# Tagged Questions

**13**

votes

**1**answer

287 views

### Curves which do not dominate other curves

Let $g>1$ be an integer. Does there exist a (smooth projective) genus g curve $X$ which doesn`t dominate a curve of positive genus and genus smaller than $g$?
Surely such curves exist. Just take a ...

**10**

votes

**0**answers

158 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?

**1**

vote

**2**answers

328 views

### Equation for simple Jacobian of a genus two curve

Let $X$ be a curve of genus two over a field $k$ with a $k$-rational point. Let $J$ be the Jacobian of $X$.
Can we write down an explicit equation for the abelian surface $J$?
I know $X$ can be ...

**6**

votes

**1**answer

261 views

### Are ranks of Jacobians over number fields unbounded?

Fix a number field $K$.
Is the rank of $J(K)$ unbounded, where $J$ ranges over the Jacobians of all smooth, projective, geometrically connected curves over $K$?
Does there exist an integer $g$ such ...

**1**

vote

**1**answer

289 views

### Does a curve over a number field have a finite etale cover of given degree

Let $X$ be a (smooth projective geometrically connected) curve over a number field $K$ of genus $g\geq 2$. Let $d\geq 2$ be an integer.
Does there exist a curve $Y$ over $K$ with a finite etale ...

**1**

vote

**1**answer

229 views

### What is the reduction of this hyperelliptic curve

Let $K$ be a number field and $E/K$ an elliptic curve with equation $Y^2Z = X^3 +AXZ^2+BZ^3$ in $\mathbf{P}^2_K$, where $A,B\in K$.
Let $S$ be non-empty finite set of finite places of $K$ and suppose ...

**2**

votes

**1**answer

237 views

### Maps of algebraic curves (and their Jacobian)

When people consider a map $\varphi: C \rightarrow C$ between algebraic curves and they mention the "associated map" on the Jacobian of $C$. Which map do they mean? Do they mean $\varphi^{\*}$ or ...

**2**

votes

**1**answer

355 views

### Correspondences on curves and their induced maps on differentials?

How does a correspondence on an algebraic curve $C$ induce a map on $\Omega^1_C$? Apparently it passes through the Jacobian of $C$ but I don't quite understand it.
More specifically, I was reading a ...

**2**

votes

**0**answers

207 views

### Reference for Jacobians in characteristic $p$

I am looking for a basic reference for Jacobians of algebraic curves in characteristic $p>0$. I just want basic facts about the relation between the curve and its Jacobian.
I dont want to assume ...

**4**

votes

**4**answers

730 views

### The use of embedding a curve into its Jacobian

I'm looking for as many examples/applications as possible of the use of embedding a smooth projective geometrically connected curve $X$ over a number field $k$ with $X(k)\neq \emptyset$ into its ...

**8**

votes

**5**answers

636 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 ...

**2**

votes

**1**answer

278 views

### Do divisors of degree g with this property exist in general

I have the following question. It's a long shot, but worth the try.
Let X be a compact connected Riemann surface of genus $g\geq 2$. Does there exist an effective divisor $D$ on $X$ of degree $g$ ...

**4**

votes

**1**answer

275 views

### What is the mod l monodromy of a generic trigonal curve?

For a hyperelliptic curve H, the mod 2 monodromy is smaller than $GSp_{2g}(F_2)$ -- since the two torsion of the Jacobian H is generated by differences of Weierstrass point, the monodromy of a generic ...

**7**

votes

**0**answers

315 views

### Defining equations for hyperelliptic Jacobians in a neighbourhood of the identity

Let $X$ be a hyperelliptic curve of genus $g \ge 2$ over a field $k$ (of characteristic not 2, 3 or 5, if you like, but could be positive in general). Let $J$ be the Jacobian of $X$, thought of as ...

**13**

votes

**1**answer

743 views

### Construction of the determinant line bundle on the degree $g-1$ Picard variety

Consider $J^{g-1}$, the variety of degree $g-1$ line bundles on a compact Riemann surface of genus $g$. Recall that $J^{g-1}$ is a torsor for the Jacobian, thus has dimension $g$. We can produce ...

**5**

votes

**2**answers

349 views

### Fano 3-fold of degree 4

Let $X$ be the intersection of two quadrics in $P^5$. It is well known that the intermediate Jacobian $J(X)$ is isomorphic to $J(C)$ for a genus 2 curve, related to the pencil of quadrics whose base ...

**1**

vote

**0**answers

282 views

### Abel-Jacobi map for regular fibered surfaces.

Let $f:C\to S$ be a regular fibered surface where $S=Spec(R)$, $R=dvr$. Assume $C$ has smooth geometrically integral generic fibre $C_K$. We also assume the existence of a section $x\in C(S)$. Let ...

**14**

votes

**4**answers

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 ...