All Questions
22,770 questions
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Which abelian varieties over a local field can be globalized?
As the title says, if $\mathcal{A}$ is an abelian variety over $\mathbb{Q}_p$, is there a criterion as to if I should expect there to exist $A$ over $\mathbb{Q}$ such that
$$\mathcal{A}\cong A\times_{\...
- 4,396
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votes
1
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59
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Is decomposability of polynomials ∈ℤ[𝑋] over ℚ an undecidable problem?
By a decomposition of a polynomial $F(x)$ over a field $K$ we mean writing $F(x)$ as
$$
F(x)=G(H(x)) \quad(G(x), H(x) \in K[x]),
$$
which is nontrivial if $\operatorname{deg} G(x)>1$ and $\...
- 240
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0
answers
18
views
Can one decompose quasi finite morphism as a composition of an open immersion and a finite morphism?
Can one decompose quasi-finite separated morphism of schemes as a composition of an open immersion and a finite morphism?
I am willing to assume that all the involved schemes are Noetherian.
- 2,639
6
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1
answer
184
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What is the smallest and "best" 27 lines configuration? And what is its symmetry group?
I was this past year working with a bright high-schooler on algebraic geometry following Reid's book Undergraduate Algebraic Geometry, and we got all the way to proving that there is at least one line ...
- 35.4k
2
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0
answers
61
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Whitney stratifications of hypersurfaces
Suppose that $X$ is a Whitney stratified algebraic variety with strata $\{S_i\}.$
Suppose that $Z$ is a hypersurface of $X$ which transversely intersects all strata of $X$, i.e. $S_i \cap Z$ is a ...
- 43
0
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0
answers
47
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Hasse principle for Brauer groups of fields of transcendence degree 2
In his paper "A Hasse principle for function fields over PAC fields" (DOI link), Ido Efrat proves the following result: Let $F$ be an extension of a perfect PAC field $K$ of relative ...
- 31
0
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0
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65
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Chow moving lemma with additional property
All varieties are over algebraically closed field of characteristic zero. Let $S$ be a smooth projective surface. Let $D$ be an irreducible divisor on $S$, $H$ be another divisor and $Z\subset S$ be a ...
- 219
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47
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A question of irreducibility of certain affine algebraic sets
Let $K$ denote an algebraically closed field of characteristic zero, and let $p_1(T), \dots, p_m(T)$ denote $m$ irreducible polynomials in $K[x_1, \dots, x_n][T]$ of degree at least $1$. Set
$$ S= \{ (...
- 6,214
1
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28
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Integral hull of a polyhedron Q is polyhedron
Let $Q \subseteq R^n$ be a rational polyhedron and let $Q_I=Convexhull(Q \cap Z^n)$. By finite basis theorem, we have $Q=P+C$ for some rational polytope $P$ and finitely generated cone $C$ where $C=R_+...
- 21
1
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71
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The definition of Hodge bundles with metric
A system of Hodge bundles is a direct sum of holomorphic vector bundles $E = \oplus_{p+q=n} E^{p,q}$ with a morphism $\theta : E^{p,q} \rightarrow E^{p-1,q+1} \otimes \Omega_X^1$ such that $\theta^2 = ...
- 11
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43
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The relation between Hodge bundles with metric and polarized variation of Hodge structures
Recently I've been reading Simpson's paper "constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, 1988, JAMS". On page 898 he mentioned about ...
- 11
4
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0
answers
97
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Dimension of the intersection of the commuting variety with a particular subspace
Let $\mathcal C$ denote the commuting variety of pairs of matrices in $M_n(\mathbb{C})$, defined as:
$$
\mathcal C = \{ (A, B) \in M_n(\mathbb{C})^2 \mid [A, B] = 0 \}.
$$
It is well known that $\...
- 309
3
votes
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answers
122
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Galois cohomology and Levi subgroups
Let $F$ a field and $G$ a smooth connected reductive group with a Levi subgroup $M$. Under what assumptions is $H^1(F, M) \to H^1(F, G)$ injective? In the case $F$ is nonarchimedean local I believe ...
- 605
-1
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41
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Is it possible to backtrack an optimization solver? [closed]
I have an optimization problem and was using a linear programming optimizer to find solutions. However, I find that past a certain size, the problem becomes "infeasible" and has no solutions....
0
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0
answers
67
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Reducible quartic space curve that is set-theoretic complete intersection
$\newcommand\P{\mathbb P} \newcommand\C{\mathbb C}$For a degree 4 curve, the Macaulay quartic curve is the only irreducible space curve (up to isomorphism) for which it remains an open problem to find ...
- 2,275
2
votes
1
answer
133
views
Properness of quotient map
I am new to algebraic spaces and stacks. My question is the following:
Let $X$ be a scheme and $G$ be a group scheme action on $X$. Let $[X/G]$ be the quotient stack. Then when the natural map $\pi: ...
- 613
3
votes
1
answer
191
views
Square root of relative Kähler differentials and families of curves
Let $f: X \to S$ be a smooth morphism of schemes of relative dimension $1$. Then $K=\Omega^1_{X/S}$ is a line bundle on $X$. I am interested in the following question:
When does $\Omega_{X/S}$ have a ...
- 6,612
1
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0
answers
113
views
Polynomial discriminant equation
This is a fairly straightforward question, and I am hoping a definitive answer exists.
Does there exist a quadratic form $A \in \mathbb{C}[x_1, x_2, x_3, x_4]$ and a cubic form $B \in \mathbb{C}[x_1, ...
- 26.9k
1
vote
1
answer
82
views
Sequence of MMP with scaling cannot be isomorphism
Let $(X,B)$ be a projective klt pair, H is an $\mathbb{R}$-divisor s.t. $K_X+B+H$ is nef. Suppose that we can run $(K_X+B)$-MMP with scaling $H$(that is, the flip exists) and denote $\alpha_i:X \...
- 11
0
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0
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83
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finiteness of quotient map
I am new to algebraic space s and stacks. My question is the following:
Let $X$ be a scheme and $G$ be a group scheme acting on $X$. We have the natural map $\pi : X \to X/G$. When $\pi$ will be a ...
- 613
5
votes
1
answer
224
views
Explicit Jacquet-Langlands correspondence for real reductive groups
Let $G$ be a connected reductive group over $\mathbb R$. Let $G'$ over $\mathbb R$ be an inner form of $G$ with ${}^LG={}^LG'$. By local Langlands correspondence over $\mathbb R$, if a $L$-packet of $...
- 6,612
3
votes
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answers
167
views
Finite generativity of algebra with valuation
Let $C$ be a commutative finitely generated algebra with no zero divisors. If necessary, we can assume it to be graded and a unique factorization domain. Let $a\in C$ be a prime element.
Let's also ...
1
vote
0
answers
79
views
Projection from a point and singularity
Let $X \subset \mathbb{P}^n$ be a hypersurface with $n \ge 3$. Let $x \in X$ be a closed point. Consider the map given by projection from $x$:
$$\phi: X \dashrightarrow \mathbb{P}^{n-1}$$
Suppose that ...
- 1,040
1
vote
0
answers
80
views
Galois group of shimura varieties with different level structure
Let $(G,X)$ be a shimura data, and $K$ an open compact neat subgroup of $G(\mathbb A_f)$. Suppose $K'\subset K$ is open and normal, then, I see in many references that the finite etale cover $Sh(G,X)_{...
- 775
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0
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39
views
Descend local system to the canonical model of Shimura varieties
Suppose $(G,X)$ is a Shimura data, and $E$ be its reflex field. In page 33 of this paper, it constructs an etale local system on the canonical model $\mathrm{Sh}(G,X)_{K,E}$ (variety over $E$) for any ...
- 775
4
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167
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Intuition on geometry of sections
Premise: I have asked the same question on math.stackexchange about a week ago, without receiving answer. Therefore I've decided to ask it also here. If this violates the rules, I apologize and I'll ...
2
votes
0
answers
91
views
Galois representations attached to discrete automorphic representations
Let $F$ be a totally real field. Let $G$ be a (split) connected reductive group over $F$. Let $\pi$ be an irreducible automorphic representation of $G$.
Recall in the work of Buzzard and Gee "The ...
- 6,612
1
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106
views
+50
About dimensions of quotients of quasi projective varieties
This question is related to this one. If I have an locally closed, quasi projective scheme $X$ contained in an affine space, and a linearly reductive group $G$ acting freely on $X$, are there examples ...
- 556
3
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109
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Bertini's theorem at a fixed point
Recently, I am learning Bertini's theorem because I encounter "generic smooth" problem during my research. I'm not an algebraic geometer and I read the Hartshorne Chapter 3 Theorem 10.8 to ...
- 51
3
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103
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Jacobian of a reducible curve with arbitrary singularities
Let X be a reduced, reducible curve over $\mathbb{C}$ with locally planar singularities, and let $\widetilde{X}$ be its normalization. I am interested in the Jacobian varieties $\mathrm{Jac}(X)$ and $\...
- 31
0
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0
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90
views
Length of generic intersection in local ring
Let $(R, \mathfrak{m})$ be a regular local ring and let $I\subset R$ be an ideal of coheight 1. Let $a \in \mathfrak{m}\setminus \mathfrak{m}^2$.
If $a$ that is not a zero divisor of $R/I$ we have ...
1
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0
answers
260
views
Are there connections between Calabi-Yau manifolds and number theory?
I am interested in understanding whether there are any significant connections between Calabi-Yau manifolds and number theory. Calabi-Yau manifolds are central objects in algebraic geometry and string ...
4
votes
2
answers
160
views
Connectedness of degeneracy loci
Let $E, F$ be vector bundles of ranks $e, f$ on a smooth variety $X$ of dimension $n$.
Let $\varphi : E \to F$ be a morphism and let $k$ be such that $n > (e-k)(f-k)$.
Fulton-Lazarsfeld's theorem ...
- 331
3
votes
1
answer
153
views
Geometry and topology of Fuchsian character varieties
Consider the hyperbolic space, $\mathbb H^2$. A Fuchsian group is a discrete subgroup of $\text{PSL}(2,\mathbb R)$. We can generate tessellations, especially $\{p,q\} \;\text{tesellations}$ of $\...
- 357
6
votes
1
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276
views
effective descent of coherent sheaves
I am new to stacks and algebraic spaces. I have the following question:
Let $X$ be a scheme and $G$ be a group scheme action on $X$. Then $X/G$ exists as an algebraic space. Let $\pi: X \to X/G$ be ...
- 613
4
votes
0
answers
86
views
Levis, parabolics and Bruhat-Tits over Henselian local rings
Let $(R,m)$ be a Henselian local ring with algebraically closed or finite residue field $k$ and fraction field $F$. For example, we may work with $R=W(\mathbb F_p^{alg})$.
The paper "Reductive ...
- 6,612
2
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answers
123
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Generalization of Koszul cohomology for wedge product with a fixed q-form: literature and references?
Let $A$ be a commutative ring. For an odd positive integer $q$ and an integer $p$ in the range $1 \leq p \leq q$, consider the cochain complex:
$0 \to \Lambda^p(A^N) \to \Lambda^{p+q}(A^N) \to \cdots \...
- 21
2
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0
answers
134
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Universal semistable curve
For the moduli of stable curve, we have the result by Deligne-Mumford-Knudsen that there's an isomorphism of moduli spaces
$$\Phi : \overline{\mathcal{X}}_{g,n} \xrightarrow{\sim} \overline{\mathcal{M}...
- 834
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1
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197
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Section 3 of Atiyah's "On analytic surfaces with double points" — some questions
I have some questions about section 3 of Atiyah's "On analytic surfaces with double points," a short 9 page paper. Section 3 is all dedicated to proving lemma 4.
Near the end of section 3, ...
- 129
2
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120
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Analogs of Plücker relations in Clifford algebras, and Bott periodicity (?)
Classical Plücker relations can be viewed as conditions on coefficients of an element $x=\sum_Sc_Se_S$, $S=(i_1,...,i_k)$, $i_1<\cdots<i_k$, $\{i_1,...,i_k\}\subset\{1,...,n\}$ of an exterior ...
- 18.4k
3
votes
1
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363
views
Variants of Grothendieck section conjecture
Let $X$ be a smooth projective variety defined over a field $k$. We fix the following notations : $\overline{k}$ denotes the algebraic closure of the field $k$, $X_{\overline{k}}$ denotes the variety $...
- 443
3
votes
1
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138
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Can the coefficients of a Taylor series be expressed as rational functions for an affine variety?
Let $k$ be an algebraically closed field and $V \subseteq k^n$ an affine variety corresponding to a prime ideal $P \subseteq k[t_1, \dots, t_n]$. For $x\in V$ let $O_x = \{p/q \mid p,q\in k[t_1, \dots,...
- 696
3
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2
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366
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Rational divisors on Calabi–Yau threefolds
Following the construction of [2], consider the full subcategory $\mathcal{D}_0 \subset D^\flat(\operatorname{Coh} \omega_{\mathbb{P}^1})$ consisting of complexes whose cohomology objects are ...
- 317
1
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108
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L.c.i locus of Hilbert scheme of points on singular varieties
Let $X$ be an algebraic variety over $\mathbb{C}$. What can we say about the l.c.i. locus of $\text{Hilb}^n(X)$?
When $X$ is smooth, it is well-known that the l.c.i. locus of $\text{Hilb}^n(X)$ is ...
- 385
4
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0
answers
237
views
What do we do when $G$ doesn't have a Shimura variety?
Let $G$ be a reductive group. If one can associate to $G$ a Shimura datum $(G,X)$, then the étale cohomology of the associated Shimura variety $\operatorname{Sh}(G,X)$ is a strong tool for the ...
- 57
0
votes
0
answers
88
views
Extend algebraic morphism to a compactification with normal crossing boundary
Suppose $X$ and $Y$ are smooth algebraic variety over a char $0$ field $k$, and $f:X\to Y$ a morphism. I want to ask whether there exists compactifications $\bar X$ and $\bar Y$ such that $\bar X\...
- 775
0
votes
0
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90
views
Compactification of smooth varieties with normal crossing boundary
I see in this paper, page 46, the second sentence of 4.1, that every smooth variety over a characteristic $0$ field can be embedded into a proper smooth variety with normal crossing boundary, and the ...
- 775
4
votes
0
answers
171
views
Why are the Hodge filtrations on cohomology canonically bounded?
If $X$ is a complex projective variety of dimension $n$ then the de Rham cohomology $H^{k}(X,\mathbb Q)$ naturally has a mixed Hodge structure with an increasing weight filtration $W_\bullet$ and a ...
- 141
2
votes
1
answer
125
views
Changing the weight space for an eigenvariety
Let $G$ be an algebraic group (like $\operatorname{GL}_2$ or $\operatorname{GL}_2 \times\operatorname{GL}_2$ for example). Assume that there exists an eigenvariety $\mathcal{E}^G$ parameterizing ...
- 93
2
votes
1
answer
184
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Number of rational points of a quotient of connected linear algebraic groups
Let $H$ be a closed connected subgroup of a connected linear algebraic group $G$ over an algebraically closed field of characteristic $p>0$, and let $\sigma=\sigma_q$ be the standard Frobenius ...
- 23