# Questions tagged [algebraic-curves]

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

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 (...