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4
votes
0answers
171 views

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
2answers
244 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 ...
4
votes
1answer
149 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
1answer
127 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
1answer
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
1answer
249 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
1answer
267 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
3answers
601 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
1answer
326 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
1answer
586 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 ...
11
votes
4answers
452 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
0answers
167 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 ...
3
votes
0answers
255 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
1answer
410 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
2answers
782 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 ...