All Questions
975 questions with no upvoted or accepted answers
2
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288
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Set theoretic complete intersections in toric varieties
Is it expected that every smooth projective variety over the complex numbers, is a set-theoretic complete intersection into a smooth projective toric variety?
Is there an example of a smooth ...
user120812
2
votes
0
answers
330
views
Algebraization of open formal subschemes
Let $\mathfrak{X}$ be a locally noetherian adic formal scheme over $\text{Spf}(A)$, with $A$ an $I$-adically complete and separated noetherian ring.
Suppose the mod $I$-fiber of $\mathfrak{X}$ is an ...
user95222
2
votes
0
answers
256
views
Neron Severi under specialization
Let $X$ be a smooth projective variety over $\mathbf{Q}$, and $\mathcal{X}$ a smooth projective model over $\mathbf{Z}[1/N]$ for $N$ large enough.
Call $\eta$ the generic point $\text{Spec}(\mathbf{Q}...
user95222
2
votes
0
answers
147
views
Genus Zero Diophantine Equations and Infinite Valuations
I'm interested in an explicit upper bound for the integral solutions of a certain genus zero curve $F(X,Y)=0$. I found some papers that address this problem:
[1] Solving genus zero diophantine ...
- 211
2
votes
0
answers
20
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Cohomology of $\mathbf{G}_a\otimes_{\mathbf{Z}}\mathbf{G}_a$
Let $X$ be a smooth projective variety over a field.
Is $$H^p(X_{Zar},\mathbf{G}_a\otimes_{\mathbf{Z}}\mathbf{G}_a)$$
at all related to $H^p(X_{Zar},\mathbf{G}_a) = H^p(X,\mathcal{O}_X)$ via tensor ...
user92332
2
votes
0
answers
110
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Dual sheaf of universal pointed unipotent connection and the canonical de Rham torsor
I'm trying to understand a particular point in the paper The Unipotent Albanese Map and Selmer Varieties on Curves by Kim.
We fix a basepoint $b \in X(L)$ on a curve $X$ over $L$ a characteristic 0 ...
- 121
2
votes
0
answers
234
views
Global minimal Weierstrass equation over function fields
Let $K$ be a field of characteristic zero, $V/K$ a smooth projective variety and $F=\overline{K}(V)$ be the function field of $V$ over $\overline{K}$. For any elliptic curve $E/F$, it has a Weiertrass ...
- 21
2
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answers
483
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Absolute Hodge cycles over $\mathbf{Q}$
In the 1986 notes by Milne "Hodge cycles on abelian varieties", Deligne defines the notion of absolute Hodge cycles.
For a smooth projective variety defined over $k\subset\mathbf{C}$ non ...
user120812
2
votes
0
answers
72
views
Support of étale sheaves
Let $X$ be a scheme, $i: Z\to X$ a closed subscheme, $j: U\to X$ its complement in $X$. Assume the codimension of $Z$ in $X$ is large (at least $2$).
Let $A$ be an étale sheaf on $U$, $B$ an étale ...
user120812
2
votes
0
answers
656
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Specialization maps for Chow groups
Let $S$ be a finite type regular integral affine scheme of finite type over $\text{Spec}(\mathbf{Z})$, and $\mathcal{X}\to S$ a smooth projective morphism. Let $\eta$ be the generic point of $S$, $s\...
user119470
2
votes
0
answers
239
views
Group completion of Chow varieties
Let $X$ be a quasi-projective variety over a perfect field $k$.
Given a projective embedding $j : X\to \mathbf{P}(\mathscr{E})$, the Chow variety $\text{Chow}_r(X, j)$ is a quasi-projective variety ...
user113393
2
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0
answers
357
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Does the pro-étale local system defined over a p-adic period domains interpolate crystalline representations?
There is a Grothendieck-Messing period morphism of rigid-analytic spaces $\pi: \mathcal{M}_\eta^{rig}\to \mathcal{Fl}$ going from the generic fiber of an EL-type Rapoport-Zinks to a flag variety. The ...
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2
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0
answers
254
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Global sections of higher direct images
If $f : X\to V$ is a smooth proper map of smooth schemes, what are the global sections of
$R^if_{fppf, *}\mu_p$
$R^if_{fppf, *}\mathbb{G}_{\rm m}$
I was reading Milne's book "Arithmetic duality", ...
user113393
2
votes
0
answers
228
views
On a class of loci in Chow varieties
Let $k$ be a field, $i:X\hookrightarrow \mathbf{P}(\mathscr{E})$ be a fixed projective embedding of a smooth projective $k$-variety $X$, whose dimension is pure and equals $d\ge 0$.
For $0\le p\le d$,...
user95222
2
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answers
69
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A question about abelian varieties
For an abelian variety $A$ over a global field $K$, $\mathcal{A}$ its Néron model on $C$, either a smooth projective geometrically irreducible curve over a finite field, or the spectrum of the ring of ...
user95222
2
votes
0
answers
519
views
Good place to learn about arithmetic schemes?
Where is a good place to learn about arithmetic schemes? There is discussion in Eisenbud-Harris's book The Geometry of Schemes (and also Mumford's red book) and I hear that there is discussion in Liu'...
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2
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304
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Surjectivity of map of Picard schemes implies abelian
Note: This question was asked on MSE first, but got zero reactions. So I deleted it there, and am now posting it here.
I am looking for a reference or explanation of the fact that is used in Mumford'...
- 203
2
votes
0
answers
121
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Global invariant cycles in positive characteristic
Let $k$ be an algebraically closed field of positive characteristic $p$ and fix a prime $\ell\neq p$. Let $X$ be a smooth connected $k$-variety and $f:Y\rightarrow X$ a smooth projective morphism. ...
- 728
2
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124
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finiteness of Abelian varieties $B$ with $T_\ell A \cong T_\ell B$ for all primes $\ell$
Let $K$ be a number field.
In Faltings' Finiteness Theorems for Abelian Varieties over Number Fields, Corollary 3:
Let $A/K$ be an abelian variety, $d > 0$. Then there are only finitely many ...
user19475
2
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0
answers
180
views
Two elliptic curves not dominated by a genus two curve
Let $k$ be a number field and let $E$ and $E'$ be elliptic curve over $k$.
There is a genus two curve $X$ over $\overline{k}$ which dominates $E$ and $E'$.
Question. Is there a genus two curve $X$ ...
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2
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0
answers
140
views
Endomorphism of a Jacobian
Let $C$ be a curve of genus $g$ defined over the finite field $k=\mathbb{F}_q$. Set $A=J(C)$ the jacobian of $C$, then $A$ is an abelian variety defined over $k$ of dimension $g$.
The algebra $End_k(...
- 389
2
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0
answers
113
views
compute conjugacy classes of matrices over $\mathcal{Z}$
Given an irreducible polynomial $f(X)\in\mathbb{Z}[X]$, do you know an efficient algorithm to compute the number of conjugacy classes of matrices $A\in M_n(\mathbb{Z})$ with characteristic polynomial $...
- 389
2
votes
0
answers
286
views
Does the sheaf of locally exact differential forms splitting in positive characteristic
Let k be an algebraically closed field of characteristic $p>2$, $X$ a smooth projective curve of genus $g>1$ over $k$, and $F_X:X\rightarrow X$ be the absolute Frobenius morphism. Let $B^1_X$ be ...
- 85
2
votes
0
answers
453
views
Exponential diophantine equation system
I noticed a strange relation months ago :
$\begin{cases}3^5+10^2=7^3\\3+7=10\\2+3=5\end{cases}$
For the sake of math, I searched for positive integer non trivial (i.e. not containing any 0) ...
- 153
2
votes
0
answers
228
views
p-divisible groups over a p-adic field
p-div gps over a finite field (e.g., $\mathbb F_p$) are classified by Dieudonne modules.
There are also results for p-div gps over integers of local fields (e.g., over $\mathbb Z_p$).
However, are ...
- 153
2
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0
answers
345
views
Examples of semi-stable models of curves
Let $R$ be a discrete valuation ring with fraction field $K$ of characteristic zero and residue field $k$ of characteristic $p>0$. Assume $k$ is algebraically closed. I want to produce examples of ...
- 2,323
2
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148
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Purely inseparable $k$-rational dominant maps between an absolutely irreducible $k$-surface and $\mathbb{P}^2$
Let $X$ be an absolutely irreducible surface over an algebraically closed or finite field $k$ of characteristic $p$. Is it true that the following conditions are equivalent?
There is a purely ...
- 1,833
2
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0
answers
93
views
Galois cohomology of $GL_n(E^s \hat{\otimes} R)$
Let $E= \mathbb{F}_p(\!(u)\!)$ and write $E^s$ for a separable closure. Write $G_E = \mathrm{Gal}(E^s/E)$ for the absolute galois group of $E$. Let $R$ be a noetherian $\mathbb{F}_p$-algebra and write ...
- 189
2
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0
answers
336
views
Reduction "modulo $p$" of $\mathfrak{p}$-torsion points of CM elliptic curves
Let $E/L$ be an elliptic curve defined over a number field $L$. Assume moreover that $E$ has complex multiplication by an imaginary quadratic field $K$. Let $\mathfrak{p}$ be a prime ideal of $K$. ...
- 7,609
2
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0
answers
101
views
Split multiplicative galois representation and specialization
My questions stems from my attempt to understand the paper of Greenberg and Stevens about the Mazur-tate-Teitelbaum conjecture (you can find the paper here). To understand this question you probably ...
- 121
2
votes
0
answers
107
views
the least point on a variety over a finite field
Let $p$ be a large prime parameter and $V\subseteq \mathbb{P}^n_{\mathbb{F}_p}$ a variety defined over the finite field $\mathbb{F}_p$ with bounded degree and dimension (w.r.t. $p$). Assume that $V$ ...
- 3,362
2
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0
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476
views
Does the numerical equivalence relation coincide with the homological one for 1-cycles (in positive characteristic)?
Is the Grothendieck's Standard Conjecture D (stating that the numerical equivalence relation for algebraic cycles with rational coefficients coincides with the homological one) known to be true for ...
- 16.9k
2
votes
0
answers
416
views
In how many ways can one extend the zero section of the affine line with a double origin
Let $X$ be the affine line with a double origin over $\mathrm{Spec}\,\mathbb Z$. Let $X_\eta$ be its generic fibre, the affine line with a double origin over $\mathrm{Spec}\,\mathbb Q$.
Let $0$ be ...
- 21
2
votes
0
answers
182
views
An elliptic curve trivial over any extension unramified outside 7 and infinity?
Is there an elliptic curve $E/\mathbb{Q}$ such that $E(K)$ is trivial for every finite extension $K/\mathbb{Q}$ with discriminant a power of $7$ ?
- 11.3k
2
votes
0
answers
128
views
Local duality for abelian varieties
Let $A$ be an abelian variety over a p-adic field $K$. Let $I$ be the inertia group of $K$. There is a Yoneda pairing $$H^n(\hat{\mathbb{Z}},A^I) \times Ext^{2-n}_{\hat{\mathbb{Z}}}(A^I,\mathbb{Z}) \...
- 213
2
votes
0
answers
389
views
Are there good properties of the divided power completion map?
Let $Y \to X$ be a closed immersion of smooth schemes over, say, the ${\rm Spec}(\mathbb{Z}_p)$. The completion map $$X_{/Y}\to X$$ is an ind-closed immersion (sometimes called pseudo-closed immersion)...
- 33
2
votes
0
answers
149
views
subschemes of abelian scheme over artinian basis
Let $R$ be an artinian thickening of a field $k$. Denote with $S=Spec(R)$. Let $A$ be an abelian scheme over $S$. Let $X$ be a closed, reduced, equidimensional subscheme of the special fiber $A_k$. I ...
- 21
2
votes
0
answers
531
views
vanishing of étale cohomology of affine surface
Let $U$ be an affine smooth surface over an algebraic closure of a finite field. Let $\mathscr{A}/U$ be an Abelian scheme and $\ell \neq \mathrm{char}(k)$ be prime.
Are there vanishing results for ...
user19475
2
votes
0
answers
119
views
Bounds for the Tamagawa number of the Jacobian of a hyperelliptic curve
Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$ and let $C$ be a hyperelliptic curve of genus $g$ defined over $K$ with Jacobian $J$.
Suppose that $C$ is given by ...
- 337
2
votes
0
answers
120
views
Benchmark problems for computing rational points on varieties
Are there standard benchmark problem sets used for empirically evaluating algorithms designed for computing rational points on (various classes of) algebraic varieties?
If so, could you please point ...
2
votes
0
answers
434
views
algebraicity of Néron-Tate canonical height for Abelian varieties over global function fields
(transcendence of canonical heights)
Is the Néron-Tate canonical height for an Abelian variety $A$ over a global function field $K$, $\hat{h}: A(K) \times A^\vee(K) \to \mathbf{R}$ known to always ...
user19475
2
votes
0
answers
244
views
Descent theory of line bundles on abelian varieties under isogenies (in char p>0)
I have a couple of questions regarding the descent theory of line bundles on abelian varieties under isogenies in positive characteristic.
Let $X$ be an abelian variety and $L\in Pic(X)$ a line ...
- 614
2
votes
0
answers
255
views
Lang isogeny for group stacks
Let $G$ be a commutative algebraic group stack over $\mathbb{F}_q$ (I don't really care about the precise definition: I'm secretly thinking about the Picard stack of a projective curve). To what ...
- 3,605
2
votes
0
answers
220
views
Bound for the degree of the field of definition for a closed point of a variety
While attempting to prove some existence theorem for matrices over $\mathbb{F}_{2^k}$ I've come across the following problem concerning fields of definition for closed point of, say affine, varieties.
...
- 351
2
votes
0
answers
606
views
Kawamata-Viehweg Vanishing Theorem for Excellent Surfaces
Is there any version of Kawamata-Viehweg vanishing theorem that holds for excellent surfaces? I will be very glad if there is one such, but if not then is it at least true for a surface over a non ...
- 165
2
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0
answers
218
views
The etale cohomology``ring" structure of torsion sheaves on varieties
For a topological manifold $M$, one can speak of the cohomology ring structure $H^*(M, k)$ where $k$ is a ring. If one replace $M$ by an arithmetic schemes $X$ over a base ring $S$, and replace $k$ by ...
- 1,160
2
votes
0
answers
143
views
Dualizing sheaf in mixed characteristic for regular schemes.
I've been looking many places, but everything I find seems to either talk about (a) varieties or (b) extremely general situations with dualizing complexes. As I am not in the situation of (a) (i.e. ...
- 419
2
votes
0
answers
489
views
what are the possible CM-fields of PEL type shimura varieties ?
In the paper "Travaux de Shimura" section 6, Deligne had defined a PEL- type shimura variety, for the following datum $(F,E,D,\psi)$, with $F$ a totally real cubic field, and $E$ a imaginary ...
- 709
2
votes
0
answers
137
views
Removing finitely many points from a Shimura curve
Let $X$ be a compact Shimura curve. If we remove finitely many points from this curve, do we neccessarily get a "non-compact Shimura curve"? I have some reasons to believe that the answer is negative, ...
- 637
2
votes
0
answers
473
views
$\sigma$-conjugate iff $p$-adically close
First some notations. Let $p$ be a prime, $k$ a perfect field of characteristic $p$, $W=W(k)$ the ring of Witt vectors over $k$, $\sigma : W \rightarrow W$ the Frobenius, $R$ a commutative $\mathbb{Z}...
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