0
votes
0answers
56 views

Do principally polarized abelian varieties enjoy a genus expansion?

This is a vague question from an interested outsider: It is well known that abelian varieties which arise as Jacobian of a curve (or a bit more general as Prym variety) are distinguished by the fact ...
2
votes
1answer
76 views

Units of Endomorphism Rings of Jacobian Varieties with Real Multiplication

Let $(A,a)$ be a principally polarised (with indecomposable polarisation) Abelian variety over $\mathbb C$. Assume that End(A) contains an order $R$ of a totally real number field of degree $>1$ ...
3
votes
3answers
312 views

General curves of genus 3 as plane sections of Kummer surfaces

Is it true that a general curve of genus 3 is a plane section of an appropriate Kummer surface in $\mathbb P^3$? By Kummer surface I mean image of a principally polarized Abelian surface w.r.t. the ...
10
votes
0answers
178 views

Infinitely many curves with isogenous Jacobians

Let $g\geq 4$. Are there infinitely many compact genus $g$ Riemann surfaces with (mutually) isogenous Jacobians? Does the situation change in positive characteristic?
7
votes
1answer
387 views

For which fields does the isogeny theorem hold

Let $k$ be a field. We say that the isogeny theorem holds over $k$ if, for any abelian variety $A$ over $k$, there are only finitely many $k$-isomorphism classes of abelian varieties $B$ over $k$ ...
5
votes
1answer
300 views

Genus 2 curves vs Abelian surfaces

In the Satake compactification of abelian surfaces we have the following degeneration of a family of abelian surfaces in $\mathbf{H}_2$ $lim_{t \to \infty}\begin{pmatrix} it & b \\\ b & ...
8
votes
5answers
657 views

Schottky locus in genus 2

Let $\phi_g : \mathcal{M}_g \rightarrow \mathcal{A}_g$ be the period mapping from the open moduli space of genus $g$ Riemann surfaces to the moduli space of $g$-dimensional principally polarized ...
9
votes
1answer
489 views

Can we always find a curve which doesn't have semi-stable reduction

Let $K$ be a number field and let $g$ be a positive integer. Does there exist a smooth projective geometrically connected curve $X/K$ of genus $g$ such that $X$ does not have semi-stable reduction ...
2
votes
0answers
227 views

quasi-trigonal curves

I have read in the literature about quasi-trigonal curves. Such a curve C is a hyperelliptic curve X with two points p,q identified (basically a pinch). They seem to be pretty important in the theory ...
14
votes
4answers
1k views

Which curves can be found on Abelian varieties ?

We know tha each genus 2 curve is embedded into its degree 1 Jacobian. Under which conditions on $C$, $A$, $g$ and $n$ is it possible for a genus $g$ smooth curve $C$ to be embedded in an Abelian ...