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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An alternative description of an holomorphic map associated to a complete linear system

I need an help with an exercise in Miranda's book "Algebraic curves and Riemann surfaces". More precisely is the exercise in Problems V.4 I. Given a Riemann surface $X$ and a divisor $D$ on $X$ ...
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1answer
101 views

Proof on page 215 of Miranda's book

At the page 215, Miranda says that the dimension of the fiber of the map: $$ \gamma: \{(X,D_{2g-1})\} \mapsto \{X_g\} $$ where $\{(X,D_{2g-1})\}$ is the space of the pairs with $X$ an algebraic curve ...
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1answer
78 views

Prove that the curvature of $\gamma$ is $\frac{\kappa_{\alpha}}{\sin^2\theta}$

Let $\alpha:I\to {\mathbb R}^3$ be a cylindrical helix with a unit vector $u$ such that $u\cdot T_{\alpha}$ is a constant for all $t\in I$. For $t_0\in I$, the curve ...
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3answers
233 views

Prove: $\kappa^2v^4=|\alpha^{''}|^2-(\frac{dv}{dt})^2.$

Given a regular curve $\alpha:\mathbb R\to {\mathbb R}^3$, Prove: $$\kappa^2v^4=|\alpha^{''}|^2-\left(\frac{dv}{dt}\right)^2.$$ ,where $\kappa$ is the curvature, $v$ is the rate of change of ...
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54 views

Examples of One dimensional fields

A one dimensional field $K$ over a ground field $k$ contains $k[x]$ for $x \in K \setminus k$ such that it is a finitely generated $k[x]$-module. The textbook I'm studying uses its geometric ...
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1answer
63 views

Prove the holomorphic line bundle $\lambda(p+q)$ is the dual of the natural projective bundle

Let $M=\mathbb{C}P^1$ be the complex projective space, $U_0=\{[z_0,z_1]:z_0\ne 0\}$, $U_1=\{[z_0,z_1]:z_1\ne 0\}$ be the coordinate charts and define ...
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1answer
92 views

Finding a curve that intersects with $V(X_{0} X_{1}^{3} + X_{1}^{4} − X_{2}^{4})$ under certain conditions.

Let $D=V(X_{0} X_{1}^{3} + X_{1}^{4} − X_{2}^{4})\subset\mathbb{P}_{\mathbb{C}}^{2}$ and $C=V(X_{0} X_{1}^{2} + X_{1}^{3} − X_{2}^{3})\subset\mathbb{P}_{\mathbb{C}}^{2}$. I have got that $C\cap ...
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60 views

A question about intersection number on surfaces

This question is from the Qing Liu's book: Algebraic Geometry and Arithmetic Curves, Exercise 9.1.6. Let $X\to S$ be an arithmetic surface and $X_s$ a closed fiber. Let $C_1,...,C_m$ denote the ...
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81 views

Help with the proof of Max Noether's Residue Theorem from Fulton's book

I'm having problems understanding one part of the proof of the Residue Theorem, on chapter 8 of Fulton's book Algebraic Curves, section 8.1 (http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf page ...
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1answer
112 views

Find cubic Bézier control points given four points

What I need is to generate an SVG file while having a series of (x,y) ready. P0(x0,y0) P1(x1,y1) P2(x2,y2) P3(x3,y3) P4(x4,y4) P5(x5,y5) ... I need to make a ...
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2answers
98 views

Singular Points on Algebraic Curves

What does it mean for a point on an algebraic curve (over any field) to be singular? Over $\mathbb{R}$ or $\mathbb{C}$ is means that all derivatives vanish, but how does this definition generalise to ...
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13 views

Determine the valuation of $\rho$ with $F = k(x,\rho)$ at the only place above $(\infty)$.

Let us assume that we have the following setup. Let $F=k(x,\rho)$ be an algebraic function field with $$f(x,y) = y^n+a_1y^{n-1}+\cdots+ a_iy^{n-i}+\cdots+a_n \in k[x][y]$$ irreducible in $y$, and ...
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1answer
56 views

suggestion for a book on algebraic curves

I don't know if i'm asking in the wrong place, but I'm studying the book Algebraic curves of Fulton, and having some problems understanding the final chapters (6,7 and 8). I've found a good help in ...
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0answers
98 views

Book/lecture notes on algebraic curves

Although there surely is plenty of references on MSE about algebraic curves, my need are very specific and so I will open this topic anyway. I follow this year a course on (hyper)elliptic curves ...
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1answer
40 views

What is the dimension of $A-B$, where $B$ is a subspace of $A$?

My question is really simple, what is the dimension of $A-B$, where $B$ is a subspace of $A$? this space is well-defined? I found this space in this paper on page 440: Following my calculations in ...
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1answer
89 views

The local rings of $xy=0$ and $xy+x^3+y^3=0$ are not isomorphic, but have isomorphic completions?

I know that if you have a commutative local ring $R$, and you take its completion $\widehat{R}$ the inverse limit of the $R/\mathfrak{m}^i$, you get another local ring. However, nonisomorphic local ...
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1answer
35 views

Problem involving distances on a parametrically defined curve

SO the problem is stated below: A curve is defined paramterically by: $$x(t)=a\cosh t$$ and $$y(t)=b\sinh t$$ where a and b are positive constants and $-\infty < t < \infty$ The expression ...
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2answers
126 views

The intersection points are collinear

It is given a hexagon inscribed in a conic section. I want to prove that the pairs of opposite site intersect at three points of the projective plane that are collinear. How could we do this? ...
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1answer
169 views

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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2answers
205 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
37 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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51 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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1answer
114 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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1answer
72 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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1answer
57 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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84 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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79 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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1answer
76 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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52 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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1answer
30 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
106 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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53 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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26 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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0answers
31 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
56 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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25 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
39 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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40 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
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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1answer
37 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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0answers
36 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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2answers
87 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 ...
3
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1answer
115 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
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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80 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 ...
4
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3answers
100 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 ...
4
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2answers
114 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 ...