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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22
votes
3answers
2k views

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 ...
5
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1answer
889 views

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
5
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1answer
296 views

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 $\{(\sin(t),t)\in A^2(\mathbb R);t\in \mathbb R \}$ ...
33
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4answers
4k views

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 ...
23
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2answers
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 ...
5
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2answers
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 ...
2
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1answer
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 ...
7
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1answer
248 views

Nonsingular projective variety of degree $d$

For each $d>0$ and $p=0$ or $p$ prime find a nonsingular curve in $\mathbb{P}^{2}$ of degree $d$. I'm very close just stuck on one small case. If $p\nmid d$ then $x^{d}+y^{d}+z^{d}$ works. If $p\...
15
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3answers
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When is a Morphism between Curves a Galois Extension of Function Fields

My apologies if this question has already been answered somewhere on this site: when I searched, I could only find specific examples rather than the general question. It is known that the category of ...
21
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1answer
1k views

What is the best way to see that the dimension of the moduli space of curves of genus $g>1$ is $3g-3$?

This fact was apparently known to Riemann. How did Riemann think about this?
20
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1answer
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Proving that the genus of a nonsingular plane curve is $\frac{(d-1)(d-2)}{2}$

I'm studying from Joseph Silverman's book The Arithmetic Of Elliptic Curves and I'm trying to do as many exercises as I can. Right now I'm trying to do Exercise 2.7 from chapter II which reads as ...
23
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2answers
4k views

Explicit Derivation of Weierstrass Normal Form for Cubic Curve

In page 22-23 of Rational Points on Elliptic Curves by Silverman and Tate, authors explain why is it possible to put every cubic curve into Weierstrass Normal Form. Here are relevant pages: (My ...
15
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3answers
1k views

Applications of Belyi's theorem

Belyi's theorem (1979) is stated as follows: A smooth projective curve over $\mathbb C$ is defined over a number field if and only if there exists a finite morphism (of varieties over $\mathbb C$...
8
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1answer
1k views

Computing the monodromy for a cover of the Riemann sphere (and Puiseux expansions)

Given an irreducible polynomial $P(X,Y) \in \mathbb{C}[X,Y]$, one obtains by analytic continuation a Riemann surface $M$ with a branched covering $f \colon M \to \mathbb{P}^1_{/\mathbb{C}}$, ...
6
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2answers
426 views

Is the circle a rational curve and what is its function field?

It does seem like the circle ($S^1=\{X^2+Y^2=1\}\subseteq k^2$ for $k$ a field) is a rational curve: it has parameterization $X=2T/(T^2+1)$ and $Y=(T^2-1)/(T^2+1)$. On the other hand, we have a ...
6
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1answer
265 views

Why is the extension $k(x,\sqrt{1-x^2})/k$ purely transcendental?

Consider the function field $k(x,\sqrt{1-x^2})$ of the circle over an algebraically closed field $k$. Is $k(x,\sqrt{1-x^2})$ a purely transcendental extension over $k$? I'm curious because I was ...
4
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3answers
593 views

Compactness of Algebraic Curves over $\mathbb C^2$

I was reading through Kirwan's Complex Algebraic Curves and I've been stuck on the following exercise: Given a (non-constant) polynomial $P(x,y)$, show that the curve in $\mathbb C^2$ defined by $P(x,...
4
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0answers
1k views

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 ...
5
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2answers
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 ...
4
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3answers
340 views

Parametrization of the intersection between a sphere and a plane

I can't find a way to get the parametric equation $\gamma(t)=(x(t),y(t),z(t))$ of a curve that is the intersection of a sphere and a plane (not parallel to any coordinate planes). That is $$\begin{...
3
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1answer
304 views

Proper curves over some field are projective

I'm looking for a reference of the statement Let $X$ be a proper curve (scheme of dimension one) over the field $k$. Then $X$ is projective. There is a some kind of guided exercise in Liu's ...
3
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1answer
417 views

General surface no lines

I've been studying surfaces recently and I came across the following statement: A general surface $S \subset \mathbb{P}^{3}$ of degree $m \geq4$ contains no lines. Does anyone have any idea how to ...
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1answer
95 views

Measuring a curve with Dirac delta function.

Formally, if I want to measure the length of a closed curve $f(x,y) = 0$, I presumed I could write: $$ L = \int^\infty_{-\infty}\int^\infty_{-\infty} \delta( f(x,y) )\, dx\, dy, $$ but trying this ...
1
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2answers
354 views

Number of solutions of an arithmetic function's equation

Say, an equation is given below \begin{equation} 2\pi(x) - \pi(2x)=\omega(x) \end{equation} where $x$ is a positive integer, $\pi(x)$ is the prime-counting function, and $\omega(x)$ is the number of ...
13
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2answers
4k views

Geometric interpretation and computation of the Normal bundle

My question concern the definition, geometric meaning, and usage of the normal bundle in algebraic geometry. Let $X$ be a nonsingular variety over an algebraically closed field $k$, and $Y\subseteq X$...
21
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3answers
3k views

What is a local parameter in algebraic geometry?

Shafarevich offers the following theorem-definition: "At any nonsingular point $P$ of an irreducible algebraic curve, there exists a regular function $t$ that vanishes at $P$ and such that every ...
14
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2answers
918 views

Why is it so difficult to find beginner books in Algebraic Geometry? [duplicate]

I don't understand why it is so difficult to find a really beginner book in Algebraic Geometry. Let's take for example Fulton's algebraic curves: An introduction to Algebraic Geometry. I've already ...
11
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1answer
678 views

How to compute the order $\text{ord}_P (f)$ for $f \in K(C)$

First lets fix some notation. Let $C$ be a projective curve (i.e. projective variety of dimension 1) defined over a field $K$. Suppose that $P \in C$ and that $P$ is a smooth point. It is known that ...
12
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2answers
339 views

Can there be a point on a Riemann surface such that every rational function is ramified at this point?

Let $X$ be a compact connected Riemann surface, and let $S\subset X$ be a finite subset. Does there exist a morphism $f:X\to \mathbf{P}^1(\mathbf{C})$ which is unramified at the points of $S$? I'm ...
16
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6answers
3k views

Trefoil knot as an algebraic curve

Is the trefoil knot with its usual embedding into affine $3$-space an algebraic curve (maybe after extending scalars to $\mathbb{C}$)? Is there even some thickening to some algebraic surface? If not, ...
6
votes
2answers
882 views

Definition of intersection multiplicity in Hartshorne VS Fulton for plane curves

In exercise I.5.2 in Harshorne there is the following definition of intersection multiplicity for two curves in $\mathbb{A}^2$: \begin{equation} \mathrm{length}_{\mathcal{O}_P}\mathcal{O}_P / (f,g) \...
9
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1answer
870 views

Conic by three points and two tangent lines

With the exception of degenerate situations, a conic is uniquely determined by five points lying on it. Likewise, five lines tangent to a conic uniquely define that conic. With four points and one ...
8
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1answer
865 views

Threefold category equivalence: algebraic curves, Riemann surfaces and fields of transcendence degree 1

According to this article, The category of curves over the complex numbers is equivalent to two other categories: Riemann surfaces and fields of transcendence degree 1 over $\mathbb{C}$. But I ...
7
votes
2answers
1k views

Abstract Nonsingular Curves

In section I.6 of Algebraic Geometry, Hartshorne establishes a that every curve is birationally equivalent to a nonsingular projective curve. To do this, he defines for any given curve $C$ with ...
7
votes
1answer
528 views

Computing Riemann surfaces of a given algebraic function

I've never seen written in a book a way or an algorithm for computing Riemann surfaces of a given algebraic function. I would like to know how to construct such Riemann surface using intuitive cutting ...
5
votes
1answer
258 views

Divisor on curve of genus $2$

I suffer from lack of concrete examples in Algebraic Geometry, so I will appreciate it if somebody can help me in understanding a bit better this one: Let $\mathcal{C}$ be a genus $2$ curve (...
5
votes
1answer
417 views

ample sheaf which is not very ample

I'm having trouble understanding a remark in Hartshorne: Let $X$ be the nonsingular projective cubic defined by $y^2z = x^3 - xz^2$ and put $P_0 = (0,1,0)$. The claim is that $\mathscr{L}(P_0)$ is not ...
11
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2answers
210 views

Families of curves over number fields

Is there known a number field $K$ and a curve $F(x, y) \in K(t)[x, y]$ such that $F(x, y)$ does not have points over the field of rational functions $K(t)$ but for all but finitely many positive ...
9
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2answers
1k views

Elementary proof of the Hurwitz formula

I am aware of two forms of the Hurwitz formula. The first is more common, and deals only with the degrees. So if $f:X \rightarrow Y$ is a non-constant map of degree $n$ between two projective non-...
9
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1answer
1k views

A problem about the twisted cubic

I have some difficulty with the following problem: Let $f : k → k^3$ be the map which associates $(t, t^2, t^3)$ to $t$ and let $C$ be the image of $f$ (the twisted cubic). Show that $C$ is an ...
8
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1answer
2k views

Basis for the Riemann-Roch space $L(kP)$ on a curve

Let $C$ be a smooth algebraic plane complex curve of degree $d$ defined as the zero locus of a homogeneous polynomial $F$. Let $P\in C$. For a positive integer $k$ consider the divisor $$D=kP$$ ...
8
votes
4answers
456 views

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 ...
6
votes
2answers
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 ...
5
votes
1answer
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 ...
5
votes
1answer
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 ...
4
votes
2answers
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 ...
4
votes
1answer
224 views

Number of tangent lines to an algebraic curve passing through a given point

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 $x$. ...
4
votes
2answers
1k views

Canonical divisor on the symmetric product of a hyperelliptic curve

Let $C$ be a hyperelliptic curve of genus $g$ and let $S = C^{(2)}$ denote the symmetric square of $C$. Let $\nabla$ be the divisor on $C^2$ defined by $\{(P, \overline{P}) \mid P \in C\}$ where $\...
3
votes
1answer
531 views

What is a field of definition of a morphism?

An algebraic variety $X$ over a field $K$ is a $K$-scheme integral separated and of finite type over $K$. Definition: If $L$ is a subfield of $K$ we say that $X$ is defined over $L$ if there ...
3
votes
1answer
502 views

Tangent lines to a curve passing through a given point

Here is a cool exercise from Shafarevich's book (Ex. 7 in Chapter 1, Section 1): Given an irreducible (affine) plane curve $C$ over a field of characteristic $0$, and point $P$ in the plane, ...