All Questions
481 questions with no upvoted or accepted answers
3
votes
0
answers
132
views
How does the number of connected components of the Néron model change in a family of abelian varieties?
Given an elliptic curve $E/\mathbb{Q}_p$, it is known that the component group of the Néron model of $E$ is cyclic of order $-v(j(E))$ when $E$ has split multiplicative reduction, and in all other ...
- 333
3
votes
0
answers
256
views
What is a twisted D-module?
Let $X/\mathbb{C}$ be an abelian variety, $Y$ be the dual abelian variety, and $P$ be the Poincaré bundle on $X\times Y$. On p.207, Correction to “Sheaves with connection on abelian varieties” (by M. ...
- 615
3
votes
0
answers
160
views
What does the Néron model of the dual abelian variety parametrize?
Let $K$ be a field which is complete with respect to a discrete valuation $v$ with ring of integers $R$ and residue field $k$. Let $A$ an abelian variety over $K$ and let $A^t$ be the dual abelian ...
- 3,031
3
votes
0
answers
112
views
What are the possibilities of the general fibres in an Iitaka fibration?
This question is motivated by complex algebraic geometry.
If $X$ is a complex algebraic variety with Kodaira dimension in $[1,\dim X-1]$, then the Iitaka fibration (the rational map induced by the ...
- 9,501
3
votes
0
answers
136
views
"Vanishing locus" of forms in the $h$-topology
Let $\Omega_{h}^p$ be the sheaf of $p$-forms in the $h$-topology defined as the sheafification for the $h$-topology of the presheaf,
$$ Y \mapsto \Omega^p_Y(Y) $$
Kebekus and Schnell show that when $X$...
- 3,730
3
votes
0
answers
94
views
Dimension of a kernel of a cocycle map
Inspired by a previous question (Dimension of a kernel of a linear map) and some of the answers I was given I thought wheter I can generalize the question to the following:
Compute the kernel (or at ...
- 911
3
votes
0
answers
122
views
Torsion of Fermat hypersurfaces
An interesting invariant of a rationally chain-connected variety $X/k$ is the exponent of the group,
$$ \ker{(\mathrm{CH}_0(X_K) \xrightarrow{\deg} \mathbb{Z})} $$
where $K = k(X)$ is the function ...
- 3,730
3
votes
0
answers
87
views
Obstruction for points to be contained in smooth hypersurfaces in tterms of inseparability degree of residue field
Let $k$ be an imperfect field of char $p>0$ and
$x \in \mathbb{P}^n_k$ be closed point of projective space.
In this discussion Qing Liu wrote that
Over an imperfect field, a reduced point can not ...
- 619
3
votes
0
answers
105
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Points with residue fields having big inseparability degree cannot be contained in smooth hypersurfaces
Let $X$ be a $k$-scheme over imperfect field $k$ and $x \in X $ some (reduced) point with residue field $\kappa(x) = \mathcal{O}_{X,x}/ \mathfrak{m}_x$. How to check that if $\kappa(x)$ has "big ...
- 619
3
votes
0
answers
110
views
Local global principle over infinite extension of $\Bbb{Q}$ which is not algebraically closed
Let $A$ be an algebraic variety over a field $K$, which is finite extension of $ \Bbb{Q}$.
We say local global principle holds if $A(K_v) \neq \emptyset$ implies $A(K) \neq \emptyset$, where $K_v$ is ...
- 1,541
3
votes
0
answers
91
views
Can you find a Darboux basis for any skew integral form on a full-rank lattice in $ℂ^n$ so that the first $n$ vectors are $ℂ$-linearly independent?
Any skew bilinear form $\omega$ on $\mathbb{Z}^{2n}$ can be brought into the form
\begin{equation}
\begin{pmatrix}
0 & \Delta \\
-\Delta & 0
\end{pmatrix},
\quad
...
- 1,141
3
votes
0
answers
127
views
Isogeny of elliptic curve over positive characteristic $p$ which does not come from characteristic $0$
Let $K$ be quadratic imaginary field. Let $E$ be an elliptic curve which has CM over $R_K$
($R_K$ is ring of integers of $K$).
According to SIlverman's ''ADvanced topics in the arithmetic of elliptic ...
- 1,541
3
votes
0
answers
120
views
Resolving the "wild" singularities of $\mathbb A^n/C_n$
Let the cyclic group on $n$ elements, $C_n$, act on $\mathbb A^n$ by permuting the co-ordinates (over a field $k$). If $n \neq 0 \in k$, we can resolve the singularities of $X = \mathbb A^n/C_n$ by ...
- 7,746
3
votes
0
answers
314
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Ampleness of the normal bundle to the Albanese image
Let $X$ be a projective surface of general type over $\mathbb{C}$, and assume that $\Omega_X$ is globally generated. Then the Albanese map $a \colon X \to \operatorname{Alb}(X)$ is a local embedding ...
- 66.3k
3
votes
0
answers
154
views
Principally polarised supersingular abelian varieties
If $A/k$ is a supersingular abelian variety, then it is $k$-isogenous to $A'/k$ a superspecial abelian variety, see "A Note on supersingular abelian varieties" by Chia-Fu Yu. And if $A'/k$ ...
- 645
3
votes
0
answers
87
views
Two pairings on the group $K(L)$ associated with a non-degenerate line bundle $L$ on an abelian variety
Let $A$ be an abelian variety over a field and let $L$ be a non-degenerate line bundle on $A$.
Then $L$ gives rise to a morphism $\lambda:A\to A^*$ from $A$ to its dual.
As usual, let $K(L):=\ker(\...
- 3,076
3
votes
0
answers
301
views
Galois invariant of Tate module
Let $K$ be a finite extension of $\mathbb{Q}_p$. Let $V$ be a de Rham representation of $G_K=\operatorname{Gal}(\overline{K}/K)$. By Corollary 3.8.4 of Bloch and Kato - L-functions and Tamagawa ...
- 247
3
votes
0
answers
149
views
What direction does the derivation of an inseparable algebraic variable point in?
I've been thinking about the geometry of inseparable field extensions lately, since I'm studying smoothness in commutative rings in an advanced topics course this semester. I've generally come to the ...
- 1,969
3
votes
0
answers
209
views
Endomorphisms ring of complex abelian variety under isogenies
I’m trying to understand if over $\mathbb{C}$ two abelian varieties have the same complex multiplication if and only if they are isogenous. Is it true?
If it is true this means that if I consider $A$ ...
3
votes
0
answers
230
views
Toric degeneration of Kummer Surface
I am wondering if there are any explicit examples of a toric degeneration of a Kummer surface (e.g. as a family of projective varieties say), and what the central fibre can look like? (I am working ...
- 205
3
votes
0
answers
145
views
Richelot isogenies in characteristic $2$
I am interested in Richelot isogenies between ordinary abelian surfaces in characteristic $2$. If I am not mistaken, the corresponding theory is developed in Article "J.-B. Bost, J.-F. Mestre, ...
- 1,833
3
votes
0
answers
197
views
How to write down an explicit equation of given degree yielding a smooth hypersurface in a projective space?
Let F be a field of positive characteristic $p$ and let $d,n$ be two positive integers.
Can we explicitly write down an equation defining a smooth hypersurface $X_d⊂\mathbb P^n_F$ of degree d ?
This ...
- 417
3
votes
0
answers
368
views
Homomorphisms of abelian varieties and Tate modules
Let $A$ and $B$ be abelian varieties over a field $k$ and $\ell$ be a prime different from the characteristic of $k$, we have an injection $Hom(A,B) \otimes_\mathbb{Z} \mathbb{Z}_\ell \to Hom (T_\ell ...
- 647
3
votes
0
answers
218
views
Is there a proper smooth variety in characteristic $p$ whose Hodge-to-de Rham spectral sequence does not degenerate at $E_1$?
By Deligne-Illusie, such a variety has no lifting to $W_2(k)$. In their paper they state that they do not know if such an example exists. Has this question been answered since then?
- 4,164
3
votes
0
answers
110
views
Using principal polarisation to "cancel" Jacobian summands in isomorphism
I'm working through the sketch proof of irrationality of cubic threefolds in Huybrechts' The geometry of cubic hypersurfaces.
Let $J(X)$ denote the intermediate Jacobian of a cubic threefold $X \...
- 305
3
votes
0
answers
133
views
Isomorphism of certain irreducible representations over finite fields
We are given a faithful representation of a cyclic group of order 5 $\rho: C_5=G \rightarrow End_{\mathbb{F}_3}(V) $ with $dim_{\mathbb{F}_3}V=8$ as vector space. It is also known that $V=U\oplus W$ ...
- 269
3
votes
0
answers
232
views
Lifting a Frobenius endomorphism under an étale morphism
Let $X$ be a smooth affine scheme over $\mathbb{Z}/{p^2}$ that is a complete intersection, say $X$ is the spectrum of $\mathbb{Z}/{p^2}[x_1,...x_n]/(f_1, ... f_r)$, where $n-r$ is the dimension of $X$....
- 111
3
votes
0
answers
352
views
Example of a non log-canonical pair for an abelian variety with polarization of degree >2
Let $(A,L)$ be a polarized abelian variety of dimension $g$, with an indecomposable polarization of degree $\chi(L)=d$. There is a theorem of Debarre and Hacon about the singularities of pairs:
...
3
votes
0
answers
212
views
Genus two curves on abelian surfaces
Considering a smooth genus two curve $C_2$, let $J(C_2)$ be its Jacobian surface, and take $p \in J(C_2)$ an $m$-torsion point. Let $A = J(C_2)/Z_m$, where $Z_m$ acts by $x \mapsto x+p$. The image of $...
- 427
3
votes
0
answers
305
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Abelian varieties by Moonen and van der Geer: proof of rigidity lemma
I try to understand a reduction step in the proof of rigidity lemma as proved in Moonen's and van der Geer's Abelian varieties (Lemma 1.11 on page 12- if the link not work the draft version is online ...
- 5,986
3
votes
0
answers
143
views
What kind of equivalences exist between categories of characteristic $0$ and characteristic $p$?
The tilting equivalence for perfectoid algebras gives an equivalence of categories $$K\text{-perf} \cong K^\flat\text{-perf}$$
where the left-hand-side are algebras in characteristic zero and the ...
- 4,164
3
votes
0
answers
186
views
The Weil restriction of an elliptic curve with respect to $\mathbb{F}_{p^2}/\mathbb{F}_{p}$
For a prime $p > 3$ consider the quadratic finite field extension $\mathbb{F}_{p^2}/\mathbb{F}_{p}$. Also, consider the elliptic curves
$$
E\!: y_0^2 = x_0^3 + ax_0 + b,\qquad
E^{(1)}\!: y_1^2 = ...
- 1,833
3
votes
0
answers
180
views
Subquotients of abelian varieties with good reduction
Let $B$ be an abelian variety over a DVR with good reduction, and let $A$
be a subquotient of $B$. Then $A$ has good reduction.
I know a proof of this statement using Neron-Ogg-Shafarevich. Is ...
- 39
3
votes
0
answers
194
views
component group of Neron models
Let $A$ be an abelian variety over a discretely valued field $K$ and $\mathcal A$ its Neron model over $R$ (the ring of integers of $K$) and $\mathcal A^0$ the identity component of $\mathcal A$.
...
- 31
3
votes
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answers
144
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Are there three ordinary elliptic curves $E$, $E_1$, $E_2$ such that $E^2 \cong E_1 \!\times\! E_2$?
Consider the elliptic curve $E\!: y^2 = x^3 + 1$ of $j$-invariant $0$ over an algebraically closed field $k$ of characteristics $p$. Let me remind that $E$ is ordinary (i.e., non-supersingular) iff $p ...
- 1,833
3
votes
0
answers
151
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Variety Isomorphism Problem for Abelian Surfaces
This is a special case of this question, where it is asked whether there exists an algorithm to determine whether two varieties are isomorphic. There, an answer by Bjorn Poonen explains how to solve ...
- 437
3
votes
0
answers
135
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Is the generalized Kummer threefold rational in characteristics 3?
Let $E_i\!: y_i^2 = x_i^3 - x_i$, $i = 1, 2, 3$ be three copies of the supersingular elliptic curve in characteristics $3$. Consider on $E_i$ the following automorphism of order $3$:
$$
\sigma(x_i,...
- 1,833
3
votes
0
answers
147
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Parallel transport for variety over finite field
I was wondering: Given a variety over a finite field, say the projective plane or sphere over $\mathbb{F}_q$. Then I can try to define parallel transport along (geodesic) curves. In particular, I can ...
- 1,846
3
votes
0
answers
409
views
Non algebraizable formal abelian schemes
I'd like to collect a good number of examples of formal schemes over $\text{Spf}(\mathbf{Z}_p)$, whose special fiber is a projective variety over $\mathbf{F}_p$, but that are not algebraizable.
If ...
user97068
3
votes
0
answers
78
views
Under what conditions are superspecial abelian surfaces isomorphic over a finite field?
Let $E_1$, $E_2$, $E_3$, $E_4$ be supersingular elliptic curves over a finite field $\mathbb{F}_{p^2}$, where $p$ is an odd prime. There is a well known theorem stating that over the algebraic closure ...
- 1,833
3
votes
0
answers
58
views
Is the isogeny class 1109.a of abelian surfaces in the LMFDB complete?
The LMFDB lists the Jacobian of the genus-2 curve 1109.a.1109.1 (http://www.lmfdb.org/Genus2Curve/Q/1109/a/1109/1) as being isolated in its rational-isogeny class. However, the LMFDB does not purport ...
- 61
3
votes
0
answers
112
views
Indecomposablity in purely inseparable extensions
Let $k$ be a field of characteristic $p$ (e.g. the separable closure of $\mathbb{F}_p(t)$) and consider the extension $k(x)/k$ where $x^p\in k$ but $x$ does not. Consider a (finitely generated) ...
- 31
3
votes
0
answers
176
views
Component groups of commutative group schemes
I'm interested in the following question.
Suppose $P$ is a smooth commutative group scheme over a global field $k$, such that $P$ is separated and locally of finite type.
Suppose, in addition, $P^0$ ...
user119470
3
votes
0
answers
255
views
What are the easiest counterexamples to Serre-Lang over non-algebraically closed fields?
Let $k$ be a field of characteristic zero. Let $A$ be an abelian variety over $k$. Let $X\to A$ be a finite etale morphism with $X$ a connected (smooth projective) variety over $k$.
Then, choosing a ...
- 161
3
votes
0
answers
307
views
Isotrivial factors of Jacobian
Let $k$ be an algebraically closed field of positive characteristic that it is not the algebraic closure of a finite field. Fix a smooth proper $k$-curve $C$ and write $J_C$ for its Jacobian abelian ...
- 728
3
votes
0
answers
113
views
Cohomologies of $[V/GL_n]$ in characteristic $p$ for a representation $V$ of $GL_n$
Let $V$ be a representation of $G=GL_n$ (or more generally any reductive group $G$) over an algebraically closed field $\mathbb k$ of characteristic $p$. Let $[V/G]$ be the corresponding quotient ...
- 790
3
votes
0
answers
111
views
Notation for theta divisors in _Tata Lectures on Theta II_
Is there anyone here who has worked through Mumford's Tata Lectures on Theta II and can help me with my difficulties in following what seems to be confusing and inconsistent notation?
I have had more ...
- 1,298
3
votes
0
answers
416
views
The final step in the proof of Neron-Ogg-Shafarevich as in the paper of Serre-Tate
I have been reading the paper "Good Reduction of Abelian Varieties" of Serre-Tate and in particular the part where they show that the Tate module being unramified implies good reduction.
As in the ...
- 7,746
3
votes
0
answers
231
views
Automorphisms of Jacobians and polarizations
In a question previously asked on MO (Units of Endomorphism Rings of Jacobian Varieties with Real Multiplication), a result from the paper ``On the fields of rationality for curves and for their ...
- 521
3
votes
0
answers
405
views
Jacobians of curves with maximal Picard number
What can be said about a complex curve $C$, if its jacobian $J(C)$ has the maximal Picard number?
It is natural to expect that for a general curve of given genus its Jacobian has Picard rank 1 (isn't ...
- 1,170