# Questions tagged [algebraic-curves]

An algebraic curve is an algebraic variety of dimension one. An affine algebraic curve can be described as the zero-locus of $n-1$ independent polynomials of $n$ variables in affine $n$-space over a field. Examples include conic sections, compact Riemann surfaces and elliptic curves. Singularities of curves are extensively studied as a basic case in singularity theory. Via algebraic function fields and modular curves they have links to number theory.

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### Degree 1 divisor on a curve

Let $C$ be a smooth projective curve of genus $g$ over complex numbers. Is it true that any degree one divisor is linearly equivalent to $1.p$ for some point $p$? I know it's true for $\mathbb P^1$. ...
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### the local behavior of a function at a place and its relation to the concept of divisors

I am studying functions in the Riemann-Roch space $L(sP)$, with $s \in \mathbb{Z}^+$ and $P$ a place of degree $r > 1$. My focus is on the local behavior of these functions at higher-degree places. ...
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### Relative dualizing sheaf of the family $\varphi: C\rightarrow \mathbb{A}^1$ where $C$ is the vanishing locus of $xy=t^k$ in $\mathbb{A}^3$.

This is an exercise from the book Moduli of Curves. Let $C$ be the vanishing locus of $xy= t^k$ in $\mathbb{A}^3$. Define a morphism $$\varphi:C\rightarrow \mathbb{A}^1, (x,y,t)\mapsto t.$$ Then $C$ ...
1 vote
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### Rank of elliptic curves does not decrease under field extension?

Let $E/\Bbb{Q}$ be an elliptic curve. Rank of elliptic curve over quadratic extension $L=K(\sqrt{D})/K$ is calculated by a formula $rank(E/L)=rank(E/K)+rank(E_D/K)$(this is easy to prove). In ...
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### Smooth part of an Affine Plane Curve defined by irreducible polynomial is a Riemann Surface

Miranda states the following. Given an irreducible polynomial in $\mathbb{C}[x,y]$, the singular points of its locus of roots $X$ forms a finite set. If we delete these points of $X$ the the ...
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### Quotient curve of Fermat curve over a finite field (Ireland -Rosen exercise)

I'm thinking a exercise 12 of Ireland-Rosen "A Classical Introduction to Modern Number Theory" p.226 as follows: But I have no idea, could you give some hints starting from (a) ?
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### Can a polynomial with even exponents be expressed as a multiple of $x^3+y^3-1$

Given a polynomial in two variables $P(x,y)$ with even exponents, can there be another polynomial $Q(x,y)$ such that $P(x,y)=(x^3+y^3-1)Q(x,y)$? The original problem I am trying to solve asks: Is ...
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### Descartes folium

The geometry of Descartes' folium, $x^3+y^3=3axy$ has been well studied. Can someone tell me which geometric property characterizes the following cubic curve: $$bx^3+y^3=3axy$$ The previous curve is a ...
1 vote
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### Necessary and sufficient conditions for the existence of a bitangent

For a 2D curve defined over a limited range, is the existence of a region over which the sign of the curvature is the opposite from the rest of the range both necessary and sufficient to prove the ...
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### Maximal number of multiple points for an irreducible quartic

I was working on this problem, but I don't see how I can solve it. I was given a hint, but I don't know how to use it. Can anyone help me? Thanks in advance! Let $f \in \Bbb C[x_0, x_1, x_2]$ be an ...
1 vote
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### Proof of Belyi's theorem with additional requiremt that ramification indices above 1 are equal to 2

I'm trying to prove an exercise problem from Szamuely's "Galois Groups and Fundamental Groups", which is a version of belyi's theorem. The exercise in question The normal version without the ...
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### Polynomial parametrization and degree nominator of parametrization of a rational curve

Let $\mathcal{C}\subset\mathbb{R}^2$ given by $f(x,y)=0$ be a rational curve. Then, there exists a rational parametrization $\phi(t)\,:\mathbb{R}\rightarrow \mathbb{R}²$. Under which circumstances is ...
1 vote