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Results tagged with ag.algebraic-geometry
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user 26737
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
15
votes
1
answer
2k
views
What are the local properties of schemes preserved under global sections?
$\newcommand{\Spec}{\mathrm{Spec}\ }$
Let $(P)$ be a property of rings. I call $(P)$ local when $(P)$ satisfy these two
conditions:
if $A$ is a ring satisfying $(P)$, then the distinguished rings $A …
- 1,176
4
votes
Motivation for the Jacobian Variety
As outlined by the other answers, the Jacobian $J_X$ of a curve $X$ defined over $\mathbb{F}_q$ indeed encapsulates all cohomology information of $X$. In particular one can read the zeta function $\ze …
- 1,176
8
votes
Defining isogenies over smaller fields
There is another obvious obstruction: by definition two twists are isomorphics above an extension, but are not isomorphic above their field of definition (and also not isogenous).
I believe that thes …
- 1,176
1
vote
Accepted
Unibranch points (definition for varieties over arbitrary field)
For a scheme $X$, say that $X$ is topologically unibranch at $x$ if $\mathop{Spec} O_{X,x}$ is geometrically unibranch (meaning that $O_{X,y}$ is geometrically unibranch at all generisations $y$ of $x …
- 1,176
4
votes
0
answers
236
views
The dual abelian scheme in derived algebraic geometry
$\def\Pic{\mathcal{Pic}}\def\Gm{\mathbb{G}_m}\def\Hom{\mathop{Hom}}\def\HOM{\mathcal{Hom}}$
If $A/S$ is an abelian scheme, the fppf sheaf $\Pic^0_{A/S}$ is representable by an abelian scheme $\hat{A}$ …
- 1,176
1
vote
Accepted
The size of endomorphism rings and the relation to ordinariness of Abelian surfaces
The general reference for this sort of questions is Waterhouse, Abelian varieties over finite fields. Your question is answered in: Theorem 7.2. If $A$ is ordinary (and simple), then $\mathop{End}(A)$ …
- 1,176