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Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

4 votes
0 answers
281 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 $\hat{A}$ …
2 votes
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 …
Damien Robert's user avatar
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 $\mathop{End}(A)$ …
Damien Robert's user avatar
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 …
Damien Robert's user avatar
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 …
Damien Robert's user avatar
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 …