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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‘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 ...
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12 views

Intersection points of two Bernoulli lemniscates

What is the maximum number of intersection points of two Bernoulli lemniscates in the real plane? A Bernoulli lemniscate is a degree four curve with two nodes on the line of infinity in complex ...
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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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17 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
53 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 ...
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74 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 ...
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31 views
+50

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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1answer
52 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 ...
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1answer
79 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 ...
3
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45 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
61 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) = ...
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35 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
59 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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1answer
77 views

How can I get smooth curve at the sigmoid function?

I'm trying to implement the sigmoid curve by using the following. Then, I got the following curve. ![enter image description here][1] But As you can see, in the red circle. it might be not smooth. I ...
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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 ...
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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 $3?$ ...
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1answer
56 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
66 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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25 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 ...
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2answers
38 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. ...
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1answer
62 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 ...
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1answer
15 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?
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1answer
30 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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26 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 $L$ ...
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2answers
106 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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0answers
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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42 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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54 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
51 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
37 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 ...
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206 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 ...
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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 ...
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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 ...
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24 views

Lissajous Curve

$$ \gamma(t)= (x(t),y(t))=(sin(2t),sin(3(t)) $$ Justify that we can reduce the domain of study to [0, $\pi/2$], by specifying the necessary symmetries to obtain the whole curve. I'm not really too ...
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34 views

Number of parameters of genus 5 curves

I want to show that up to automorphisms, the number of parameters of the intersection of three general quadrics in $\mathbb{P}^4$ is exactly $12$. I want to do it using the tools of chapter $7$ by ...
4
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1answer
90 views

A cubic hypersurface in $\mathbb{P}^{4}$ that passes through $7$ points in general position with multiplicity $2$.

I am reading Rick Miranda's "Linear systems of plane curves". A cubic hypersurface in $\mathbb{P}^{4}$ that passes through $7$ points in general position with multiplicity $2$ is not expected to ...
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1answer
40 views

$d+1$ distinct points of a rational normal curve in $\mathbb{P}^{d}$ are linearly independent

Let $X\subset\mathbb{P}^{d}$ be a rational normal curve. After a change of coordinates, it is the image of the map: $\nu:\mathbb{P}^{1}\rightarrow\mathbb{P}^{d}, (a_{0}:a_{1})\mapsto ...
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97 views

Weil does not imply Cartier on variety $X$.

Show that the divisor $D$ defined by $a = b = 0$ in the variety $X \subset \mathbb{A}^4$ defined by $ad - bc = 0$ $($the cone on a smooth quadric surface$)$ is not locally principal. My attempt ...
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1answer
51 views

Proof of Proposition IV 6.1 (Hartshorne page 349)

There has already been a question on this exact Proposition in Hartshorne (Proof of Halphen's Theorem), but there was not a satisfactory answer, and I'll try to make more precise the source of my ...
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26 views

The function field of the quotient of an algebraic curve

Let $X$ an algebraic curve and $G$ a finite group acting effectively on $X$. Then, $G$ also acts on the function field of $X$ by: $g(f)(x)=f(g^{-1}(x))$. It's easy to see that it is an action. ...
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35 views

About Runge's method

I have been reading about some Diophantine equations (like Runge's theorem and Cassel's theorem) and in the text says that these theorems are solved using Runge's method, but it doesn't say what ...
2
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0answers
49 views

Effective divisor vs curve on surface

Hartshorne in his book, with the term "a curve $C$ on a surface $S$" (over an algebraically closed field $k$) means that $C$ is an effective divisor on $S$. So, can I conclude that a "a curve $C$ on ...
2
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2answers
38 views

Locally free sheaf generated by global sections and vanishing cohomology on curves

Let $C$ be a smooth projective curve. Let $\mathcal{F}$ be a locally free sheaf on $C$ satisfying $H^1(\mathcal{F})=0$. Is it then true that $\mathcal{F}$ is generated by global sections?
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1answer
44 views

What is the symmetry group of the circle as a complex curve?

In a complex (projective) plane CP2 using homogeneous coordinates $(x, y, z)$ what is the group of (projective) transformations that leave the complex circle $$x^2 + y^2 = R^2 z^2 $$ invariant as a ...
2
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3answers
67 views

Existence of covers for given genus and degree.

Suppose we are given a genus $g$ and degree $n$; under which circumstances is there a curve $C$ of that genus admitting a map of degree $n$ to $\mathbb{P}^1$? For example: If $n = 2g-1$ there is no ...
3
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0answers
50 views

Qing Liu's definition of an algebraic variety, a non-separated line

First, a little reminder. In Qing Liu's Book on algebraic Curves, algebraic varieties are defined as Let $k$ be a field. An affine variety over $k$ is the ...
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2answers
98 views

Relation between intersection and product of ideals

Let $C$ be a smooth projective (irreducible) curve in $\mathbb{P}^n$ for some $n$. Denote by $I_C$ the ideal of $C$. Let $g \in I_C\backslash I_{C}^2$, an irreducible element. Is it true that for any ...