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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Invariant point (flex) invariant under projective transformations.

Let $C$ be a projective curve in $\mathbb{P}_2$ defined by a homogeneous polynomial $P(x, y, z)$ and let $\alpha$ be a linear transformation of $\mathbb{C}^3$. Let $Q$ be the homogeneous polynomial $Q ...
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20 views

Calculating eigenvalues of the induced action on $H^0(2 K_C)$

Given a (smooth) curve $C$ and an automorphism $\phi$ of $C$. In the first part of their paper On the Kodaira dimension of the moduli space of curves Harris and Mumford calculate the eigenvalues of ...
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27 views

field of rational functions of a curve

Let $C$ be the algebraic curve defined by the modular polynomial $\phi_N$ of order $N>1$ over the rational numbers, i.e. \begin{equation}C:=\text{specm}(\mathbb{Q}[X,Y]/\phi_N(X,Y)). \end{equation} ...
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vote
1answer
28 views

Benefit from local coordinates

I am reading Elliptic Curves by Anthony Knapp. Its the first time that I am dealing with local coordinates. In page 21 he introduces them as follows: Let $[x_0,y_0,w_0]\in \mathbb P_2(k)$ where $k$ ...
0
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1answer
24 views

Singular plane cubic curve birational to $\mathbb{P}^1$

Is it true that every singular plane cubic curve over an algebraically closed field is birationally equivalent to $\mathbb{P}^1$? I know that such a curve has to have only one singular point and that ...
2
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44 views

Divisor on curve of genus $2$

I suffer from lack of concrete examples in Algebraic Geometry, so I will appreciate it if somebody can help me in understanding a bit better this one: Let $\mathcal{C}$ be a genus $2$ curve ...
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29 views

Mapping a curve into projective space

Let $\mathcal{C}$ be a (smooth, complex, projective) genus 2 curve. Take two different points $p,q\in\mathcal{C}$ and let $K$ be the canonical divisor class. I know (by means of Riemann-Roch) that the ...
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20 views

Model for a Smooth Curve

If $K$ is a finite Galois extension such that $\mathbb{F}_q(x)$ such that $K\cap\bar{\mathbb{F}}_q = \mathbb{F}_q$ then there exists a smooth projective curve $C$ such that $\mathbb{F}_q(C) = K$. My ...
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1answer
35 views

Resolving a node singularity on a plane curve

So, I am trying to solve Exercise 1.5.6 b) on page 37 in Hartshorne's Algebraic Geometry. For completeness I will include the exercise in my post: If $P$ is a node on a plane curve $Y$, show that ...
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46 views

Genus of the product of two elliptic curves

In trying to understand the trichotomy of the genus of algebraic curves, I first consider the following two elliptic curves (over $\mathbb{Q}$), well-known to be of rank $2$, $ y^2 = x^3+17$ and $ ...
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1answer
35 views

Dual bundle of a stable vector bundle.

Given a vector bundle $E \to X$ over a complex curve $X$, let its rank and degree be $r$ and $d$ respectively. Then the slope of $E$ is defined to be $ \mu(E):= \frac{d}{r}. $ $E$ is defined to be ...
3
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0answers
36 views

Use of the Bezout's theorem in Abstract Algebra

The Bezout's theorem: Let $C$ and $D$ be two plane curves described by equations $f(X,Y) = 0$ and $g(X,Y) = 0$, where $f$ and $g$ are nonzero polynomials of degree $m$ and $n$, respectively. ...
0
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0answers
16 views

Consider the Plane Curve?

Consider the plane curve $$\gamma(t) = \left( \cosh(t) \cos(t), \cosh(t) \sin(t) \right), \;\; t \in \mathbb R.$$ Is $\gamma$ regular? If $\gamma$ is not regular, can you restrict the parameter ...
0
votes
1answer
18 views

Rate of Change of a Multivariable Function

The problem says, Find the rate of change of $$(x,y,z) = x/z + y/z$$ with respect to t along the curve $$r(t) = \sin^2{t}[ i] + \cos^2{t}[j] + 1/(2t)[k]$$ The answer is apparently ...
5
votes
1answer
81 views

‘Integral’ of a Weierstrass $ \wp $-function.

I'm revising for my finals and I've seen a question which asks: Is there a meromorphic function $f: \mathbb{C}/\Lambda \to \mathbb{P}^1$ such that $f' = \wp$? There is a hint which says consider the ...
12
votes
1answer
129 views

Maximum number of intersection points of two different Bernoulli lemniscates

What is the maximum number of intersection points of two different Bernoulli lemniscates in the real plane? (Of course two identical lemniscates share an infinite number of points.) Here are ...
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vote
1answer
33 views

Number of inflection points of an algebraic projective curve

I' m trying to prove that a curve in $\mathbb{P^{2}(\mathbb{C})}$ of degree $d$ has an infinite number of inflection points or it has at most $\le 3d(d-2)$ inflection points. Let be $C$ the curve and ...
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0answers
20 views

Reflexive sheaves on stable curves-II

This is an extension of Reflexive sheaves on stable curves. Let $C$ be a stable curve and $\mathcal{F}$ a reflexive sheaf on $C$ supported on the whole of $C$. Is the projective dimension of ...
3
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2answers
63 views

Rational parametrization of circle in Wikipedia

In http://en.wikipedia.org/wiki/Circle but also in the corresponding article in the German Wikipedia I find this formulation ( sorry, I exchange x and y as I am accustomed to it in this way ) : "An ...
9
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77 views

Multiplicity of point as a zero of polynomial.

Let $C \subset \mathbb{P}_2$ be a projective curve defined by a homogeneous polynomial $P(x, y, z)$, and let $L \subset \mathbb{P}_2$ be the line $\{z = 0\}$. Assume $L$ is not a component of $C$. If ...
2
votes
1answer
60 views

Laguerre's theorem on power of a point w.r.t. an algebraic curve

So on Wikipedia article for a power of a point there is a short section about Laguerre's theorem. The problem is, the article has no references, and whenever I'm trying to Google it the only things I ...
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vote
1answer
53 views

Euler characteristic of a singular fiber

I am trying to understand Kodaira's classification of fibers. In the table at page 41 of Miranda's book http://www.math.colostate.edu/~miranda/BTES-Miranda.pdf there is given the Euler number of the ...
3
votes
1answer
83 views

How to curve fit an unknown function?

I have data which can be described by $y=f(x,z)$ where $z$ varies from 170 ~ 154. Now values given by $ks$ are known sample values that equals value given in the table header, $uks$ are unknown ...
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46 views

Is the union of two projective curves in the projective plane a projective curve?

As the title suggests, is the union of two projective curves in the projective plane a projective curve? Any help would be appreciated, thanks.
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1answer
63 views

the Galois closure of $\mathbb{F}_4(x,y) / \mathbb{F}_4(x)$

I want to show that $\mathbb{F}_4(x,y) / \mathbb{F}_4(x)$ is NOT a Galois extension. Consider the field extension $\mathbb{F}_4(x,y) / \mathbb{F}_4(x)$ where $y$ is root of the polynomial $g(T) = ...
6
votes
0answers
36 views

Affine curve is union of $d$ lines through point of multiplicity $d$. [closed]

Let $C$ be an affine curve defined by a polynomial of $P(x, y)$ of degree $d$. Show that if $(a, b)$ is a point of multiplicity $d$ in $C$ then $P(x, y)$ is a product of $d$ linear factors, so $C$ is ...
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1answer
61 views

$\mathbb{F}_4(x,y) / \mathbb{F}_4(x)$ is a Galois extension, with $\mathbb{F}_4(x,y)$ a function field of an algebraic curve

Consider the field extension $\mathbb{F}_4(x,y) / \mathbb{F}_4(x)$ where $y$ is root of the polynomial $$ f(T)= x^4 + x^2T^2 + x^2T + x^2 + xT^2 + xT + T^4 + T^2 + 1 \in \mathbb{F}_4(x)[T]. $$ I ...
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votes
1answer
85 views

How can I get smooth curve at the sigmoid function?

I'm trying to implement the sigmoid curve by using the following function. A is 3.2505508013 B is 1.5223545069 and K is 0.56. ...
0
votes
1answer
28 views

What does the power of an ideal *mean*?

I am stumped trying to understand Silverman's definition of $\operatorname{ord}_P(f)$, the (normalized) valuation on $\bar K[C]_P$ (which denotes the localization of a curve $C$'s coordinate ring at ...
0
votes
1answer
34 views

Cuve length - Vector

If we have $C: r(t) = (t + t^2 , t^3, t^4)$, how can we then calculate an approximation of the curve length to $C$ by adding the length of the line connecting $0$ to $1,$ $1$ to $2$ and $2$ to $3?$ ...
0
votes
1answer
59 views

How to show that an object is a discrete valuation ring? (Fulton, Exercise 2.14)

I need some help to solve the following problem that appears on page 31 of the book of William Fulton entitled Algebraic Curves. Exercise : Let $ V = \mathbb{A}^1 $, $ \Gamma (V) = k[X] $, $ K = ...
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0answers
70 views

Understanding the Falting's Theorem

I'm an undergraduate student of mathematics, but soon I'll graduate, and as a personal project I want to understand Falting's Theorem, specifically I want to understand Falting's proof; but yet I have ...
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votes
0answers
26 views

$\mathcal{I}(\mathcal{Z}(f)) = \langle f \rangle$

Let $f(x) \in k[x]$. Show that $\mathcal{I}(\mathcal{Z}(f)) = \langle f \rangle$ if and only if $f$ is the product of distinct linear factors in $k[x]$. Here, $\mathcal{Z}$ is the zero locus and ...
0
votes
2answers
39 views

$V=\mathcal{Z}(xy-z) \cong \mathbb{A}^2$.

This question is typically seen in the beginning of a commutative algebra course or algebraic geometry course. Let $V = \mathcal{Z}(xy-z) \subset \mathbb{A}^3$. Here $\mathcal{Z}$ is the zero locus. ...
6
votes
1answer
63 views

Visual understanding for “the genus” of a plane algebraic curve

I am trying to understand the genus of an algebraic curve in the complex plane $\mathbb{C}P2$. I am looking for a visual or intuitive understanding. The difference between a sphere and a torus as a ...
0
votes
1answer
16 views

Relation between curvature of curve and dual curve?

For a plane algebraic curve, does there exist a relationship between the curvature of the curve and the curvature of its dual curve?
0
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1answer
31 views

What do the conics $x^2=-1$ and $0=1$ correspond to in $P_{\mathbb{R}}^2$?

I read this in Reid's book Undergraduate Algebraic Geometry. I don't even know whether my question is worded correctly. It says "In a suitable coordinate system, any conic in $P_{\mathbb{R}}^2$ is one ...
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1answer
24 views

A proof on the center of curves I am unsure of

Here is a proof in a book I am reading. It seems fairly short, but I kind of got lost. Especially when $\lambda$ was introduced. I usually get ideas after awhile of staring at it, but I am getting ...
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votes
0answers
27 views

Question about a notation of line bundle

If $C$ is a complex Riemann Surface with positive genus, $D$ a divisor on $C$, $L$ a line bundle of $C$, with the term $L(D)$ what do we mean? I have this idea: $L(D)$ set of all sections of $L$ ...
1
vote
2answers
109 views

Tangential space to the rational normal curve

Exercise 15.5 (Harris, Algebraic Geometry: A First Course): Describe the tangential surface to the twisted cubic curve $C \subset \mathbb P^3$. In particular, show that it is a quartic surface. What ...
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9 views

Parametrization of a Conic to Compute Multiplicity of Intersection

Given the two curves $C: y=x^2$ and $D: y=2x^2$ how can I use the parametrization to show that the multiplicity at (0,0) is 2?
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1answer
30 views

What points help identify a cubic and an exponential graph?

What are the different points on the graph we need to know which would help determine the equation of the curve. For example in a quadratic graph, we can determine the equation if we know either any ...
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22 views

Morphism from the projective line to an algebraic group

Let $F$ be a field (if require can assume of characteristic $0$) and $\mathbb{P}_F^1$ be the projective line. Suppose $G$ is a connected algebraic group over $F$. We denote the set of $K$-rational ...
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43 views

Suppose nine distinct points in $\mathbb P^2(\Bbb C )$ do not all lie on any one line and any line through two of them passes through a third. Show…

(From Kirwan's 'Complex Algebraic Curves', chapter 2 q10. Had a search around but can find no solution!) Show there's a projective transformation taking these points to the points $$\begin{array} \\ ...
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0answers
57 views

Do retractions exist only on rational curves?

I read the following in Eisenbud's Commutative Algebra with a view .... Let $k$ be an algebraically closed field. Recall that a retract is a morphism which is a retraction of the inclusion, and $X$ is ...
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1answer
54 views

Prove that an isolated point of $C: (f=0)\subset \mathbb{R}^2$ must be a max or min of $f: \mathbb{R}^2 \rightarrow \mathbb{R}$.

Let $f\in \mathbb{R}[x,y]$ and let $C: (f=0)\subset \mathbb{R}^2$; we say that $P\in C$ is isolated if there is an $\epsilon >0$ such that $C\cap B(P,\epsilon)=P$. Prove that if $P\in C$ is an ...
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1answer
39 views

Dual curve of the lemniscate of Bernoulli?

Does anybody happen to know the homogeneous equation $F(u,v,w) = 0$ for the algebraic curve dual to the lemniscate of Bernoullli: $$ F(x,y,x) = (x^2 + y^2)^2 - A (x^2 -y^2)z^2 = 0?$$ I looked at the ...
0
votes
2answers
211 views

Number of solutions of arithmetic funtion's equation.

Say, an equation is given below \begin{equation} 2\pi(x) - \pi(2x)=\omega(x) \end{equation} where $x$ is a positive integer, $\pi(x)$ is the prime-counting function, and $\omega(x)$ is the number of ...
0
votes
0answers
48 views

Why is $div(z)$ well-defined?

I'm reading Fulton's algebraic curves book and I didn't understand why divisors of $z \in K$ are well-defined (page 97): The problem he mentioned: I didn't understand why solving this problem we ...
1
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0answers
35 views

homomorphism inducing Galois cover

We are given a homomorphism $\rho: \pi_1(\Sigma_g - \{p_1,...,p_k\}) \rightarrow G$ , where $\Sigma_g$ is genus g Riemann surface and $p_i$'s are points on it, G is a finite group. Then it is claimed ...