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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Flexes of cubic curve

Which are the flexes of the cubic curve of Fermat $$x^3+y^3+z^3=0$$ at $\mathbb{P}^2(\mathbb{C})$ ? Could you give me a hint how we could find the flexes? Do we have to use maybe the following ...
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163 views

Does the projective Fermat curve have singular points?

I want to check if the projective Fermat curve, $$X^n+Y^n+Z^n=0, n \geq 1$$ has singular points. Could you give me a hint how we could do this? Do we have to check maybe if the curve is ...
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1answer
33 views

Definition of generic point over finite set

I am currently reading David Marker's paper "A Remark on Zilber's Pseudoexponentiation" (J. Symb. Logic 71 (2006), no. 3). He describes the axioms for Zilber's pseudoexponential fields. The Strong ...
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36 views

Number of rational points on a curve and genus of a curve

I've just started with algebraic geometry, so i apologize in advance if my question is too easy to show. Given is a curve $\Gamma $ in $\mathbb{P}^{2}(\mathbb{F_{q^{m}}})$ defined by ...
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79 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 ...
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68 views

Calculating of genus of a curve

Let $C$ be a curve over $\mathbb{F}_q$ in projective plane. So $C$ can be done as zeroes of some gomogeneous polynomial $\in \mathbb{F}_q[x,y,z]$ with degree $n$. Whether is there algorithm which is ...
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44 views

Intersection Number of $B = Y^2 - X^3 + X$ and $F = (X^2 + Y^2)^3 - 4X^2Y^2$ using the fact $I(P,F \cap B) = ord_P^B(F) $.

Firstly, I apologise that this question is specific to two polynomials. I understand that this post will not help a lot of people, and for that I am sorry. I will make it up to you all by anwsering ...
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67 views

Proof of Halphen's Theorem

I am struggling with details of the proof of Halphen's Theorem in Hartshorne's Algebraic Geometry (chapter 4.6, Proposition 6.1, page 349). The statement of the theorem is: A curve $X$ of genus ...
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28 views

could someone help me with this particular detail in this article?

I'm reading this article and I was stuck in this part: I didn't understand his trick computing the orders at $P$. Remark: He defines $D=\inf\{div(\omega_{g-1}),div(\omega_g)\}$, i.e., ...
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62 views

Fixed point of curve automorphism

Exercise I.F-8 from "Arbarello, Cornabla, Griffiths, Harris: Geometry of algebraic curves" states that for a complex algebraic genus $g$ curve and its automorphism $\varphi$ of order $n$ the number of ...
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67 views

Bounds on eigenvalues of Hecke operator on the Jacobian

Let $p$ be a prime not dividing $N$. Consider the Hecke operator $T_p$ on the Jacobian $\text{Jac}(X_0(N))$. I'm thinking of $T_p$ as coming from a correspondence. If I understand correctly, the ...
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62 views

When is an holomorphy ring a PID?

I posted this question on mathstackexchange but I realized it is probably more suitable for mathoverflow. I will use the notation and language of Stichtenoth, Algebraic Function Fields and Codes. ...
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50 views

When is a holomorphy ring a PID? [duplicate]

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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21 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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2answers
59 views

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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39 views

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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22 views

$V^T=V(F_1^T,\ldots,F_r^t)$

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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28 views

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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41 views

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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30 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
45 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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22 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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38 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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39 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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24 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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32 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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35 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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20 views

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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54 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 ...
2
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1answer
78 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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60 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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3answers
90 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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72 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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27 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
60 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
68 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
36 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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29 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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39 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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18 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 ...
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108 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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43 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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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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48 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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44 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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49 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
21 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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17 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 ...