# 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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### Is there something similar to a discriminant for algebraic curves of the form $y^2 = f(x)$ (similar to the discriminant of elliptic curves)?

Consider an algebraic curve $X$ defined by an equation $y^2 = f(x)$ where $f$ is a polynomial of degree $d \geq 3$ with coefficients in a field $k$. For $m = 2$ and $d = 3$, we can get an equation of ...
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### Is there a rational parameter for the cylindrical curve?

Is there a rational parameter for the cylindrical curve? \begin{align*} \left\{ \begin{split} x^2-y^2=2\\ y^2-z^2=3 \end{split} \right. \end{align*} It seems to have something to do with the topic of ...
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### Galois covers of curves of arbitrary degree.

For smooth projective algebraic curves over finite fields. Do they admit a finite Galois cover of any degree, that is not induced by just base extension?
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### Fulton's Algebraic Curves Exercise 7.2.3

I have a (hopefully) quick question about this. Fulton's Algebraic Curve, exercise 7.2.3 on page 85 says: Let $F$ be any plane curve with no multiple components. Generalize the results of this ...
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### Kodaira dimension of an elliptic curve is zero

Let $C$ be an elliptic cure, i.e. a projective curve over $\Bbb{C}$ having genus $1$. I'm trying to understand why the Kodaira dimension of $C$ is zero (according to this article). I know that the ...
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### Sketches of stable 6-pointed curves

I was wondering if I could ask about stable $n$-pointed curves (essentially intersecting copies of $\mathbb{P}^1$ with at least $3$ special points on each copy - these may be either a point of ...
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### Liu's “Algebraic Geometry and Arithmetic Curves” - Proposition 4.4. (Ch. 7) and Exercises 5.1.29 and 5.1.30

In Liu's book, Chapter 7, Proposition 4.4, the question is about closed embeddings (or: closed immersions) coming from very ample divisors. The theorems are quite well-known when the ground field $k$ ...
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### Computing basis for $\mathscr{L}(D)$, $D$ a divisor on a algebraic curve

In reading Hartshorne IV.1.3 on Riemann-Roch, I wonder if there is a general method (an algorithm) for finding a basis for $\mathscr{L}(D)$, where $D$ is a divisor on a (complete, nonsingular corve ...
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### Surjectivity of the degree homomorphism

Regarding the following question and discussion: Is the degree homomorphism $\text{deg}: \text{Pic}(X)\to \mathbb{Z}$ surjective? We agree that if $X$ is a curve over an algebraically closed field the ...
### Genus of $y^3=(x^2+1)^2(x^3-1)$
I'm trying to proving that the genus of the normalization of $C:y^3=(x^2+1)^2(x^3-1)$ is 4 by using the Riemann-Hurwitz formula on the projection map $\pi:C\rightarrow\mathbb{P}^1$ which sends \$[x:y:z]...