# 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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### When do equations represent the same curve?

Suppose we have two sets of parametric equations $\mathbf c_1(u) = (x_1(u), y_1(u))$ and $\mathbf c_2(v) = (x_2(v), y_2(v))$ representing two 2D planar curves. When I say "2D planar curves" I mean ...
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### Rational map on smooth projective curve

Let $f:C \rightarrow C'$ be a rational map, here $C$ and $C'$ are smooth projective curves. I cannot understand how is $f$ a morphism? (This is lemma from book by Klaus Hulek) Thanks
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### Prove that a set in $\mathbb R^3$ is not an algebraic set

I want to prove that the set $\{(\cos(t),\sin(t),t)\in A^3(\mathbb R); t\in \mathbb R \}$ is not an algebraic set. I already proved that the set $\{(\sin(t),t)\in A^2(\mathbb R);t\in \mathbb R \}$ ...
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### Intuitive explanations for the concepts of divisor and genus

When trying to explain AG-codes to computer scientists, the major points of contention I am faced with are the concepts of divisors, Riemann-Roch space and the genus of a function field. Are there any ...
11k views

### How does one calculate genus of an algebraic curve?

I've been reading about parametrization of algebraic curves recently and the idea of the "genus of a curve" appears quite often (my impression is that a curve is parametrizable exactly when it has ...
328 views

### Transforming quadratic parametric curve to implicit form

I have a algebraic parametric curve $$\mathbf{p}(t) = (x(t), y(t))$$ where $x$ and $y$ are both polynomials of degree $\leq p$. Now, I want to find the implicit form $f(x, y) = 0$. A document I'm ...
149 views

### All polynomial parametric curves in $k^2$ are contained in affine algebraic varieties

I have started working through the textbook Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea and I am stuck on one part of an introductory question. The question begins by getting one to ...
248 views

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### Cardinality of the Fiber of a Finite Morphism Vs. Degree (via Vakil)

I would like to show the following (note: it is not an assigned problem, so it may be false) (EDIT: Indeed it is false, see end of post): Suppose $f:X \rightarrow Y$ is a finite, surjective morphism ...
3k views

### Rational parametrization of circle in Wikipedia

In http://en.wikipedia.org/wiki/Circle but also in the corresponding article in the German Wikipedia I find this formulation ( sorry, I exchange x and y as I am accustomed to it in this way ) : "An ...
340 views

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### Is there a name for the curve $t \mapsto (t,t^2,t^3)$?

Is there a name for the curve given by the parametrization $\{(t,t^2,t^3); t\in\mathbb R\}$? Here is a plot from WA. An another plot for $t$ from $0$ to $1$. This curve is an example of a subset of ...
375 views

### Normal bundle of twisted cubic.

Let $C$ be a twisted cubic in $\mathbb P^3$. I'd like to compute the splitting type of normal bundle $N_{C/\mathbb P^3}$? I understood that $T_{\mathbb P^3}|_C=\mathcal O(4)^{\oplus 3}.$ So we have an ...
822 views

### Roadmap to Riemann hypothesis for curves over finite fields

I am a beginning graduate student with (almost) no background in algebraic geometry. I would like to learn the proof of the Riemann hypothesis for curves over finite fields, including all ...
1k views

### The canonical divisor of the projective line

Let $A$ be a ring. Assume it has some nice properties if necessary, e.g., $A$ is a Dedekind domain. Let $\mathbf{P}^1_A$ be the projective line over $A$. I want to show that $-2 [\infty]$ is a ...
449 views

### Importance of Riemann-Roch theorem

I read yesterday the statement of Riemann-Roch theorem and I didn't actually detect the huge importance that anyone tells me it has... So, can anyone provide me with some examples or reasons for being ...
Let $C=V(f)\subset \mathbb{P}^2$ be a smooth plane algebraic curve of degree $d$. For all $x\in \mathbb{P}^2\setminus C$ there are at most $d(d-1)$ tangent lines to $C$ passing through $x$. ...