All Questions
8,187 questions with no upvoted or accepted answers
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131
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Hodge numbers and weight filtration
Let $X$ be a complex smooth projective variety and $D$ a divisor on $X$ with normal crossings. As usual, denote by $D(m)$ the disjoint union of all possible intersections of $m$ irreducible components ...
- 1
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142
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Are fixed points of automorphisms rational?
Fix an integer $d$ and a number field $K$ containing the $d$-th roots of unity. Let $X$ be a variety over $K$ such that $Aut(X / K)$ has an element $\varphi$ of order $d$. Is it true that the fixed ...
- 9
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238
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How to show integrally closed implies topologically unibranch
On p.52 of Mumford's book Algebraic Geometry: Complex projective varieties, he states that
$$\mathcal{O}_{x.X} \text{is integrally closed} \ \Rightarrow X \ \text{is topologically unibranch at } \ ...
- 953
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89
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Degree of permutation of hypercube
Given $S_0\cup S_1=T_0\cup T_1=\{0,1\}^n$, $S_0\cap S_1=T_0\cap T_1=\emptyset$, with $|S_i|=|T_i|$ for both $i\in\{0,1\}$, what is degree of transformation that simultaneously maps $S_i$ to $T_i$ for ...
- 13.9k
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183
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If the quotient of an algebraic space $X$ by a finite group is a scheme, is $X$ a scheme?
If the quotient of an algebraic space $X$ by a finite group $G$ is a scheme, is $X$ already a scheme? Here $G$ is just a finite group, but I'd like to know the answer when $X$ is defined over Spec(Z)....
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717
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Complete Intersection
Let $I$ be an ideal of the polynomial ring $P=K[x_{1},...,x_{n}]$ that is generated by degree two polynomials ${f_1,...,f_k}$.
The zero set $\mathcal{Z}(I)$ is isomorphic to an affine space of
...
- 1
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243
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Determine existence of irreducible variety in given homology class
Given a homology class $\alpha \in H_k(X,\mathbb{Z})$ on a variety $X$, is there a way to determine if there exists an irreducible subvariety $Y \subset X$ that has that class, i.e. $[Y] = \alpha$?
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125
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Is there an explicit way to glue a stable map in projective space by writing down the family of maps explicitly in terms of polynomials?
Let $v_1:\mathbb{P}^1 \longrightarrow \mathbb{P}^2$
and $v_2:\mathbb{P}^1 \longrightarrow \mathbb{P}^2$ be two holomorphic maps
of degree $d_1$ and $d_2$ respectively. Suppose they agree at some ...
- 3,245
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389
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Locally free sheaves and flat families of projective scheme
Let $f:X \to Y$ be a flat proper morphism of noetherian projective schemes and $\mathcal{F}$ is a coherent sheaf on $X$. Suppose for all $y \in Y$, $\mathcal{F} \otimes_{\mathcal{O}_Y} \mathcal{O}_y$ ...
- 1,593
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95
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Rationality of intersection of algebraic groups
Suppose that $G$ (defined over $\mathbb{Q}$) and $H$ (defined over $\mathbb{R}$) are two algebraic subgroups of a larger algebraic group defined over $\mathbb{Q}$. Assume that $G(\mathbb{R})$ and $H(\...
user70017
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197
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Cohen-Macaulay fibers
Let $Y$ be a set of points in $\mathbb{P}^n$. Then we can write a resolution
$$0\rightarrow P_n \rightarrow \cdots \rightarrow P_0\rightarrow \mathcal{O}_Y$$
where each $P_i=\bigoplus_j\mathcal{O}_{\...
- 39
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166
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For which pairs of distinct positive primes $p$ and $q$, the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[\sqrt{pq}]$ is a UFD?
For which pairs of distinct positive primes $p$ and $q$, the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[\sqrt{pq}]$ is a UFD? I've proved that neither $p$ nor $q$ can be congruent to $1$ modulo $...
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227
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What is the meaning of "field of constants"?
Let $X$ be a smooth proper variety over a field $k$, and let $D \subset X$ be a smooth irreducible divisor.
What is the meaning of "the field of constants of $D$"?
Here is my guess: $D$ is proper, ...
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161
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Rational group scheme
Suppose $G$ is a group scheme over a field $k$, i.e., $G$ is a functor from the category $\text{Alg}_k$ of unital commutative, associative $k$-algebras to the category of $\text{Groups}$. Suppose that ...
user69002
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79
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Terminology regarding divisor on a curve
Suppose that $D = \sum n_i P_i$ is a divisor on a curve $C$, say, over a field. Is there a standard algebraic geometry terminology referring to the set $\{ P_i : n_i \neq 0 \} \subset |C|$? Support of ...
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238
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Pro-constructible subset of scheme intersects very dense subsets?
Let $X$ be a scheme, let $D$ be a very dense subset of $X$ and let $Y$ be a pro-constructible subset of $X$. Is it true that $Y \cap D \neq \emptyset$?
If $Y$ is just constructible, this is true.
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119
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A simple question about a resolution of a conifers singularity
Let $X$ be a conifold defined by the equation $xy-zw=0$ in $\mathbb{C}^4$ and $\tilde{X}$ its crepant resolution, which is isomorphic to $\mathcal{O}_{\mathbb{P^1}}(-1)^{\oplus 2}$. Then there is a ...
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276
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Cohomology of pushforward under the double cover
Given a double cover $\pi: C \to \mathbb P^1$, where $C$ is a genus $g$ curve over algebraically closed field, I want to compute the group $\mathrm H^1(\mathbb P^1, \pi_*\mathbb G_m)$ in flat topology....
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123
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A quantity associated with an algebraic variete
Let $P:\mathbb{C}^{n}\to \mathbb{C}$ be an irreducible homogenous polynomial.
Is there a geometric or algebra geometric interpretation for the following quantity:
The maximum number $k$ such that ...
- 366
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196
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Hermitian metric on conic Kaehler-Einstein setting
I have a technical question :
Consider the triple $(M,D,\omega)$ where $M$ is a Fano manifold, $D$ is a smooth divisor whose Poincare dual is $\lambda c_1(M)$ and $\omega$ is a conic Kaehler -...
user21574
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619
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Homogeneous polynomials of degree 3 in two variables
Let $V$ be the four-dimensional real vector space consisting of all homogeneous polynomials of degree 3 in two variables with coefficients in $\mathbb{R}$.
Let $U$ be the set of all elements of $V$ ...
- 2,125
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261
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Ring of integers in Artin-Schreier extension
Question put in mathstackexchange but received no answer.
It is well-known( see Goldschmidt book: Algebraic Functions and Projective Curves)
that for $q$ a power of $2$ a quadratic separable ...
- 3,998
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187
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projective map from $\overline{\mathcal{M}}_{0,n}$
Suppose I have a morphism $f:\overline{\mathcal{M}}_{0,n} \to \mathbb{P}^N$ birational onto its image, and I know exactly what $F$-curves are contracted (or "dually", what divisors are contracted). ...
- 3,779
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1k
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Twisting sheaf of Serre
I'm sorry if my question is rather trivial, but I can't figure it out.. Given $A$ a ring and $P=Proj(A[X_0,\cdots,X_n])$, I know that $\oplus_n H^0(P,\mathcal{O}(n))=A[X_0,\cdots,X_n]$. This equality ...
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221
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Why is the set of geometrically irreducible curves of degree d in P^2 open in P^((d+2 2)-1)?
Let $X$ be a $k$-scheme. We say that $X$ is geometrically irreducible if $X\times_k \mathrm{spec}{K}$ is irreducible for all algebraically closed extensions $K$ of $k$.
Ravi Vakil states in his ...
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109
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pushing out families of curves
Let $f:X\rightarrow Y$ be a morphism of schemes with smooth curves as fibers. Let $g:X\rightarrow Z$ be a family of smooth or nodal curves with $Z$ a regular scheme. Does the push-out $Z\coprod_X Y$ ...
- 1
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234
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Triangulated category of singularities of quotient
Let $X$ be a variety, the triangulated category of singularities $D_{sg}^b(X)$ is obtained by taking the quotient of $D^b(X)$ by the category of perfect complexes. Suppose there is a group $G$ acting ...
- 3,472
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74
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Upper bound on number of cells created by varieties of co-dimension 1
Say I have polynomials $p_1,p_2,\dots,p_m$ in $\mathbb{R}^n$ (ie. over $n$ variables), each of degree $d$. Is there an upper bound on the number of "regions" created by the surfaces $p_i = 0$? Let's ...
- 143
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115
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Dimensions of two spaces
Let $R=\Bbb K[x_1,\dots,x_n]/I$ and $Z=\mathsf Z(I)$ where $\Bbb K$ is any field with $char(\Bbb K)\neq 2$.
Is there a way to describe space of $f_1,f_2\in R$ that satisfies $$f_1+f_2\neq 0,\mbox{ }\...
- 13.9k
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288
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Homology class of variety defined by an ideal
if a subvariety of codimension n is given by an ideal of polynomials with n generators, then the homology class of the variety is given by the intersection product of the classes of the individual ...
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160
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Closed Invariant Forms on Complex Projective $k$-Space
Considering complex projective $k$-space as the homogeneous space $SU_k/U_{k-1}$, is it true that every $SU_k$-invariant form is closed?
- 167
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210
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Rational multiple of a line bundle
In the paper http://arxiv.org/pdf/1207.5011.pdf of Chi Li and Song Sun, they say that "$D$ is a smooth divisor which is $\mathbb{Q}$-linearly equivalent to $−\lambda K_X$ for some $λ \in \mathbb{Q}$", ...
- 303
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226
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Regular point of a map in algebraic geometry
What is the correct definition of a regular point of a map in algebraic geometry?
More specifically, let $f:X\to Y$ be a map of varieties with $f(p) = q$, and let $Z=f^{-1}(q)$. Let $\hat{X}$ be the ...
- 2,537
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183
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When can one find holomorphic sections vanishing at a point to a certain order?
Let $X$ be a compact complex manifold (say of dimension $2$) and $L \rightarrow X $ a holomorphic line bundle. Consider the following statements:
Statement $A_0$: Given any point $p\in X$, there ...
- 3,245
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209
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Resolution of singularities of projective varieties
Let $X\subset\mathbb{P}^n$ be an irreducible variety, and let $Sing(X)$ be its singular locus. Let $Y$ be the blow-up of $\mathbb{P}^n$ along $Sing(X)$. Assume that we know that the strict transform ...
user58018
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101
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on lifting elements in a tangent space
Let X a normal integral scheme over a base field scheme, assumedd to be singular and an integer $n$
Let $\mathcal{O}=k[[t]]$, we consider the arc space $X(\mathcal{O})$ which is a $k$- pro-scheme and $...
- 3,472
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133
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Reciprocity laws in different dimensions
Let $M/L/Qp$ be a finite galois abelian extension of local fields and define
$\mathcal{M}=M\{\{T\}\}=\{\sum_{i\in \mathbb{Z}}a_iT^i:a_i\in M,\min_{i\in \mathbb{Z}}, v(a_i)>−\infty , \lim_{i\to −\...
- 33
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288
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What can one say about zero-cycle groups for products of Chow motives
What can one say about the Chow group of zero-cycles (up to rational equivalence) for a product of smooth projective varieties and Chow motives (so, I am interested in the kernel $Chow_0(P)\otimes ...
- 16.9k
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109
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Vanishing of the module of differentials of a extension of perfect fields
Let $L|F$ be a extension of perfect fields of characteristic $p$, $\phi_F:F \to F_{\phi}$, $\phi_L:L \to L_{\phi}$ the Frobenius isomorphisms ($F_{\phi}=F$ but considered as $F$-algebra via $\phi_F$). ...
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335
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Descent datum for a line bundle
Let $\pi:C \to \mathbb P^1$ be a double cover branched at $r$ points. To understand the theory of descent better, I would like, if possible, to construct by hands the descent datum of a line bundle ...
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444
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"Spreading out" locally free sheaves
Let $Y$ be a regular surface flat, projective over $R$, where $R$ is complete DVR. Let $X$ be the generic fiber and $F$ be a locally free sheaf on $X$. We know that if the rank of $F$ is $1$ then we ...
- 2,032
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161
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birational equivalence of linear sections of algebraic varieties
Let $X$ be an irreducible algebraic variety and suppose that $L$ is a linear space defined by the linear forms $l_1,l_2,\ldots,l_k$. I want to study $L\cap X$. I would like to know whether the ...
- 325
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268
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Holomorphic vector bundles over $\mathbb{CP}^1$ and elliptic curves
Can anyone suggest a good exposition of the classification of holomorphic vector bundles over $\mathbb{CP}^1$? Also does there exist any analytic or more elementary proof of Atiyah's classification of ...
- 2,106
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274
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path integral and index theorem
I actually have an integral which is used to prove Atiyah-Singer index theorem for spin complex in a path integral fashion. The integral I need to evaluate is following (in simplified form)
$\int \...
- 121
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92
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Is the Nisnevich topology quasi compact?
We only consider schemes that are smooth and separated over a field $K$. Let $\{X_i\longrightarrow X\}_{i\in I}$ be a Nisnevich covering of a scheme $X$ (all $X_i$ and $X$ are smooth and separated). ...
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237
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excess intersection theory
Can the excess intersection theory be applied to the following problem:
I have a non-singular irreducible variety $X$ of dimension $k$ and degree $d$ and $k+1$ hyperplane sections of $X$, $H_1,H_2,\...
- 325
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79
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Stable analytic manifold under simple action
For an integer $m > 1$, let us define the action
$$
f: X_i \to (1+X_i)^{m} - 1
$$
on $C[[X_1,...,X_N]]$, where $C$ is the complex number field. Consider the analytic manifold $V(I)$ defined by the ...
- 1
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105
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$\Gamma_Z(\widetilde M)\cong\widetilde{ \Gamma_Z(M)}$
Let $R$ be a Noetherian ring and let $M$ is an $R$-module. Consider the associated affine scheme $(\text{Spec R},\mathcal{O}_{\text{Spec R}})$ and Suppose $Z\subset X$ is a closed subset of $\text{...
- 21
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182
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Zariski open set of linear forms
Let $I$ be a graded homogeneous ideal over $k[x_1, ... ,x_n]$ and $h$ a linear form, let $H$ be the corresponding hyperplane and $I_H$ the restriction of $I$ to $H$.
I am looking for a Zariski open ...
- 73
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239
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Cohomology group vs sheaf of cohomology group
Suppose $F$ is a coherent sheaf on a smooth (algebraic or complex) variety $X$. Then we can consider the cohomology groups $$H^p(X,F)$$ for all $i$. Now, let we consider the sheaf $$\mathcal{H}^p(X,F)$...
- 401