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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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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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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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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10 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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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 ...
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44 views
+100

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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24 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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32 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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algebraic varieties with log terminal singularities

I am looking for some non-trivial examples of algebraic varieties which have log-terminal singularities.
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$(-1)$-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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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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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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17 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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16 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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20 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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15 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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47 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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37 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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30 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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52 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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47 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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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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30 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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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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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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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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3 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
26 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 ...
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66 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 ...
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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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28 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 ...
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3answers
207 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 ...
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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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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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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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Determine the number of real roots of the system.

Determine the number of real roots of the system,$1.$$x^3y - y^4 =a^2$ $2.$$x^2y+2xy^2+y^3=b^2$ where $a$ and $b$ are real parameters.
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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 ...
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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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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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2answers
64 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 ...
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Algebraic curves, problem of cubics, with intersection, parabolic branch…

Find a cubic that intersect V(X^2-XY+Y^3)with at least multiplicity 6 in the origin, and besides it has an horizontal parabolic branch and the point (0,1) belongs to that cubic. First of all, to get ...
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When are the coordinates of the intersection points of plane curves actually algebraic conjugates

Suppose $f(x,y)\in\mathbb{Z}[x,y]$ is an irreducible polynomial defining a plane curve. Say I want to find the intersection of this plane curve with the line defined by g(x,y)=y-ax+b. One way to do ...
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An Estimation for Multiple Points in Fulton's Curve Book

I am learning basic algebraic geometry by following Fulton's "Algebraic Curves". In 5.4, the auther stated the following theorem: Theorem If $F$ is an irreducible projective plane curve ...
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Isomorphism between $Pic_X[2]$ and $(\mathbb{Z}/2\mathbb{Z})^{2g}$

Let $X$ a Riemann surface of genus $g$. How could I prove that there is an isomorphism between $Pic_X[2]$ (i.e the line bundles $L$ such as $L^2 \simeq \mathcal{O}_X $) and ...
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69 views

Embedding an affine curve in a proper curve.

I am trying to figure out the following problem from Q.Liu's algebraic geometry in chapter 4. Let $U$ be an integral affine algebraic curve over $k$. (a) show that there exists a proper curve $ ...
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Is it important to study plane algebraic curves before read Fulton's book

I'm studying Fulton's algebraic curves book and I would like to know how important study plane algebraic geometry before read Fulton's book. Example of books on this subject: Algebraic Curves - ...
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What does linearly equivalent mean in this context

I'm trying to understand this proof of Fulton's algebraic curves book page 107: I didn't understand what does linearly equivalent mean in this context and why this implies it suffices to show that ...
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Why this order is well-defined?

I'm trying to understand why this order defined in Fulton's algebraic curves book is indeed well-defined: I have two questions: Why $u\in \mathcal{O_P}(X)$ (he is implicitly assuming this fact so ...