All Questions
8,187 questions with no upvoted or accepted answers
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Reference: A nowhere vanishing section of a vector bundle is locally split
Well-known fact:
If $(A, \mathfrak{m})$ is a local Noetherian ring, $E$ is a finitely generated free $A$-module, and $e\in E$ is an element not contained in $\mathfrak{m}E$, then $E/eA$ is also a ...
- 7,338
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304
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Flat family of Hilbert scheme of points
Given $X$ a k3 surface and $X^{[r]}$ the Hilbert scheme of points on it, I know that the Hilbert-Chow morphism $\rho:X^{[r]}\rightarrow X^{(r)}$ to the symmetric product is birational and, said $D\...
- 13
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133
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complementary bundle for a divisor
For a divisor in a complex manifold, what is known about a complementary bundle to the divisor in the manifold (either for the tangent or the cotangent bundle). Is there a description in terms of ...
- 1,143
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230
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Toric morphism fiber and kernel dimensions
Given a morphism between two smooth toric varieties $f: X \rightarrow Y$, is the dimension of the kernel of $\mathrm{d}f$ at any point $p \in X$ equal to the dimension of the fiber at $f(p) \in Y$?
...
- 1,719
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145
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Zero Dimension Intersection
Let $M$ be smooth and purely $r$-dimensional, $E$ be a vector bundle of rank $r$ over $M$, $s$ be a regular section of $E$ and $Z$ the zero scheme of $s$. Then $[Z]$ is dual to the top Chern class $...
- 105
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536
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splitting of the Hodge filtration
Let $X$ be a smooth projective variety over a subfield $k$ of $\mathbb{C}$. Then one has a short exact sequence
$$
0 \to H^0(X, \Omega^1_X) \to H^1_{dR}(X/k) \to H^1(X, \mathcal{O}_X) \to 0
$$
One ...
- 9
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153
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torsors on quasi-split groups
Let $\mathbf{G}$ be a split connected reductive group scheme over a scheme $X$.
Let $X'\rightarrow X$ an étale Galois cover of group $\Gamma$.
We consider $G$ a quasi-split group scheme over $X$ ...
- 3,472
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127
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Geometric interpretation of table with permutations and inversions
Let $T(n,k)$ is the number of permutations of numbers $1, ..., n$ and each of the permutations has $k$ inversions. We can consider a table for $T(n,k)$ for some $n$ and $k$. For eg.
$n=1,...,6$, $k=1,....
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99
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Unions of orbits of dimension $\leq n$
Let $G$ be a complex linear algebraic group acting on a smooth complex projective variety $X$ with finitely many orbits. Note that each $G$-orbit is a smooth locally closed subvariety of $X$.
For a ...
- 4,920
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199
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Question about the "middle" intermediate Jacobian
Suppose $Y$ is a smooth projective variety of dimension $2p-1$ over $\mathbb{C}$. I have a few questions about the $p^{~\text{ th}}$ intermediate Jacobian $J^p(Y)$ of $Y$.
Does it come from (i.e. is ...
- 299
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268
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Birational contraction to a $\mathbb{Q}$-Gorenstein Variety
Given a birational contraction morphism $X\rightarrow Y$
of complex normal algebraic varieties.
If $Y$ is a smooth variety, what kind of singularities can appear
on $X$?
I would be grateful of any ...
- 1,595
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122
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Is it possible for P to have degree less than 2n?
Let $P\in\mathbb{R}[x, y]$ be a polynomial such that there are exactly $n$ pairs $(x, y)$ in $\mathbb{R}^2$ such that $P(x, y)=0$. Is it possible for $P$ to have degree less than $2n$?
- 545
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245
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A kind of Stein factorization for non-proper morphisms
Let $S$ and $T$ be a noetherian connected normal scheme, say. Let $K$ (resp. $L$) be the function field of $S$ (resp. $T$). Assume that $char(L)=0$. Let $f: S\to T$ be a smooth surjective morphism. Is ...
- 1,673
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138
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Profinite Local Ring inside Polynomial Ring
This is a "technical" question that I came across in my research.
Let $A = \textbf{Z}_{p}[\![t_1, \cdots, t_a ]\!]<z_1, \cdots, z_b>$ be the $(p, t_1, \cdots, t_a)$-adic completion of the ...
- 61
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101
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Degree and quasi projective family
Let $V$ be a quasi-projective variety in $\mathbb{P}^{n}\times\mathbb{P}^m$. If $p\in \mathbb{P}^m$, we define the degree of $V_p$ as the degree of its closure in $\mathbb{P}^n$.
Question : $\exists ...
- 1
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150
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Explicit calculation of module of derivations on noncommutative polynomial ring
Let $R$ be a commutative unital associative ring and set $R<x,y>$ to be the $R$-algebra of non-commuting polynomials in two variables over $R$.
Explicitly how would one go about computing ...
- 5,405
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203
views
Compactification of the affine space with a del Pezzo surface
Is there some nice compactification of the affine space with a del Pezzo surface of degree $\le 4$ (for example with a cubic surface) ?
More precisely, I would like a projective algebraic variety $X$ ...
- 7,722
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119
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Is the singularity of a secant variety in codimension 2 Kleinian?
Let $X\subset{\mathbb P}^d$ be a rational normal curve, $d\gg0$. Then the $k$-th secant variety $Sec^k(X)$ is smooth away from $Sec^{k-1}(X)$, and I wonder what is the singularity of $Sec^k(X)$ at a ...
- 267
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408
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Parabolic bundle and chern class
I'm a physicist and I'm trying to figure out some things about parabolic bundles.
In particular I'd like to understand which is the relationship, if present, between the parabolic degree and the first ...
- 113
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117
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Consistency of the u-invariant under field extension
A algebraic field extension L/k induces of homomorphism between the Wittrings. We get
$\phi: W(k) -> W(L)$. If every anisotropic isometry class of $W(k)$ stays anisotropic, the kernel of $\phi$ ...
- 193
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183
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Embed the normalization of a curve in a larger space
I'd like to believe that this problem has a positive answer, but I don't know a nice reference. Actually I've never worked with embedded curves, so I apologize in advance if the question is too silly.
...
- 165
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1k
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Finite separable extension of fields imply the number of intermediate subfield is finite
The proof of statements either uses Galois theory or Artin primitive element theorem.I would like to know whether there is a proof without using these.The reason to avoid using Galois theory is that ...
- 1,982
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117
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properties for DM stacks
Let $X$ be a smooth projective variety over $k$ and $G$ a finite group acting on $X$. Furthermore $\mathrm{char}(k)$ does not divide the order of $G$. Consider the quotient stack $[X/G]$. Is it right ...
- 165
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161
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birational invariants of projective surfaces
I am studying Castelnuovo's rationality criterion for surfaces.
Let $S$ be a projective surface and $K$ a canonical divisor on $S$.
Let's use the notation $h^i(S,\mathcal{O}_S)=dimH^i(S,\mathcal{O}_S)...
- 109
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100
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divisor class group with modulus
Let $C$ be a smooth projective curve over a field $k$ and $S \subset C$ a finite number of points. A modulus is simply a divisor supported on $S$. What is the divisor class group with modulus?
I ...
- 1
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320
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Invariants of the Determinant Form
Consider a form of degree $r$ in $n$, that is, a homogeneous polynomial
$$f(x_1, \ldots, x_n)=\sum_{i_1+\ldots i_n=r}\alpha_{i_1 ... i_n}x_1^{i_1} ... x_n^{i_n}
$$
After the linear change of ...
- 801
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247
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Hodge structure of abelian surfaces
In my case, I have an abelian surface $A$ of (2,8)-polarization, and I have some finite group (may not be abelian group) $G$ acting on $A$ without fixed point. I want to understand when there is a ...
- 3,472
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130
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Tilting for quadric
Kapranov showed that for a quadric $Q$ in $\mathbb{P}^n$ there is a tilting bundle. For example if we consider a vector space $V$ of dimension $n=4$ and $\mathbb{P}(V)$ then the tilting bundle is $\...
- 741
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194
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Schur functor for sheaves
When one is given a partition $\lambda=(\lambda_1,...,\lambda_r)$ and a locally free sheaf $\mathcal{E}$ on for example a Grassmannian variety one can apply the Schur-functor $\Sigma^{\lambda}(\...
- 741
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128
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Fitting ideals and a Grassmannian construction
Let $L$ be a locally free and finitely presented sheaf over a Noetherian scheme $X$ and
$$ E\overset{\varphi}\to F \to L \to 0$$
a free presentation of $L$, where $E$ and $F$ have finite ranks $n$ and ...
- 251
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549
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Fitting ideal sheaves and determinant bundles
I am working on a problem in algebraic geometry which comes down to a fact in commutative algebra that I am hoping is well-known.
Suppose $F$ is a coherent sheaf on a smooth variety $S$, and that the ...
- 5,857
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382
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Hypersurfaces with Picard group generated by classes of lines on the same plane
For which values of $d$ is the following possible: There exist a smooth hypersurface $X$ in $\mathbb{P}^3$ of degree $d$ with Picard number $d$, containing $d-1$ lines $l_1,...,l_{d-1}$ on the same ...
- 1,070
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0
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115
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invariance of the dimension of severi varieties of surfaces
Suppose I have a smooth projective surface $S\subset P^n$ embedded by a very ample linear system $|L|$. Consider now the generalized Severi varieties that parametrize curves on $S$ belonging to $|L|$, ...
- 3,779
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394
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Birational map of non-singular projective curves
Is it true that a birational map of non-singular projective curves is an isomorphism?
- 713
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0
answers
200
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canonical bundle of the relative spectrum
maybe it is a very trivial quetion but:
suppose we have a smooth projective variety $X$ over $k$ and $\mathcal{A}$ an $\mathcal{O}_X$ algebra. We have the relative spectrum $Spec(Sym(\mathcal{A}))\...
- 1
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0
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102
views
Is equivariant homology class preserved in the limit?
Suppose $C \to (S,0)$ is a family of curves (with at most nodal singularities) parametrized by a pointed curve $(S,0)$, that is proper and flat. Let $u:C \to X/G$ be a family of maps to a quotient ...
- 778
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168
views
semisimple conjugacy classes over general bases
Let $k$ be an algebraically closed field, $G$ a connected reductive group, $T$ a maximal torus, $W$ the Weyl group and $\chi:G\rightarrow T/W$ the Steinberg morphism.
We know that if $\gamma,\gamma'\...
- 3,472
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180
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A quadratic form pair
Let $Q_s(x)\in\Bbb Z[x_1,x_2,\dots,x_s],\hat{Q}_{\hat s}(y)\in \Bbb Z[y_1,y_2,\dots,y_\hat s]$ be pair of homogeneous purely non-diagonal (every term of form $x_ix_j$ or $y_iy_j$) quadratic forms and ...
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97
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Embeddings of curves and an intrinsic description of the normal bundle
Assume we have a smooth algebraic curve $C$ over complex numbers. Consider then all possible embeddings of $C$ into a smooth algebraic surface $X$.
Here are two questions:
1) How many are there ...
- 127
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255
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Image of critical points
Let $K$ be a field of characteristic $0$, $f:K^n\rightarrow K^n$ be an algebraic function, that is, $n$ polynomial functions in $n$ variables. Let $S$ be the set of critical points of $f$. If $K=\...
- 3,741
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324
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Quotienting a scheme by finite group
Can one take quotient by finite group in the category of schemes? Will the singularities be visible? For example, it looks like $\mathbb{C}/\{z \sim -z\}$ is isomorphic to $\mathbb{C}$.
What about ...
- 778
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218
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is this a simplicial model category?
A basic result in the theory of model categories is that simplicial sets form a simplicial model category. The same is true for simplicial $k$-algebras. I have two questions related to this. One ...
- 761
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67
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open subset in constructible set of divisors
Let a smooth projective curve $X$ over $\mathbb{C}$.
Let a pair $(x, D)$ a pair xith a closed point $x$ and $D$ an effective divisor on $X$, such that $d_{x}:=m_{x}(D)\neq 0$.
Let $N=\deg (D)$ and $X^...
- 3,472
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71
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Compatibility of maps on points under base change
Let $S$ be an arbitrary scheme, and let $X,S'$ be $S$-schemes.
Using e.g. EGA I (3.3.9), (3.3.14), one obtains for any $S'$-scheme $T$, viewed as an $S$-scheme via $S'\to S$, a canonical bijection ...
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0
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85
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$\mathcal{F}$-- 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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113
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Question on a paper of Montesano
Does anyone know what are the results in the paper
"D. Montesano, Rendiconti della Reale Accademia di Napoli, Ser. 3, Vol. 13 (1907), Su novi tipi di superficie razionali di 5 ordine"? I have one ...
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161
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Is there a reference book for the duality between the genus of function fields and the discriminant of number fields?
Bjorn Poonen mentions in his "Lectures on rational points on curves" the analogy between the genus of a function field and the discriminant of number fields. I'm looking for a reference book for this ...
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217
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on the fibers over closed points
Let $X$ and $S$ $k$-schemes of finite type . ($k$ a field) and $U$ an open subset of $X$
Let $f:X\rightarrow S$ a $k$-morphism of finite type.
We assume that for any closed point $s\in S(\bar{k})$, $...
- 3,472
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0
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124
views
field of definition of abelian varieties with extra endormorphism
Let $A$ be a complex abelian variety such that $\mathrm{End}(A)$ is strictly bigger than $\mathbb{Z}$.
Question: Is is true that $A$ is defined over $\overline{\mathbb{Q}}$?
This is of course what ...
- 1
0
votes
0
answers
307
views
local systems, duals, cohomology
Let $U=\mathbb{P}^1-\{p_1, \ldots, p_n\}$ be a Zariski open subset of the projective line. Consider a rank $r$ local system of complex vector spaces $V$ on $U$ and assume that the monodromy ...
- 11