# Questions tagged [algebraic-curves]

for questions on one dimensional algebraic varieties over any field, including questions of moduli, and questions about specific curves.

751
questions

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### Under what conditions is the compositum of two rational cubic Kummer extensions a rational function field?

Let $k$ be an algebraically closed field of characteristic $p > 3$ and $F = k(x)$ be the rational function field in the variable $x$. Consider two Kummer extensions $F_1 = F(\sqrt[3]{g_1})$, $F_2 = ...

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39 views

### A question about principal divisors and poles [closed]

Let be $C$ a smooth curve, $P\in C$, $n\in\mathbb{N}$ and $f\in K(C)^*$.
If $f$ has a pole in $P$ of order at most $n$ then $div(f)\geq -n(P)$.
Is it true that if $div(f)\geq -n(P)$ then $f$ has a ...

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154 views

### Let $\phi$ be the Frobenius morphism; why is $1-\phi$ separable? [closed]

Let $C$ be an elliptic curve defined over $\mathbb{F}_p$. We know that the Frobenius morphism $\phi:C\longrightarrow C$ is inseparable, i.e, $K(C)/\phi^*K(C) = K(C)/K(C)^q$ is an inseparable extension....

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48 views

### Question about the Frobenius morphism [closed]

I read the proof of the Hasse Theorem at page 138 in "Arithmetic of elliptic curves" by J.Silverman and i don't understand why the Galois group of the extension $\overline{\mathbb{F}}_q/\mathbb{F}_q$ ...

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85 views

### Dimension of a linear system of divisors on singular curve

Consider an singular irreducible plane curve $C \subset \mathbb{P}^2_k$ of degree $d>1$ over algebraically closed field $k$ which is given as vanishing locus $C=V(f(x,y,z))$ of a $f \in k[x,y,z]$ ...

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99 views

### Monodromy group of the generic plane curve

Let's work over $\mathbb{C}$. The degree $d$ curves in $\mathbb{P}^2_{\mathbb{C}}$ are parameterized by a projective space $|\mathcal{O}_{\mathbb{P}^2}(d)|$. Let $U_d\subset |\mathcal{O}_{\mathbb{P}^2}...

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240 views

### Why $E_1(\mathbb{Q}_p)\cong\mathbb{Z}_p$

I read an article where it is said: $E_1(\mathbb{Q}_p)\cong \mathbb{Z}_p$ where $E$ is an elliptic curve over $\mathbb{Q}_p$ and $E_1(\mathbb{Q}_p)=\{P\in E(\mathbb{Q}_p):\tilde{P}=\tilde{O}\}$.
The ...

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93 views

### very ample line bundle on singular curve

Let $X$ be a singular reduced irreducible projective curve over $k$(algebraically closed).
How to show that if $X$ has arithmetic genus 1,then for any smooth (closed) point $p\in X$,$\mathcal{ O}(3p)$...

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**1**answer

190 views

### Surjectivity of the Abel-Prym map

It is well known that the Abel-Jacobi map restricted to $\text{Eff}_g(C)$ surjects onto the Jacobian $\text{Jac}(C)$, since every divisor of degree $g$ is effective.
Is there an analogous statement ...

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74 views

### Finding the nearest quadratic Bézier curve

Given a set of three-dimensional quadratic Bézier curves.
I'm looking for some analytical solution to find the nearest curve to an arbitrary point in space.
Example
I already have a brute force ...

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109 views

### singularities of the theta divisor $\Theta$

By $\Theta_{sing}$ we denote the singularities of the theta
divisor $\Theta$ of the Jacobian variety $J(R)$ of a compact Riemann surface
R of genus $g\ge 4$. Then
$$
\text{dim} \Theta_{sing}=
\{\...

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54 views

### What conditions are sufficient for two points to be independent in the Mordell-Weil group?

Consider a finite field $\mathbb{F}_{q}$ and an elliptic surface
$$
\mathcal{E}\!: y^2 + a_1(t)xy + a_3(t)y = x^3 + a_2(t)x^2 + a_4(t)x + a_6(t),
$$
where $a_i(t) \in \mathbb{F}_{q}[t]$. I am mainly ...

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189 views

### Linear system on singular plane curve

Let $C \subset \mathbb{P}^2_k$ an irreducible plane curve of degree $d >1$
over algebraically closed field $k$. That is $C=V(f(x,y,z))$ where $f \in k[x,y,z]$ homogeneous of degree $d$. Let $\{...

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votes

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106 views

### Supersingular elliptic curves and their automorphisms

If $E$ is a supersingular elliptic curve over a finite field of characteristic $p$, what is known about its automorphism group (i.e the stabilizer of a point in algebraic curves terminology).
Do all ...

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104 views

### For which (g,q) does there exist a supersingular curve?

We say a curve over a finite field $\mathbb F_q$ is supersingular if it's Frobenius eigenvalues (on $H^1(X,\mathbb Z_\ell)$) are of the form $q^{1/2}\alpha$ for $\alpha$ a root of unity.
As far as I ...

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113 views

### Separable morphism of curves

A proof from Janos Kollar's Lectures on Resolution of Singularities Kollar (p 37) works as follows:
Theorem 1.58 (M. Noether, 1871). Let $k$ be an algebraically closed
field and $C \subset \...

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votes

**1**answer

296 views

### Family of elliptic curves in $\mathbb P^3$

For points $p_1=[1,0,0,0], p_2=[0,1,0,0], p_3=[0,0,1,0]$, $p_4=[0,0,0,1]$ and $p_5=[1,1,1,1]$
in the projective space $\mathbb P^3$, Let $l_{ij}$ be the line through $p_i, p_j$.
Let
$$C=l_{12} \cup ...

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76 views

### Two components on the moduli of stable maps $\overline{\mathcal{M}}_{2,0}(\mathbb{P}^1,3)$

I have two questions adapted from Exercise 24.3.1 in the book Mirror Symmetry.
Consider the moduli of stable maps $\overline{\mathcal{M}}_{2,0}(\mathbb{P}^1,3)$, where a typical element is a degree $...

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152 views

### Is the Cassels “$x - \theta$” map algebraic in some sense?

Setup: Let $k$ be a field of characteristic $0$, let $f(x) \in k[x]$ be a monic separable polynomial of degree $n \geq 4$, and let $\theta$ denote the image of $x$ under the map $k[x] \to K_f := k[x]/(...

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134 views

### Smooth affine curve with no immersion in projective plane

(1) I am trying to find an example of a smooth affine curve $C$ over $k$ with no immersion $C \to \mathbb{P}^2_k$ (for me a curve is an integral separated dimension one scheme of finite type over $k$)....

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**1**answer

120 views

### Rational Peano curves

An rr function (i.e. rational rational function) is a quotient
$$ \frac fg\,:\, \Bbb Q\ \to\ \Bbb Q\cup\{\infty\} $$
such that $\ f,g\,\in\,\Bbb Z[X],\ $ where $\ g\ne 0.$
QUESTION Do there exist ...

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**1**answer

186 views

### The $S$-unit equation for functions on curves

Let $X$ be a smooth projective connected curve over a number field $k$, and let $S \neq \emptyset$ be a finite set of closed points of $X$. The curve $Y = X \setminus S$ is affine, and we denote by $R$...

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**2**answers

183 views

### Simple proof that the arithmetic genus is non-negative

I take an irreducible and reduced closed curve $C\subseteq \mathbb{P}^n$, defined over an algebraically closed field $k$ and define the arithmetic genus $p_a(C)$ as the integer such that the Hilbert ...

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179 views

### Modern reference for Andre Weil's 'Sur les courbes algébriques et les variétés qui s'en déduisent'

I'm currently interested in the cardinality of the set of values of a polynomial over a finite field.
I found a paper
Saburo Uchiyama, Sur le Nombre des Valeurs Distinctes d'un Polynome a ...

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**1**answer

87 views

### Can the dimension of Hom space between vector bundles on an algebraic curve predicted by Riemann-Roch type formula be the minimal possible?

Let us study vector bundles $E$ and $F$ on a smooth projective curve $C$. There is a Riemann-Roch type formula for the Euler characteristic $\chi(E,F)=dim\, Hom(E,F)-dim\, Ext^1(E,F)$ in terms of ...

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462 views

### Torsion in the jacobian of a super elliptic curve

Let $y^n = f(x)$ define a smooth projective curve $C$ over some field $k$ with $\deg f \geq n$ and odd and with $f(x)$ having no repeated roots. Let $J$ be the Jacobian of $C$ and $J[n]$ it's (...

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178 views

### A problem on curves of genus 3

Let $Y_0$ be a smooth projective complex curve of genus two
and $Y_1\to Y_0$ a non trivial etale map of degree two
(so $Y_1$ has genus three, and it is hyperelliptic). Call $\sigma$ the involution of
$...

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116 views

### non-singular divisors of the jacobian variety

Let $X$ be a smooth, projective curve of genus at least $4$. The well-known divisor $\theta$ of the associated Jacobian variety is $\mathrm{Jac}(X)$ is singular and also ample. The $\theta$ divisor ...

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54 views

### Given an abelian galois map of curves, what are the principal divisors on the source fixed by the galois group?

Let $f:X\rightarrow Y$ be an abelian galois map (not necessarily unramified) of nonsingular complete curves over algebraically closed $k$, where the order of the galois group $A$ is coprime to the ...

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**1**answer

302 views

### Very weak Riemann-Roch on curves (by J. Kollar)

I have a question on an unequality used in the proof of the Very weak Riemann-Roch on curves in Janos Kollar's Lecture on Resolution of Singularities (page 14):
1.13 (Very weak Riemann-Roch on ...

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votes

**2**answers

429 views

### Examples of plane algebraic curves

There are many interesting sequences of polynomials which contain
polynomials of arbitrarily high degree, for example classical
orthogonal polynomials. Most of them arise as characteristic polynomials
...

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55 views

### Glueing local systems over union of compact Riemann surfaces

Let $X,Y$ be two connected, non-singular compact Riemann surfaces such that $X$ intersects $Y$ transversely at two distinct points. Let $L$ be a $\mathbb{C}$-local system on $X$. Let $L'$ be the ...

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**1**answer

208 views

### Why does MAGMA claim that the automorphism group of a curve is trivial?

I have been trying to compute the Automorphism group of a curve using MAGMA with no success. This is what I have tried: I have tried to compute the Automorphism group of the curve $y^3=x^4-x$ and no ...

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105 views

### Unirationality of universal Jacobian over special strata of moduli space of pointed genus 3 curves

Let $M_{3,1}$ be the (coarse, non-compactified) moduli space of genus $3$ curves with a marked point over a field $k$ of characteristic zero. Throwing away the hyperelliptic curves, take the open ...

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43 views

### Largest collection of pairwise relatively prime polynomials with bounded individual degree

Suppose $\mathcal{P}$ is a set of polynomials in $\mathbb{F}_p[x_1, \dots, x_m]$ with the properties that
any $f\in \mathcal{P}$ has degree at most $d-1$ in each variable $x_i$, and
any distinct $f,g\...

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54 views

### What fraction of multivariate polynomials with bounded individual degrees are irreducible?

How many polynomials in $\mathbb{F}_p[x_1, \dots, x_m]$ with degree at most $d-1$ in each variable $x_i$ are irreducible?
Here $m$ and $d$ are positive integers, $p$ is a prime, and $\mathbb{F}_p$ ...

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72 views

### Integral kernels for geometric langlands

My apologies for the imprecise question(s), it should be clear enough that I´m a complete beginner in this subject.
The (de Rham) Geometric Langlands Conjecture over $\mathbb{C}$ takes as input a ...

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**1**answer

212 views

### Multisections of the universal curve

Fix some $g \geq 2$. Let $\mathcal{M}_g$ be the moduli space of smooth genus $g$ curves over $\mathbb{C}$. For some $d \geq 1$, let $X_{g,d} \rightarrow \mathcal{M}_g$ be the family whose fiber over ...

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87 views

### Embedding of a Compact Riemann Surface into a Projective Space

Trying to fill up details of a proof I learnt some time ago: $X$ a compact Riemann surface. We want to show that for large enough degree of a divisor $D$, the map
$X\rightarrow \mathbb{P}(H^0(X,\...

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**2**answers

1k views

### How did Riemann prove that the moduli space of compact Riemann surfaces of genus $g>1$ has dimension $3g-3$?

Consider the moduli space $M_g$ of compact Riemann surfaces (i.e., smooth complete algebraic curves over $\mathbb{C}$) of genus $g$ for some $g>1$. I'm interested in knowing how Riemann proved that ...

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100 views

### Eichler Shimura in higher genera

Let $\mathcal{M}$ denote the moduli stack of smooth elliptic curves over $\mathbb{C}$. There is a local system, $\mathcal{H}$, on $\mathcal{M}$ with fibre $H^{1}(E,\mathbb{C})$ at the $\mathbb{C}$-...

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114 views

### Geometric meaning of Koszul modules

Assume $C$ is a smooth projective curve and $L$ is a line bundle on it. We say $L$ is $p$-very ample if for any effective divisor $D$ of degree $p+1$, the evaluation map $H^0(C,L)\to H^0(C,L\otimes\...

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**1**answer

198 views

### Fundamental group of a smooth projective curve of char $0$

In this note of Akhil MATHEW, when he proves the fundamental group of a smooth projective curve over a algebraic closed field $k$ of characteristic $0$ admits $2g$ topological generators, there are ...

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132 views

### Genus two curves on abelian surfaces

Considering a smooth genus two curve $C_2$, let $J(C_2)$ be its Jacobian surface, and take $p \in J(C_2)$ an $m$-torsion point. Let $A = J(C_2)/Z_m$, where $Z_m$ acts by $x \mapsto x+p$. The image of $...

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76 views

### Points on hyperelliptic curves coming from an orbit of an algebraic group

Consider a hyperelliptic curve $C_F$ defined over $\mathbb{P}(1,1,g+1)$ by the equation
$$\displaystyle C_F: z^2 = F(x,y),$$
where $F \in \mathbb{Z}[x,y]$ is a non-singular binary form of degree $2g+...

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122 views

### Scheme-theoretic image and delta-invariants

Let $(X,o)$ be an affine, isolated, normal, Gorenstein singularity. Let $f$ and $g$ be two morphisms from $\mbox{Spec}(\mathbb{C}[[t]])$ to $X$ (also known as formal arcs) such that the closed point ...

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**1**answer

99 views

### Special secants to curves

Let $X\subset\mathbb{P}^n$ be a smooth nondegenerate (i.e. not contained in any hyperplane) curve over $\mathbb{C}$. Is it possible that every collection of $n-3$ points on $X$ lies on a $n-1$-secant (...

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51 views

### Spaces of Bordered (Ordinary, Spin- or Super-) Riemann Surfaces

It is known from the work of Deligne and Mumford that the "space" of punctured/marked Riemann surfaces is a Deligne-Mumford stack. I have few questions regarding similar statement for the spaces of ...

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75 views

### Normalization of affine curves in singular surfaces

Let $X$ be a normal, isolated surface singularity with $x_0 \in X$ the unique singularity. Let $C \subset X$ be a hyperplane section i.e., defined by a single equation. Denote by $n:\widetilde{C} \to ...

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76 views

### Infinitesimal neighbourhoods and simultaneous normalization

Let $B$ be a local, complete, integral $\mathbb{C}$-algebra of Krull dimension $1$ and $n:B \to \mathbb{C}[[t]]$ the normalization map. Given any local artinian $\mathbb{C}$-algebra $A$, we say that ...