[\pi]]$.
We consider $ X'=Spec (A[t]/(P'))$ and $X=Spec (A[t]/(P)$.
Let $d=val(\Delta(P))$ where $\...
- 3,472
2
votes
1
answer
349
views
When does the global sections of a prescheme X over an other S equals those of S?n
Let $f=(\varphi,\theta):X\longrightarrow S$ a morphism of preschemes whith $\varphi$ surjective. Let $\theta(S):\Gamma(S,O_S)\longrightarrow \Gamma(S,f_* O_X)=\Gamma(\varphi^{-1}(S),O_X)=\Gamma(X,O_X)$...
- 411
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0
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170
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Galois cohomology of generic points of formal completions (of components of a hypercovering of the subvariety): examples or general statements?
Let $Y$ be a closed smooth subvariety in a (smooth) affine variety $X$. What can one say about the etale cohomology of the generic points of the formal completion of $X$ along $Y$ i.e. about the ...
- 16.9k
0
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0
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79
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A question on counting non-leading monomials
To state the question I have in mind, it is necessary to define a few things. Let < be a graded monomial ordering on the set of monomials $\textbf{x}^\alpha = x_0^{\alpha_0} \cdots x_n^{\alpha_n}$, ...
- 26.9k
1
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0
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182
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Does obstruction class for deformations of a pair $(X,D)$ lie in ${\rm Ext}^2(\Omega^1(\log D), \mathcal{O}_X)$ when $X$ is singular?
This question is related to this one and this one.
Let $X$ be a normal variety over algebraically closed field $k$ and $D$ be an effective Cartier divisor. Let $\zeta := (\mathcal{D} \subset \...
- 909
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0
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97
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Does a (NOT necessarily positive) current have a decomposition formula?
It is well-known that for any positive (1,1)-current $T$, there is a decomposition formula according to [Siu74]. That is, $T$ can be written as an infinite sum of prime divisors plus an extra part. In ...
- 393
3
votes
1
answer
240
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sheafifying a projective limit of presheaves
Let $F=(F\_n)\_n$ be an $\ell$-adic sheaf on $X\_{et}$, for a variety $X$ over an algebraically closed field $k$ of characteristic not equal to $\ell$. Does the presheaf sending $U$ to $H^i(U,F):=\lim\...
- 4,265
1
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1
answer
338
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Power series for meromorphic differentials on compact Riemann surfaces
Suppose I have a compact Riemann surface of $g>1$ given by the quotient $H/\Gamma$ where I do know $\Gamma$ explicit. Is there a way to write down the power series of meromorphic functions, ...
- 6,825
4
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0
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215
views
vanishing of automorphic bundles
Let $S _K = S_K(G,X)$ be a Shimura variety of dimension $n$. Let $\xi$ be a (finite-dimensional) representation of $G$, which gives rise (by a construction of Harris) to an automorphic bundle $V(\xi)$ ...
2
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0
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321
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Dimension of fibres of moment maps in characteristic $p$
Suppose $G$ is a connected semisimple linear algebraic group with Lie algebra $\mathfrak{g}$ and $X$ is a homogeneous $G$-space with isotropy subgroup $H$ (associated Lie algebra $\mathfrak{h}$) that ...
- 3,545
3
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0
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201
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Refinement mapping
Good evening,
When I read Cech cohomology in Riemann surfaces written by Otto Forster, I see the following proposition : Let $X$ be a topological space and $\mathfrak{F}$ a sheaf of abelian groups on ...
- 758
7
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0
answers
401
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Finite subgroups of the symplectic group that don't act on simple principally polarized abelian varieties
I'm sorry if this question is awkwardly phrased -- I'm very much an amateur at algebraic geometry, but this question came up in my research.
Here goes. Let $\mathbb{H}^g$ be the genus $g$ Siegel ...
- 44.8k
2
votes
1
answer
262
views
Criterion for open morphisms without constructible sets?
The following theorem is proved in EGA IV 2.4.6:
Every morphism of schemes, which is flat and locally of finite presentation, is open.
I've already seen some applications of this theorem, so I want ...
- 63.1k
2
votes
0
answers
311
views
volume under induced metric of preimage set of a regular value under a polynomial map
I am interested in the following kind of polynomial map:
$$ f: \mathbb{T}^k \to \mathbb{R}^n$$
where $f$ is a polynomial map of a certain maximum degree $d$, in the sense that if we imbed $\mathbb{T}^...
- 4,466
2
votes
0
answers
298
views
Induced groupoid schemes
This is a more direct version of this question, which was perhaps a bit obtuse. This is a more elementary formulation.
Recall that for a groupoid scheme (or indeed any internal groupoid) $X = (X_1 \...
- 35.5k
3
votes
0
answers
128
views
Finite weight spaces for coherent sheaf cohomology
Given a smooth quasiprojective variety $X$ with a coherent sheaf $E$, if $X$ is not projective, then the sheaf cohomology of $E$ may not be finite-dimensional. However, if we also have the action of ...
- 61
2
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0
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526
views
How much of math could be taught without using mathematical notation? [closed]
Given that mathematics is not about number, and that it is not even about the cryptic notation used to describe mathematical problems, how much of mathematics could be taught without reference to ...
2
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0
answers
265
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On base locus of canoncal linear system on surfaces
Let $S$ be a minimal surface of general type over $\mathbb{C}$ with $p_g=h^0(K_S)>1$. As a convention, we can write $|K_S|=|M|+F$ such that $F$ is the fixed part. We know that $K_SM \le K_S^2$.
...
- 575
1
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0
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163
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Can one recover an A-algebra from its cotangent complex?
Given an A-algebra B, one can define the cotangent complex $L_{B/A}$ as $\Omega^1_{P/A}\otimes_PB$, where $P$ can be taken as the canonical resolution of $B$ associated to the pair of adjoint functor (...
- 5,052
6
votes
0
answers
261
views
Moduli of convergent isocrystals
Let $X$ be a smooth projective curve over the finite field $\mathbb{F}$. Fix $n\in \mathbb{N}$. Presumably we can naturally define the moduli functor of rank $n$, convergent isocrystals (for some ...
- 491
4
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0
answers
168
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A fast way to test whether a partial function can be extended to a chirotope of rank 3 ?
Hello,
What is a fast way to test whether a partial function can be extended to a chirotope of rank 3 ? That is: I have a domain
$E = \{1,...,n\}$
and a partial function
$f: E^3 \to \{-1, 0, 1\}$
...
- 191
3
votes
0
answers
294
views
Monomorphisms in geometry
What is known about monomorphisms in the following categories:
Schemes
Complex manifolds
$C^\infty$--manifolds
and any other kinds of geometric objects that you might think of.
How do we choose a ...
- 123
4
votes
0
answers
156
views
Characterizing non-singularity of varieties through properties of their derivations
I am interested in knowing about the possible implications between the following properties of a commutative, complex algebra:
Its spectrum is non-singular.
Its derivation module is projective and ...
- 393
3
votes
0
answers
301
views
Is $\mathcal{O}(1)$ canonical only up to something ?
Let $S$ be a scheme, $\mathcal{E}$ a locally free $\mathcal{O}_S$-module of finite rank $r$ and $\pi:P=Proj(\mathcal{E})\to S$ the projective bundle. There is a canonical surjective map of sheaves $\...
- 4,492
2
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0
answers
214
views
structure of $T_\ell A$ for $A/\mathbf{F}_q$ an abelian variety
Can someone give me references for the structure of the $G_{\mathbf{F}_q}$-module $T_\ell A$, $A/\mathbf{F}_q$ an abelian variety?
- 417
2
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0
answers
168
views
selfintersection of curves inside
What are all the possibilities of the self intersection number of a smooth curve inside an Enriques surface?
- 645
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votes
0
answers
221
views
Representable morphisms of Artin stacks.
Given a representable morphisms $f:\mathcal{X}\to\mathcal{Y}$ of Artin stacks, do the fibers of $f$ have always nonnegative dimension? If not, can you give me some examples of what can happen?
- 773
3
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1
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155
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Epimorphisms between line objects
Let $\mathcal{A}$ be a $k$-linear abelian symmetric tensor category with unit $\mathcal{O}_A$; here $k$ is a comm. ring. By that I assume implicitly that $\otimes$ is finitely cocontinuous in each ...
- 63.1k
0
votes
0
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297
views
higher direct image: proof of the proper case.
Hi.
Let $f:X→S$ be a proper, open, surjective morphism of complex reduced spaces with constant fiber dimension n (or universally open morphism with n-fibers between locally noetherian excellent ...
- 435
2
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0
answers
261
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On a characterization of the symbolic square of prime ideals in polynomial rings
If $R=k[x_1,...,x_n]$ is a polynomial ring in $n$ indeterminates over a field $k$ of characteristic $0$, there is a characterization of the symbolic square of a prime ideal $P$ (the $n$-th symbolic ...
- 805
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0
answers
333
views
Is there a reference showing that the space $\bar{M_{g,n}}$ is a closed oriented orbifold and it is hausdorff
Is there a reference showing that the space $\bar{M_{g,n}}$ is a closed oriented orbifold and it is Hausdorff? Note: here $\bar{M_{g,n}}$ is not the Deligne-Mumford space in the usual algebraic ...
- 1,499
0
votes
1
answer
200
views
Fiberproduct of normal varieties over curves
Suppose $X$ is smooth variety over an algebraically closed field and $Y \rightarrow X$ some finite generically Galois normal cover. If $C \rightarrow X$ is some smooth curve (by which I mean $C$ is a ...
4
votes
0
answers
427
views
Singularities of Hilbert scheme of points on a surface
A theorem of Fogarty states that if $S$ is a smooth algebraic surface, the Hilbert scheme $S^{[n]}$ of length $n$ subschemes of $S$ is smooth for every $n$.
Does anybody know a description of the ...
- 14.7k
0
votes
0
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69
views
Embdding dimension
Take an open neighborhood $U$ of $p\in X$ which is a closed complex subvariety of a domain $D\subset \mathbb C^m$ with coordinates $z_1,\dots,z_m$.
Let $f_1,\dots,f_k$ be functions on $D$ such that $...
4
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0
answers
129
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Moderate growth functions on complex algebraic varieties.
Does anybody know a reference that discusses moderate growth (continuous or measurable) functions on complex algebraic varieties?
I'm interested in such a discussion in the generality of varieties ...
- 2,649
6
votes
0
answers
303
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Cohomology of Zariski neighborhoods
Do there exist smooth compact (=complete) connected complex algebraic varieties $X\subset Y$ and a Zariski neighborhood $U$ of $X$ in $Y$ such that the image of $H^{\ast}(U,\mathbf{Z})$ in $H^{\ast}(X,...
- 23.5k
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0
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140
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Obstruction theories on non-smooth spaces with smooth fibres
Given a perfect obstruction theory $E^\bullet$ over a space $X$, we know that if $X$ is smooth, that the virtual fundamental class $[X, E^\bullet]$ is given by
$$[X, E^\bullet] = c_{top}\big((E^{-1})^...
- 6,290
2
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0
answers
86
views
Universal family of Hibert shemes of Points
My question is how to explicitly compute the Chern class of the universal sheaf of the moduli space of ideal sheaves of two and three points on a given smooth projective variety X (no need to be ...
- 399
5
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0
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190
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"Unknot" algebraic set defined by two mutually dependent set of variables
Let $n$ be an integer $\geq 4$, and let $V \subseteq {\mathbb C}^{2n-1}$ be the set of all
$(a_1,a_2, \ldots ,a_n,b_1,b_2, \ldots ,b_{n-1}) \in {\mathbb C}^{2n-1}$ such that the derivative of the ...
- 3,595
1
vote
0
answers
276
views
Generalizations of divided-power algebras over finite fields
In Andrews, Askey, and Roy's Special Functions, the authors state that Gauß sums are finite field analogs of the $\Gamma$-function as Jacobi sums are to B-function. The $\Gamma$-function is well-...
- 1,049
4
votes
0
answers
262
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semisimple orbifold quantum cohomology
Dubrovin conjecture says, roughly speaking, that the quantum cohomology of a variety $X$ is semisimple if and only if it is a good Fano [good means that there exits a full exceptional collection in ...
- 41
5
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0
answers
269
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Unicity of branched covering of sphere, and Hurwitz numbers
Hurwitz's encoding counts the number of branched self-coverings of a sphere, with prescribed ramification degrees at the critical points, as numbers of factorizations of the identity in a symmetric ...
- 2,519
2
votes
0
answers
450
views
Rosenlicht differentials for possibly non-reduced curves
Let $X$ be proper Cohen-Macaulay scheme of pure dimension 1 over an algebraically closed field $k$. When $X$ is moreover reduced, Rosenlicht's theory of regular differential forms gives a beautiful ...
- 1,609
3
votes
0
answers
356
views
Colimit of an etale diagram of schemes
It is known that the category of schemes is not cocomplete (e.g. see this question: Colimits of schemes). However, do diagrams of schemes for which every morphism is etale have colimits? More ...
- 15.5k
5
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0
answers
86
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Category of the smooth formal p-groups over a local ring
Fontaine showed in Asterisque 47-48 that the category of finite dimensional smooth formal $p$-groups over the ring $A=W(k)$ of the Witt vectors over a finite field $k$ is anti-equivalent to the ...
- 51
2
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0
answers
151
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Intersections of components of 'simple' ('local") Zariski coverings
I would like to study the ordered Cech cohomology with respect to a Zariski covering of a variety. I can pass to the limit with respect to refinements; the components of the 'limit covering' will be ...
- 16.9k