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 ...

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Why does this construction give a proper curve?

Let $k$ be algebraically closed. The claims is there is a functor $\{$ Finitely generated extensions of $k$ of transcendence degree $1$ $\} \rightarrow \{$ Smooth, connected, proper, integral curves ...
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85 views

(Reference Request) Desingularization of Fibrations

I am interested in morphisms from a projective surface to a projective curve $p: X \longrightarrow C$ such that the fibres are singular curves on $X$. In the best case scenario, I'd like to replace ...
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1answer
102 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 ...
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62 views

Help to translate this theorem to a more accessible language

I'm trying to understand the chapter 2 of this article. I'm stuck in this part: The theorem he mentioned is from this book and it is the following: I need help to translate this theorem to a ...
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rational quartic in $\mathbb{P}^3$

According to Hartshorne (exercise IV.6.1), a rational curve of degree 4 in $\mathbb{P}^3$ is contained in a unique smooth quadric surface. If this is the case, then it must define a divisor on it. My ...
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96 views

Why are $xy=0$ and $x^2-x=0$ not isomorphic?

I'm not sure how to rigourously back up my intuition that the curves given by $x^2-x=0$ and $xy=0$ in $\mathbb{A}^2_k$ are not isomorphic. If I denote the varieties as $V_1=V(xy)$ and ...
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95 views

The arithmetic genus of non-reduced curves

Let $(X,h)$ be a smooth projective variety, and let $C\subset X$ be a smooth rational curve. Then $C$ has arithmetic genus $0$. (That $p_a(C)=0$ is not important, just to fix ideas). But if I am ...
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30 views

Definition of multiplicity of intersection

I'm reading this paper and I don't know this definition in page 3: What is the definition of multiplicity of the intersection of a hyperplane $H$ at a point $P$ in a curve $X$? Remark: My only ...
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63 views

Computing these multiplicities

I'm trying to use some Algebraic Geometry techniques to check my understanding on them. I'm using the most stupid of all the examples: trying to compute the multiplicities of the intersections of the ...
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2answers
73 views

Why aren't those Cartier Divisors equivalent?

Please refer to Gathmann's notes http://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/main.pdf at Example 9.3.6 for context. It's trying to give an example that the map between $Div(X)$ and ...
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37 views

Fulton 8.17 ¿$\Gamma(X) = k$?

Problem 8.17 Let $X, Y$ be nonsingular projectives curves, $ f:X\to Y $ a dominating morphism. Prove that $ f (X)=Y $. Im trying to solve the problem 8.17 of Algebraic Curves Book of Fulton, there ...
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33 views

Rational functions over variety X

I 'm trying to solve this exercise of Fulton Algebraic Curves: Let $D=\sum n_P P$ be an effective divisor, $S=\{P \in X: n_P>0\}$, $U=X\setminus S$. Show that $L(rD)\subset ...
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47 views

Definition of intersection multiplicity of a curve with some hyperplanes

I'm studying the chapter 2 of this paper and I have the following doubt: What is the definition of intersection multiplicity of a curve $C$ with some hyperplanes at a point $P$? Remark: My only ...
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26 views

Loxodrome equation

On Mathworld web site, the loxodrome is defined in oblate spheroid coordinates (even though with a peculiar "minus" sign in z, see here http://mathworld.wolfram.com/SphericalSpiral.html) where they ...
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15 views

different data fitting methods

I have a list of 2-dimensional points and I want to know different techniques of arriving at an approximate analytic relation between y and x. Also important would be to understand how one technique ...
5
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1answer
109 views

Number of fibrations over a curve.

Fix a non-singular complex projective curve $C$. I would like to know how many non-singular complex projective surfaces $S$ have the following properties (up to isomorphism): There is a fibration ...
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1answer
48 views

Quaternions $\Leftrightarrow $ Rotations -Conceptual question

I have an angular velocity vector analogue defined as(called Darbaoux vector some times in curve) $ \omega(s)=\frac{\mathrm{d}\theta(s) }{\mathrm{d} s}=\delta\theta(s)_x \vec{i}+\delta\theta(s)_y ...
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1answer
34 views

Why this application is well-defined?

Let $C$ be a curve. An application $\phi:C\to \mathbb P^n$ is called regular in a point $P\in C$ if there are regular functions $f_0,\ldots,f_n$ defined in a neighborhood $V$ of $P$ in $C$ such ...
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50 views

$(-1)$-curves and base change along a field automorphism.

Suppose that $E\subseteq S$ is a $(-1)$-curve inside a non-singular complex projective surface. By a $(-1)$-curve $E$, I mean that $E\cong\mathbb P^1_\mathbb C$ and $E^2=-1$. Now consider a field ...
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48 views

Rotating Frames on Curve

We are describing a curve with a moving frame. Figure 1 says about the definition of curve Figure 1 Specifications and Proprieties of the curve We can make a rotation 3D Matrix $R(s)_{3 ...
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123 views

Canonical embedding

I'm reading the second chapter of this paper and I need help to understand what exactly this canonical embedding is: Remark: $C$ is a smooth non-hyperelliptic complete irreducible algebraic curve ...
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1answer
22 views

Question on a function defined on some plane curve.

Let $C$ be the plane curve defined by $y^d =f(x)$, where $f(x) \in \mathbb{C}[x]$ is a polynomial. Let $g(x,y)$ and $h(x,y)$ are polynomials relatively prime with each other. We consider $r(x,y) ...
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21 views

Quartic curves with four connected components

A quartic plane curve in $\mathbb{RP}^2$ can be defined by a quartic equation $F(x,y,z)=\sum a_{ijk}x^iy^jz^k$ with 15 coefficients. Now let's focus on smooth quartics that have a maximal number of ...
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1answer
42 views

Poles of functions defined on hyperelliptic curves

Consider the equation $y^2=P(x)$, where $P$ is a polynomial over a closed field $\mathbb{k}$ without multiple roots. Let $Y$ be the corresponding affine curve, $X$ - its nonsingular projective model. ...
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28 views

Suggestions for algebraic function fields papers

My professor of algebraic function fields class gave me a paper to make a project (give the proof details, fill some gaps, etc). As my previous question here suggests, the paper he gave me is hard for ...
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59 views

Find point on rotated curve

I have a curve $f(t)$ that has been rotated through an angle $\theta$, and also have defined a given offset $Y$ from the curve origin. Using the equation $Y=x*sin(\theta)+y*cos(\theta)$ which ...
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1answer
55 views

Books which defines higher differentials in algebraic curves context

I'm reading an article which mentions a lot about higher differentials: I don't know what is $\Omega^n(F)$, my background is just Fulton's Algebraic curves book which defines just $\Omega(F)$. I ...
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1answer
42 views

What kind of a curve can represent a physical trajectory

It is very well known that conics, spirals, etc. can represent a realistic trajectories of point particles. However, a physical trajectory can also intersect itself, have a cusp, and other kinds of ...
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1answer
65 views

Why this is true using Riemann-Roch theorem

Let $C$ be a curve of genus $g$ over an algebraically closed field $k$ and $K=k(C)$ the field of rational functions of $C$. Consider $P$ a point at $C$. What I know: For each $r\in \mathbb N$, we ...
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53 views

Pushing forward vector bundles on a plane curve via projection from a point

Let $C \subset \mathbb{P}^2$ be a smooth plane curve, $P \in \mathbb{P}^2$ is point not on $C$, consider projection from this point $$ \pi :\mathbb{P}^2 - \{P\} \to \mathbb{P}^1, $$ and restrict this ...
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1answer
33 views

The elements of the coordinate ring can not be regarded as functions (projective case)

I'm reading Fulton's algebraic curves and I have questions on page 46: 4 I know these fact are very basic, but I didn't understand why no elements of $\Gamma_h(V)$ can not be regarded as functions ...
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40 views

Siegel's theorem and singular curves

I notice that often Siegel's theorem (there are only finitely many integral points on a curve of genus greater than 0) is stated with the requirement that the curve be smooth. Other times the ...
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2answers
135 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) ...
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1answer
84 views

Loxodrome parametric equations

I have been trying to understand HOW one arrives at the equations $x=cos(t)cos(c)$ $y=sin(t)cos(c)$ $z=−sin(c)$ of the loxodrome. I can see that if the transformation to spherical coordinates is ...
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2answers
39 views

Create paramatric shape wihtout 'dents'

I am plotting a shape with the following equation $$\left\{ \begin{array}{c} x=r_{in} \cos(4 t)+r_{out} \cos(t)\\ y=r_{in} \sin(4 t)+r_{out} \sin( t) \end{array} \right. $$ Given various parameters ...
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Newton's Investigation of Cubics: Generalization?

I recently read about Newton's investigation of cubic curves, and how, like for quadratic curves we can classify them into parabolas, ellipses and hyperbolas, Newton was able to classify cubic curves ...
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8 views

solution to curve limits (concept question)?

I had a question about how the limits work in that 4pi would not give the correct circle distance. I understand that if it has a radius 1 that the distance would be farther but that is only for a ...
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1answer
39 views

Existence of a rational function on a nonsingular curve with a simple zero at P and order 0 at Q

Let $X$ be a nonsingular curve over an algebraically closed field $k$ (by curve over $k$, I mean an integral separated scheme of finite type over $k$ which has dimension 1). Let $P$ and $Q$ be ...
3
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2answers
81 views

What does it mean for a point to be “inside” a plane curve?

Suppose that we have a simple closed curve in the $xy$-plane: $f(x,y)=0$ (for example, $f(x,y) = x^2 + y^2 - 4$. If I give you the point $(x,y)$, is there an analytical method to determine whether ...
3
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2answers
243 views

Calculate arc length of a logarithmic spiral between two points.

Its hard for me to put into words so please bear with me. Given a line of a certain length, how could I calculate the the arc length of a logarithmic spiral given that it intersects the line at two ...
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32 views

Common zeros and GCD of polynomials

Facing another algebraic geometry problem: Let $p,q \in T[x,y]$. Prove the set $V(p,q)$ is finite if and only if set $V(GCD(p,q))$ is finite. ($V(p)$ of course meaning the subset of $A^2(T)$ where p ...
3
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3answers
786 views

Proof that two simultaneous line equations do not intersect?

Apologies if this isn't at the level of questions expected here! I've got two simultaneous equations to solve. (Equation 1): $ x y = 4 $ (Equation 2): $ x + y = 2 $ They produce the following ...
3
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49 views

Where do divisors on curves come from?

I've been learning about Riemann surfaces and the Riemann-Roch theorem, and I've been able to experience some of its great power. However, I'm curious as to what the history behind divisors on curves ...
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1answer
34 views

Decomposition into irreducible algebraic sets

I am facing following problem and would really appreciate anyone's help: I am to find decomposition of $V(x^2 - y^4, x^3 - xy^2 + x^2y^2 -y^4)$ into irreducible algebraic sets in $A^2( \mathbb{C})$. ...
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Singular Points on Irreducible Cubic Curves Defined over Not Necessarily Algebraically Closed Fields

Let $C$ be a cubic curve defined over a field $k$. Take, for example, an affine curve: $$ C = \{(x,y) \in k\times k : a x^{3} + b x^{2} y + c x y^{2} + d y^{3} + e x^{2} + f x y + g y^{2} + h x + i y ...
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2answers
76 views

Algebraic variety in $\mathbb{C^{2}}$

would somebody algebraic-geometry-savvy please help me with this problem, as I am pretty new to algebraic geometry: I have to find smallest algebraic variety (irreducible algebraic set) in ...
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30 views

What is the definition of Riemann surface of an algebraic function?

What does it mean by the Riemann Surface of a function $y=\sqrt{x^3}$? I saw how to use the cut and glue method to obtain a sphere where $y=\sqrt{x}$ can be defined. But I was not clear in what sense ...
2
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1answer
45 views

Solutions to a system of equations

I know that Bezout's theorem says that if you take two plane curves, then their maximal number of intersection points is the product of their degrees. However, assume that I have two irreducible ...
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48 views

A question about hyperelliptic curve

This question is from the Qing Liu's book Algebraic Geometry and Arithmetic Curves 7.4.10 Let P(t) $\in$ k[t] be a seperated polynomial of even degree $\geq$ 2 over an algebraically closed field ...
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71 views

$\dim (D-P)=\dim (D)-1$

I'm trying to prove this question: Let $D$ be a divisor in $F|K$ such that $\dim (D)\gt 0$ and $0 \neq f\in \mathscr L(D)$. Thus $f\notin \mathscr L(D-P)$ for almost all $P$. Then show that ...