Tagged Questions

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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When is a holomorphy ring a PID?

I will use the notation and language of Stichtenoth, Algebraic Function Fields and Codes. Let $F$ be a function field over a finite field $\mathbb F_q$, $S$ a non empty set of places (possibly ...
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12 views

Question about affine coordinate changes

Fulton in his book defines affine coordinate changes: I'm trying to prove the item (b) of this question: Let's prove using the induction suggestion. Suppose $V=V(F_1)$, where ...
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maximum of a 5th order bezier curve with restrictions

Say you have a Bézier Curve of the 5th order with restrictions on the Control points: P0 & P1 are on a horizontal line P2 & P3 are on a horizontal line P4 & P5 are on a horizontal line ...
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The definition of codimension

I reading this article and on the page 438 the author says: What is the definition of the codimension in this case? is the codimension of $\Omega^{n-1}(F-D)\omega_{g-1}+\Omega^{n-1}(F-D)\omega_g$ ...
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$V^T=V(F_1^T,\ldots,F_r^t)$

I'm reading Fulton's Algebraic Curves book on page 19 he defines $V^T$: I want to prove if $V=V(F_1,\ldots, F_r)$, then $V^T=V(F_1^T,\ldots,F_r^T)$, Is this true? I need help Thanks a lot!
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Why does Riemann-Roch theorem implies the following characterization of $\Omega^2(D)$?

I'm reading this article and I didn't understand this part in the second page of the second chapter: Why this is true using Riemann-Roch theorem? ($D\doteqdot ...
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33 views
+50

Why is this kernel isomorphic to $\Omega^{n-2}(E-2D)$?

I'm reading this article and I didn't understand the proof of the item (1) of this proposition on page 225 (see below): I have the following questions: I didn't understand how these inclusions ...
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29 views

The basis of this particular tensor product

I'm studying Fulton's Algebraic Curves book and he defines the module of differentials $\Omega$ in the following manner (R is a ring containing an algebraically closed field $k$): Let's define the ...
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1answer
36 views

Why can't we have the equality in Clifford's theorem

I'm studying this article and in the second page of the second chapter I didn't understand why we can have a strict sign $\lt$ instead of less equal sign $\le$ in Clifford's theorem. We know that ...
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18 views

Fubini-Study form and homology class of curve

bit of a computation question here. Let $C$ be a (smooth) curve in $\mathbb{C}$P$^2$ (or more generally $\mathbb{C}$P$^N$) of degree $d$. Then the homology class $[C]$ is $d \cdot ...
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1answer
36 views

How to find certain quadratic curves over $\mathbb{Q}$

Given a quartic curve C: $x^4+y^4=1$, how can I find a quadratic curve over $\mathbb{Q}$ intersecting $C$ at four points, while the intersection multiplicity of each point is 2?
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35 views

Help in this easy equivalence

If $C$ is a curve with genus $g$ and $k$ a field, I'm stuck in something I'm sure easy, I think I'm forgetting some basic things. Define $\Omega(D)=\{\omega\in\Omega;div(\omega)\ge D\}$ and ...
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1answer
22 views

$l(rP)\le l((r-1)P)+1$

If $C$ is a curve of genus $g$, I'm trying to prove the dimension of the divisor $rP$ associated to this curve is less or equal than the dimension of the divisor $(r-1)P+1$, where $r\in \mathbb N$. ...
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1answer
27 views

Why this dimension is $0$ using Riemann-Roch theorem?

If $C$ is a curve of genus $g$, I'm trying to prove the dimension of the divisor $(2g-1)P$ associated to this curve is $g$. I'm using the Riemann-Roch theorem which says: Let $W$ be a canonical ...
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33 views

Why does this map is well-defined?

I didn't understand this proof from Fulton's Algebraic curves book: Why $ord_P(f)\ge -r-1$ in order to this map be well-defined? Thanks
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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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46 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
49 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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1answer
55 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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2answers
64 views

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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3answers
82 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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2answers
56 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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25 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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1answer
57 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
58 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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1answer
33 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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1answer
26 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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30 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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0answers
12 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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13 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 ...
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87 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
34 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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1answer
42 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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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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2answers
33 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
18 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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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
32 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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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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52 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
48 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
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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1answer
54 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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49 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
28 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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34 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
78 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
30 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
32 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 ...