Questions tagged [higher-genus-curves]
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17
questions
2
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Curves having only one linear system realizing its gonality
$\DeclareMathOperator\gon{gon}$Let $C$ be a smooth irreducible projective curve defined over complex numbers. Recall that the gonality of $C$, $\gon(C)$, is defined to be the minimal possible degree ...
3
votes
0
answers
58
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Is the isogeny class 1109.a of abelian surfaces in the LMFDB complete?
The LMFDB lists the Jacobian of the genus-2 curve 1109.a.1109.1 (http://www.lmfdb.org/Genus2Curve/Q/1109/a/1109/1) as being isolated in its rational-isogeny class. However, the LMFDB does not purport ...
3
votes
1
answer
252
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Higher Genus Surfaces That "Look Like" Genus-1 Surfaces
It is my understanding that a genus-$g$ Riemann surface has $2g$ independent cycles that satisfy the usual intersection rules:
$$a_i \cap a_j = 0$$
$$b_i \cap b_j = 0$$
$$a_i \cap b_j = \delta_{ij}$$
...
8
votes
0
answers
644
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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 ...
4
votes
2
answers
1k
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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 ...
7
votes
1
answer
342
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
179
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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
239
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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 ...
4
votes
1
answer
565
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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 \...
7
votes
2
answers
488
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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
1k
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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
377
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 "...
-1
votes
1
answer
797
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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 ...
12
votes
4
answers
613
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 ...
3
votes
0
answers
366
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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 ...
3
votes
1
answer
538
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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
1k
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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 ...