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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_{\...
curious math guy's user avatar
0 votes
1 answer
59 views

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 $\...
SARTHAK GUPTA's user avatar
0 votes
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.
Rami's user avatar
  • 2,639
6 votes
1 answer
184 views

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 ...
David Roberts's user avatar
  • 35.4k
2 votes
0 answers
61 views

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 ...
user535880's user avatar
0 votes
0 answers
47 views

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 ...
aspear's user avatar
  • 31
0 votes
0 answers
65 views

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 ...
Galois group's user avatar
0 votes
0 answers
47 views

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= \{ (...
Keivan Karai's user avatar
  • 6,214
1 vote
0 answers
28 views

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_+...
Sowbarnika R's user avatar
1 vote
0 answers
71 views

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 = ...
Kimoji's user avatar
  • 11
0 votes
0 answers
43 views

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 ...
Kimoji's user avatar
  • 11
4 votes
0 answers
97 views

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 $\...
darko's user avatar
  • 309
3 votes
0 answers
122 views

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 ...
C.D.'s user avatar
  • 605
-1 votes
0 answers
41 views

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....
Bamboozle's user avatar
0 votes
0 answers
67 views

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 ...
Jose Capco's user avatar
  • 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: ...
KAK's user avatar
  • 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 ...
Zhiyu's user avatar
  • 6,612
1 vote
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, ...
Stanley Yao Xiao's user avatar
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 \...
Chi-siu's user avatar
  • 11
0 votes
0 answers
83 views

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 ...
KAK's user avatar
  • 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 $...
Zhiyu's user avatar
  • 6,612
3 votes
0 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 ...
Sasha Kucherenko's user avatar
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 ...
Naga Venkata's user avatar
  • 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)_{...
Richard's user avatar
  • 775
0 votes
0 answers
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 ...
Richard's user avatar
  • 775
4 votes
0 answers
167 views

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 ...
YetAnotherMathStudent's user avatar
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 ...
Zhiyu's user avatar
  • 6,612
1 vote
0 answers
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 ...
User43029's user avatar
  • 556
3 votes
0 answers
109 views

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 ...
MATHQI's user avatar
  • 51
3 votes
0 answers
103 views

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 $\...
Grotherd's user avatar
0 votes
0 answers
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 ...
Serge the Toaster's user avatar
1 vote
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 ...
Abdullah M Al-jazy's user avatar
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 ...
Cob's user avatar
  • 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 $\...
user82261's user avatar
  • 357
6 votes
1 answer
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 ...
KAK's user avatar
  • 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 ...
Zhiyu's user avatar
  • 6,612
2 votes
0 answers
123 views

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 \...
user47660's user avatar
2 votes
0 answers
134 views

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}...
E. KOW's user avatar
  • 834
2 votes
1 answer
197 views

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, ...
maxo's user avatar
  • 129
2 votes
0 answers
120 views

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 ...
მამუკა ჯიბლაძე's user avatar
3 votes
1 answer
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 $...
random123's user avatar
  • 443
3 votes
1 answer
138 views

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,...
kevkev1695's user avatar
3 votes
2 answers
366 views

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 ...
cdsb's user avatar
  • 317
1 vote
0 answers
108 views

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 ...
Chan Ki Fung's user avatar
4 votes
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 ...
Loading's user avatar
  • 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\...
Richard's user avatar
  • 775
0 votes
0 answers
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 ...
Richard's user avatar
  • 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 ...
D. Brogan's user avatar
  • 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 ...
BanAna's user avatar
  • 93
2 votes
1 answer
184 views

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 ...
aliquot's user avatar
  • 23

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