The higher-genus-curves tag has no usage guidance.

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### Non-hyperelliptic families of curves with trivial Ceresa class (or Gross-Schoen class)

Suppose X/K is a curve over a field K, which we want to think of as non-algebraically closed, and let x be a point of X(K). The Ceresa cycle is defined as follows; you can embed X in Jac(X) by sending ...

**2**

votes

**2**answers

355 views

### Etale covers of a hyperelliptic curve

Let $X$ be a hyperelliptic curve of genus at least two.
Let $Y\to X$ be a finite etale morphism with $Y$ connected.Then $Y$ is a smooth projective connected curve.
Is $Y$ hyperelliptic?
More ...

**5**

votes

**1**answer

197 views

### Higher Weierstrass points on curves of genus 3

So this question is directly related to a comment made by David Mumford in his
Lecture 1 given at U. Michigan in 1974 entitled: What is a curve and how explicitly can we describe them ?
Mumford ...

**1**

vote

**1**answer

142 views

### Hodge bundle on F-curves

Let $\mathbb{E}\rightarrow\overline{M}_{g,n}$ be the Hodge bundle. Let us cosider an $F$-curve of type $\overline{M}_{1,1}\subseteq\overline{M}_{g,n}$. Is the degree of the restriction of $\mathbb{E}$ ...

**3**

votes

**1**answer

230 views

### Why these two propositions are equivalent ?

" Every curve of the n-th order is in a flat space of n dimensions or less "
and
" If there be a system of $n + m + 1$ quantities $x$ connected by $n + m - 1$ homogeneous equations ; and if this ...

**3**

votes

**1**answer

296 views

### Igusa invariants of genus 2 curves as Siegel modular functions?

Hi,
Are the Igusa invariants (defined in Igusa's oringal paper) also classical Siegel modular forms? I read from somewhere that
$\psi_4=\frac{1}{4}I_4, \quad \psi_6=\frac{1}{8}(I_2I_4-3I_6), \quad ...

**6**

votes

**1**answer

300 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 ...

**5**

votes

**3**answers

746 views

### Monodromy group of 1-dimensional families of hyperelliptic curves

If $f: \mathcal{C} \rightarrow \mathcal{P}_{2g+2}$ is a general family of hyperelliptic curves (defined over $\mathbb{C}$), we know that the algebraic connected monodromy group
$Mon^{0}$ of this ...

**2**

votes

**1**answer

343 views

### Families of three dimensional algebraic curves

Let's consider spatial algebraic curve $C\subset \mathbb P^3$.
How could I describe a family of such curves, for example the set of all curves genus $g$ passing through $k$ points?
I'd like to some ...

**0**

votes

**1**answer

750 views

### Genus of algebraic curves with unknown degree

I am not sure if this is a valid question but posting any way:
Say I am over $\mathbb{F}_{p}$ for a prime $p$.
I have a curve of form $x^{2} = f(y)$ where $f(y)$ has an unknown form (and hence ...

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votes

**4**answers

467 views

### Why does the parameterization (F:F':1) happen?

1) To parameterize the conic $x^2+y^2=1$, we can use $(x,y)=(\sin t,\sin't)$ ($\sin'$ meaning the derivative of $\sin$, namely $\cos$).
2) To parameterize an elliptic curve $y^2=4x^3-g_2x-g_3$, we ...

**0**

votes

**0**answers

187 views

### Does there exist a fibration of genus two over P^1 with only 3 singular fibres but two are semi-stable fibers for algebraic surfaces?

We will work over the complex numbers C.
there exist a fibration of genus two over P^1 with only 3 singular fibres but one is semi-stable fiber.
there not exist a fibration of genus two over P^1 ...

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**0**answers

297 views

### Does there exist a non-isotrivial fibration of genus two over P^1 with only 3 singular fibres of general type surfaces?

We will work over the complex numbers C.
This question is based on Beauville's article :
there exist a non-isotrivial fibration of genus 2 over P^1 with only 3 singular fibres.
but not know for ...

**2**

votes

**1**answer

452 views

### Does there exist a non-trivial semi-stable curve of genus >1 with only 4 singular fibres

This question is based on Beauville's article in Szpiro's asterisque Seminaire sur les pinceaux de courbes de genre au moins deux from 1986.
We will work over the complex numbers $\mathbf{C}$.
Let ...

**3**

votes

**2**answers

832 views

### The arithmetic of higher genus curves

Genus 0 curves are well understood in number theory. There is also are rich theory a bunch of conjectures about the arithmetic of elliptic curves.
This leads me to the question, what we know about ...