All Questions
481 questions with no upvoted or accepted answers
2
votes
0
answers
194
views
Examples of semi-abelian schemes over a curve
Let $C$ be a nice curve, i.e. $C$ is a smooth, projective, geometrically integral scheme of dimension $1$ over a field $k$. For example, (assuming the characteristic of $k$ is neither 2 or 3) an ...
- 452
2
votes
0
answers
117
views
Splitting of prime and order of reduction of point of infinite order in an abelian variety
I have already asked this question on stackexchange without much luck. I apologize if the question is too trivial to be asked here.
Let $A$ be an abelian variety defined over a number field $K$, $P \...
- 443
2
votes
0
answers
96
views
Linear forms and the second Voronoi decomposition
This is not my area of expertise, so forgive me if the question is a bit naive. Given a collection of vectors $v_1,\ldots,v_d$ in $\mathbb{R}^n$ (with $d\geq n$), there is a corresponding set of ...
- 147
2
votes
0
answers
274
views
Generic rank of proper pushforward of the trivial line bundle
Given a proper surjective morphism $f:X\rightarrow Y$ where $X$ and $Y$ are smooth projective varieties. The proper pushforward $f_!$ is the homomorphism that sends the class of a coherent sheaf $M$ ...
- 5,901
2
votes
0
answers
175
views
Covering abelian varieties over finite fields with the product of curves
Question. Given an $n$-dimensional abelian variety $X$ over a finite field, is it possible to find smooth projective curves
$C_1,\ldots, C_n$ such that there exists a finite regular map
$C_1\times \...
- 5,901
2
votes
0
answers
61
views
The intersection form on a Jacobian
$\DeclareMathOperator{\End}{End}$
Let $J=\mathop{Jac}(C)$ be a Jacobian, $\Theta$ the theta divisor. Since it is principal, it induces a bijection between the Néron-Severri group $NS(J)$ and the ...
- 645
2
votes
0
answers
540
views
Polarizations in algebraic and symplectic geometry
In context of Abelian varieties there are a couple of equivalent ways to
introduce the polarization of a algebraic variety. One way is to
choose a line bundle $\mathcal{L}$ which satisfies certain ...
- 5,986
2
votes
0
answers
162
views
Faltings' height theorem for isogenies over finite fields
For an Abelian scheme over a ring of integers in a number field, Faltings has a theorem that describes how the Faltings' height changes through an isogeny. There are multiple references for this ...
- 7,746
2
votes
0
answers
176
views
Outer Galois representations and Tate modules of Jacobian varieties
Let $X$ be a proper smooth curve over a field $k$. Then we have an exact sequence of profinite groups
\begin{equation*}
1 \to \pi_1(X_{\overline k}) \to \pi_1(X) \to G_k \to 1,
\end{equation*}
...
- 988
2
votes
0
answers
100
views
Composing equal characteristic and mixed characteristic deformations
Suppose $k$ is an algebraically closed field of characteristic $p>0$ and $A$ is a $k$-point on a moduli space of certain objects over $k$. Suppose, moreover, that $A$ deforms to a $k[[t]]$-point $B$...
- 245
2
votes
0
answers
122
views
A Lefschetz style formula for the $\ell^\infty$ torsion of an Abelian variety over a finite field
Let $A/\mathbb F_q$ be an abelian variety over a finite field. Define $A_\ell = A[\ell^\infty](\mathbb F_q)$, the $\ell^n$ ($n\geq 0)$ torsion points defined over the base field. I can assume $\ell \...
- 7,746
2
votes
0
answers
167
views
Chow variety of 1-cycles on abelian surface
It is an easy exercise to show that on a K3 surface, a smooth genus $g$ curve moves in a $g$-dimensional linear system. Nearly the same exercise shows that on an abelian surface, the corresponding ...
- 1,701
2
votes
0
answers
195
views
Structure of non-big divisors in an abelian variety
Let $A$ be an abelian variety over $\mathbb{C}$. If $A$ has an effective non-big divisor, then $A$ is not simple. (In a simple abelian variety, every non-zero effective divisor is ample.)
What can ...
- 513
2
votes
0
answers
476
views
Uniqueness of theta divisor
Let $A$ be an abelian variety (at least over $\mathbb{C}$). Suppose we have two theta divisors $\Theta_1$ and $\Theta_2$ on $A$, which give two principal polarizations on $A$.
In general, are those ...
2
votes
0
answers
244
views
Standard application of Oort-Tate classification theorem
$\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\tors}{tors}$In Mazur's paper “Modular curves and the Eisenstein ideals”, on the bottom of page 159, it says that if $T$ is a open subscheme of $...
user141691
2
votes
0
answers
177
views
vanishing of higher algebraic de Rham cohomology and sheaves of differentials for singular curves
I'm looking for some results or references about de Rham cohomology of curves in less-than-optimal cases. The two vanishing results I care about are:
$H^{k}_{\text{dR}}(X) = 0$ for $k>2$, since $H^...
- 1,142
2
votes
0
answers
277
views
Which endomorphisms of the Tate module of an abelian variety are "algebraic"?
For an abelian variety $A$ over a field $k$ with characteristic different from $\ell$ and Galois group $G = Gal(\overline k/k)$, there is always an injective map of the form:
$$\mathbb Q_\ell\otimes ...
- 7,746
2
votes
0
answers
261
views
Vector extension for p-divisible group
Background:
I am trying to understand a proof of Messing's book at Page 120. My goal it to understand the universal vector extention.
Reference:
Messing, The crystals associated to Barsotti-Tate ...
- 397
2
votes
0
answers
141
views
Is there a way to explicitly find any rational $\mathbb{F}_p$-curve on the Kummer surface?
Consider a finite field $\mathbb{F}_p$ (where $p \equiv 1 \ (\mathrm{mod} \ 3)$, $p \equiv 3 \ (\mathrm{mod} \ 4)$), $\mathbb{F}_{p^2}$-isomorphic elliptic curves (of $j$-invariant $0$)
$$
E\!:y_1^2 = ...
- 1,833
2
votes
0
answers
409
views
Riemann Surfaces of Infinite Genus and Transcendental Curves
I'm a graduate student struggling to find a topic for his doctoral dissertation which hasn't already been explored. At present, my hope was to see if classical results about Jacobian Varieties, ...
- 1,284
2
votes
0
answers
297
views
Property of Complete Variety
I have a question about a step in the proof from Lang's "Abelian Varieties" (page 20):
By definition an abelian variety $A$ over field $k$ is a proper smooth $k$-group scheme that is irreducible.
...
- 5,986
2
votes
0
answers
247
views
field of definition of CM abelian varieties
When $A$ is a CM abelian variety of dimension $1$ (i.e., an elliptic curve), then we have a result that if it has CM by a maximal order then it has a model over a number field $F$ where $F$ is the ...
- 443
2
votes
0
answers
60
views
A conjectural formula for the "minimal degree function", $k\rightarrow d\min(k)$, attached to recursion, $f\rightarrow A(f)$, in char $3$
THE RECURSION: $f\rightarrow A(f)$
$A: t(\mathbb{Z}/3)[t^3]\rightarrow t(\mathbb{Z}/3)[t^3]$ is the $\mathbb{Z}/3$-linear map with $A(t)=0, A(t^4)=t, A(t^7)=t^4$, and $A((t^9)f)=(2t^9)A(f)+(t^3)A((t^...
- 5,422
2
votes
0
answers
89
views
Weil pairings on abelian varieties restricted to subgroups of a given order
Let $A$ be an abelian variety of dimension $g$ defined over a number field $K$. Suppose $A$ has a principal polarization and $\ell$ is a prime number. We have a Weil pairing:
$$
e_\ell: A[\ell]\times ...
- 389
2
votes
0
answers
263
views
Polarization of the Jacobian in Torelli's theorem
I'm studying an example in book Yuji Shimizu and Kenji Ueno. Advances in Moduli Theory. Translations of Mathematical Monographs, vol. 206, that shows the importance of isomorphism as principally ...
- 560
2
votes
0
answers
78
views
Singularities of quotient of a vector bundle by a lattice
Let $V$ be a complex vector bundle of rank $g$ over an open unit disc $\Delta$ and $H$ be a integral local system of rank $2g$ over $\Delta$ i.e., for every $t \in \Delta$, $H_t \cong \mathbb{Z}^{2g}$....
- 1,593
2
votes
0
answers
94
views
The quotient of a superspecial abelian surface by the involution
Let $E_i\!: y_i^2 = f(x_i)$ be two copies of a supersingular elliptic curve over a field of odd characteristics. Consider the involution
$$
i\!: E_1\times E_2 \to E_1\times E_2,\qquad (x_1, y_1, x_2, ...
- 1,833
2
votes
0
answers
198
views
schemes vs varieties in abelian varieties and maximal subscheme where line bundle is trivial
This is a very elementary question as I am just learning about abelian varieties by reading Mumford's book,
Let $X$ be a complete variety (irreducible and reducible over an algebraically closed field)...
- 785
2
votes
0
answers
98
views
Why do "large" opens of abelian surfaces have "small" canonical bundle?
Let $A$ be an abelian surface, and let $S$ be a finite set of points. Let $U=A\setminus S$. Note that $U$ is a "large" open of $A$.
Let $B\to A$ be a proper birational surjective morphism with $B$ ...
- 57
2
votes
0
answers
176
views
Trivial Tate modules
Let $A$ be an abelian group, and $p$ a prime.
I'll call $$T_p(A) := \text{Hom}_{\mathbf{Z}}(\mathbf{Q}_{p}/\mathbf{Z}_{p}, A).$$
If $A$ is finite, then $T_p(A)$ is trivial, but the converse is not ...
user119470
2
votes
0
answers
254
views
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
69
views
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
146
views
Are general ample divisors on abelian varieties smooth?
It is well known that in $\mathcal{A}_g$, the space of principally polarized abelian varieties, the general element has a smooth theta divisor (see Andreotti-Mayer, for example).
Now let $\mathcal{A}...
- 663
2
votes
0
answers
304
views
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
views
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
votes
0
answers
622
views
How do I check if an abelian variety is principally polarized?
Let $V$ be a complex vector space of dimension $g$, and let $\Lambda\subseteq V$ be a full rank lattice endowed with a Riemann form $E\colon \Lambda\times\Lambda\to \mathbb Z$. Then the pair $(V/\...
- 509
2
votes
0
answers
124
views
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
votes
0
answers
167
views
The scheme structure on the Hilbert scheme of an Abel-Jacobi curve
Let $C$ be a smooth curve of genus $g\geq 3$, embedded in its Jacobian $X=\textrm{Jac } C$ via an Abel map. Let $\textrm{Hilb}_1(X)$ be the Hilbert scheme of curves in $X$, and let $[C]\in\textrm{...
- 430
2
votes
0
answers
121
views
Direct factors of Jacobian
Is there any characterization of abelian varieties appearing as direct factor of the Jacobian of some curve?
Are there some special kind of abelian varieties that are known to be direct factor of ...
- 1,463
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
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
votes
0
answers
248
views
What are the point-like objects in $D^b(X)$ when $X$ is an abelian variety?
Let $X$ be a projective variety over an algebraically closed field $k$ and $D^b(X)$ be the derived category of bounded complexes of coherent sheaves on $X$. Let $S$ be the Serre functor on $D^b(X)$. ...
- 9,019
2
votes
0
answers
187
views
How can I find a basis of $H^0(Y, -K_Y)$ for the del Pezzo surface $Y$ of degree 5?
Consider the surface $X \subset \mathbb{P}^2_{x, y, z}\times\mathbb{P}^1_{t, s}$:
$$
s^3y^2 + t^3yz = (t+s)x^2 + tsxz + t(t^2 +s^2)z^2
$$
over a finite field $k = \mathbb{F}_{2^d}$, $\mathrm{gcd}(d, 6)...
- 1,833
2
votes
0
answers
148
views
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
votes
0
answers
137
views
Is a supersingular Kummer surface $k$-unirational in characteristic 2?
Let $k$ be a perfect field of even characteristic. Consider the simplest example of a supersingular genus 2 curve, i.e.,
$$
C\!: y^2 + y = x^5.
$$
By the article of J. S. Müller "Explicit Kummer ...
- 1,833
2
votes
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
votes
0
answers
128
views
Characters on lattices and isogenies of Abelian varieties
Let $V:=\mathbb{C}^g$ and $\Lambda \subset V$ be a lattice, i.e. a discrete subgroup of rank $2g$. Then $A:=V/ \Lambda$ is a complex torus of dimension $g$. We moreover assume that $A$ is algebraic, ...
- 66.3k
2
votes
0
answers
212
views
What is $\mathrm{Num}(X)$ for the canonical cover $X$ of a bielliptic surface $S$?
A bielliptic surface $S$ is a smooth projective complex surface of Kodaira dimension 0 with $h^1(\mathcal O_S)=1$ and $h^2(\mathcal O_S)=0$. It is well known that $S=(A\times B)/G$, where $G$ is a ...
- 2,108
2
votes
0
answers
112
views
Polarization of the Prym variety
Let $X\rightarrow Y$ a ramifield double cover of curves, $J_X, J_Y$ their jacobians, $P\subset J_X$ the Prym variety, for any line bundle $L$ on $X$ of degree $g_X-1$, denote by $\Theta_L$ the ...
- 1,891
2
votes
0
answers
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