7
votes
Accepted
Néron model, torsion and ramification
If, for example, $A[n]$ is already contained in $A(K)$, then there will be no ramification, regardless of whether or not $A'$ is an abelian scheme (although that may well force the special fiber to be ...
- 47.4k
5
votes
Voronoi and Delaunay
I'd recommend the following two books. In the first one there are some connections to toric varieties. (Chapter 9.3: Lattice polytopes and unimodular triangulations; also they deal with regular aka ...
- 6,337
5
votes
Do abelian varieties have Neron models over arbitrary valuation rings?
As a partial answer to your question, let me sketch a proof of the following result found in 2015 in collaboration with David Holmes.
Proposition. For a valuation ring $V$, every abelian $V$-scheme $...
- 5,009
4
votes
Accepted
Does torsor of an elliptic curve extend to torsor of its Neron model?
Any non-trivial $E$-torsor over $\mathbb{Q}_p$ will give you an example. Let me be more precise.
Let $E$ be an elliptic curve over $K=\mathrm{Frac}(R)$. Assume that $E$ has good reduction over $R$. ...
- 9,497
4
votes
Accepted
Surjectivity of specialization map
Since $\dim X_s = 1$, we have $H^2(X_s, \mathcal{O}_{X_s}) = 0$. Therefore, by deformation theory every line bundle on $X_s$ extends to a line bundle on the formal scheme $\widehat{X}$ (completion ...
- 16.1k
3
votes
Accepted
Complexification of Néron models of Abelian varieties
No. Take an abelian variety $A_0$ over $\mathbf{Q}$ and let $A = A_0\otimes K$ where $R = \mathbf{Q}[\pi]_{(\pi)}$ and $K = R[1/\pi]$. Then $K=K'$, $T'=0$, $N(A')_s^\circ = A\otimes R$. Then you are ...
- 16.1k
3
votes
Accepted
Extending line bundle to regular model
An similar, but slightly better approach is to first extend $\mathcal{L}_\eta$ to a coherent sheaf $\mathcal{F}$ on $S$ and then define
$$
\mathcal{L} := \mathcal{F}^{\vee\vee}
$$
to be the reflexive ...
- 39.3k
3
votes
Dimension of Zariski closure of a locally closed subscheme
Let $f: X \rightarrow Y$ be a proper dominant morphism between two locally Noetherian integral schemes.
By Stacks Lemma 02JX, we have $\dim{X}=\dim{Y}+\delta$, where $\delta$ is the transcendence ...
- 385
3
votes
Accepted
Néron models vs integral models
Note that sheaf pushforward and preheaf pushforward agree, so this a question about categories, not sites.
Lemma. If $X$ is a finite type $k$-scheme with $\dim X > 0$, then $j_*h_X$ is not ...
- 28.7k
3
votes
Surjectivity of map between Néron models $\mathcal{E} \to \mathcal{E}'$
I get that the Kodaira types of $E$ is II and for $E'$ it is IV${}^{*}$. This means that $\mathcal{E}$ is connected (or in simple terms no point over $\mathbb{Q}_3^{\text{unr}}$ reduces to the ...
- 8,945
2
votes
Do abelian varieties have Neron models over arbitrary valuation rings?
No, they needn't.
See: David Holmes: Neron models of jacobians over base schemes of dimension greater than 1., to appear in Journal fur die reine und angewandte Mathematik.
and
Giulio Orecchia: A ...
- 29
2
votes
Voronoi and Delaunay
This question is much too broad. However, a good introduction (to various generalizations, as well) is in Edelsbrunner's little book.
Edelsbrunner, Herbert, Geometry and topology for mesh generation.,...
- 96.4k
2
votes
Voronoi and Delaunay
Maybe this PhD thesis could help:
Anton, Francois. Voronoi diagrams of semi-algebraic sets. Diss. University of British Columbia, 2003.
PDF download.
"The theoretical purpose of this thesis ...
- 151k
2
votes
Accepted
Dimension of Zariski closure of a locally closed subscheme
I'll interpret your terminology "Dedekind scheme" to mean "regular integral locally Noetherian scheme of dimension one" (or dimension $\leq 1$ if you replace $d + 1$ with $d + \...
- 261
1
vote
Accepted
Descent for étale covers of proper regular models of elliptic curves
Okay let me try my hand at an answer. Specifically, I'll address the final question, which is whether there exists a unique descent of a geometrically connected finite etale connected cover $D_{\...
- 10.7k
Only top scored, non community-wiki answers of a minimum length are eligible