All Questions
Tagged with ag.algebraic-geometry arithmetic-geometry
1,279
questions
3
votes
0
answers
199
views
Is it always true that the complement of an ample divisor is affine?
Consider a proper and integral scheme $X\rightarrow\operatorname{Spec}(A)$ over a Noetherian ring $A$ and $D\in\operatorname{Div}(X)$ an effective ample Cartier divisor on $X$. Is it true that its ...
- 31
2
votes
0
answers
127
views
A relative cycle class map
Suppose I have a smooth projective morphism $p: X \to S$ between varieties, and a relative cycle $Z \subset X \to S$ which is assumed to be as nice as can be (rquidimensional with fibers of dimension $...
- 7,716
0
votes
0
answers
77
views
Potential typo in "Complete Systems of Two Addition Laws for Elliptic Curves" by Bosma and Lenstra
Here is a link to the article: https://www.sciencedirect.com/science/article/pii/S0022314X85710888?ref=cra_js_challenge&fr=RR-1.
Pages 237-238 give polynomial expressions $X_3^{(2)}, Y_3^{(2)}, ...
- 538
3
votes
1
answer
216
views
"General position" on $\mathbb{P}^1\times\mathbb{P}^1$
On $\mathbb{P}^2$ we have the notion of general positions: no 3 points on a line, no 6 on a conic, etc. In particular, blowing up points (up to 8) in general positions give ample anti-canonical class, ...
- 101
3
votes
0
answers
225
views
A Brauer group of a double covering of a "well-understood" variety
Let $k$ be a field (it is possible to assume that $k = \mathbb{Q}$ or $= \overline{\mathbb{Q}}$) and $X, Y$ nice varieties over $k$.
Let $f \colon Y \to X$ be a finite flat surjective morphism of ...
- 1,352
3
votes
0
answers
156
views
Computing basis of $\mathrm{Pic}(\bar{X})$ for a Del Pezzo surface
Say we are given a degree 2 del Pezzo $X$ given by $w^2=Q(x,y,z)$ where $Q(x,y,z)$ is degree 4. We can compute the exceptional lines by computing the 28 bitangent lines of $Q$ and look at the ...
- 31
2
votes
0
answers
207
views
Using the Dold-Thom Theorem to define \'etale cohomology
For reasonable spaces $X$, the Dold-Thom Theorem states that $\pi_i(SP(X)) \cong \tilde{H}_i(X)$ where $SP(X) = \bigsqcup_i \mathrm{Sym}^i(X)$. There is a purely algebro-geometric realization of this ...
- 7,716
1
vote
0
answers
251
views
A hard-Lefschetz theorem with torsion coefficients?
Let $X$ be a smooth projective surface over $\overline{\mathbb{F}_{q}}$. Let $\ell$ be a prime distinct from the characteristic.
Assume we have a Lefschetz pencil of hyperplane sections on $X$. Let $...
- 11
1
vote
0
answers
67
views
Simplicity of abelian varieties and localization
Let $A$ be an abelian variety defined over a number field $K$. Let $v$ be a place of $K$ and denote by $K_v$ the $v$-adic completion of $K$ with respect to $||\cdot||_v$.
Assume $A$ is simple, is it ...
- 2,113
3
votes
1
answer
139
views
Formal étaleness along Henselian thickenings
Assume that $f:X\to Y$ is an étale map between smooth varieties and $(S,I)$ is a Henselian pair. Let $\alpha\in X(S/I)$. Can we say that the lifts of $\alpha$ to $X(S)$ are in bijection with the lifts ...
- 65
0
votes
1
answer
336
views
Tate–Shafarevich group and $\sigma \phi(C)=-\phi \sigma(C)$ for all $C \in \operatorname{Sha}(E/L)$
$\DeclareMathOperator\Sha{Sha}\DeclareMathOperator\Gal{Gal}$Let $L/K$ be a quadratic extension of number field $K$.
Let $\sigma$ be a generator of $\Gal(L/K)$.
Let $E/K$ be an elliptic curve defined ...
- 1,467
4
votes
1
answer
467
views
Is there an elliptic curve analogue to the 4-term exact sequence defining the unit and class group of a number field?
Let $K$ be a number field. One has the following exact sequence relating the unit group and ideal class group $\text{cl}(K)$:
$$1\to \mathcal{O}_K^\times\to K^\times \to J_K\to \text{cl}(K)\to 1$$
...
- 221
3
votes
0
answers
261
views
Is the weight-monodromy conjecture known for unramified representations?
Let $X$ be a smooth proper variety over a number field $K$, $v$ a place of $K$ lying over a prime number $p \neq \ell$, and $V := H^n(X_{\overline{K}};\mathbb{Q}_{\ell})$. Suppose $V$ is unramified at ...
- 15.5k
1
vote
0
answers
222
views
Deformations over $A_{\inf}$
Setup:
Let $K$ be a perfectoid field of characteristic $0$ with tilt $K^{\flat}$.
Let $A_{\inf}=W(\mathcal{O}_{K^{\flat}})$ be the infinitesimal period ring.
Let $\mathcal{X}$ be a flat, projective $\...
- 282
2
votes
1
answer
286
views
An example of a geometrically simply connected variety with infinite Brauer group (modulo constants)
$\DeclareMathOperator\Br{Br}$Let $X$ be a smooth, geometrically integral, geometrically simply connected variety over a numberfield $k$. Is it possible to have $\Br(X)/{\Br(k)}$ being an infinite ...
- 137