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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18 views

computing the full constant field of an algebraic function field

Let $K$ be a field such that char$(K) \neq 2$. Let $F=K(x,y)$ be an algebraic function field field of one variable $x$ where $$y^2 = f(x) \in K[x]$$. We want to compute the full constant field of $F$ ...
4
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0answers
21 views

Conic by three points and two tangent lines

With the exception of degenerate situations, a conic is uniquely determined by five points lying on it. Likewise, five lines tangent to a conic uniquely define that conic. With four points and one ...
2
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0answers
33 views

Rational functions over curves define regular maps?

I've a rather simple question that is strucking me, perhaps because I'm a newbie in algebraic geometry. If we have a projective irreducible non-singular curve $C\subseteq \mathbf{P}^n_k$ where $k$ is ...
2
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0answers
34 views

Dessins d'Enfants and Real Algebraic Curves

I wrote a thesis on the Grothendieck theory of Dessins d'Enfants (after some articles by Leila Schneps). In Shafarevich, vol.2, there's a section on real algebraic curves. Is it possible to formulate ...
1
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1answer
25 views

Mordell's theorem-Finitely generated abelian group

In my lecture notes we have the following: Mordell proved the following: Let $C$ be a nonsingular cubic curve with rational coefficients. Then the abelian group of rational points on $C$ is ...
1
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2answers
23 views

Equation of a non-singular cubic curve

The equation of a non-singular cubic curve in affine coordinates is $$y^2+a_1 xy+a_3 y=x^3+a_2x^2+a_4x+a_6 .$$ If $\text{ch } K \neq 2, 3$ then it is written $$y^2=x^3+ax+b .$$ Why do we write it ...
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1answer
68 views
+50

Program to find closest function to fit arbitrary data

I've wanted this for years, but have never come across anything; a program for Windows to find the closest function to fit arbitrary data. The data I feed it is simple: A table with two columns ...
0
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0answers
57 views

Curves that don't have lines as components

In my lecture notes we have the following: A point $P=\left [x, y, z\right ]$ of an algebraic curve $C_F=V(F)$ is called an inflection point of $C_F$ when $P$ is not a singular point of $C_F$. ...
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0answers
36 views

Principal divisor of rational functions over nonsingular curves and pullback

I'm studying the theory of divisor over algebraic varieties for a seminar and I came across a problem that I think I solved almost completely except for a point that I'm missing. Let be $k$ an ...
3
votes
1answer
53 views

Computation of the global sections of a normal sheaf

Let $Y\subset X=\mathbb{P}^r$ be the image of the Veronese embedding $\mathbb{P}^1\rightarrow\mathbb{P}^r$. I want to calculate $dim$ $H^{0}(C,\mathcal{N}_{Y|X})$, where $\mathcal{N}_{Y|X}$ is the ...
0
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0answers
14 views

Points at infinity correspond to asymptotic slopes

Let $ P^2\mathbb{C} = \{ [a, b, c] | a,b,c \in \mathbb{C}^* \} $ the complex projective plane. So $ [a,b,c] \sim [x,y,z] $ iff $ \exists \lambda \in \mathbb{C}^* \colon \lambda(a,b,c) = (x,y,z) $. In ...
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0answers
10 views

The Cusp $w^2 + p(z,w)=0$ is desingularizable in the origin $O \in \mathbb{C}$

I have just studied a method in projective geometry over complex numbers on how to desingularize a curve in a point but i'm a little bit confused. I don't know the name of this classical method in ...
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1answer
26 views

Property of the intersection multiplicity: $I(P, y \cap x)=1$

How can we show the following property of the intersection multiplicity? $$I(P, y \cap x)=1 , \text{ where the point } P \text{ is at } (x, y)$$ Edit: My try: $$I(P, f \cap g ) \geq m_P(f) ...
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0answers
15 views

Properties of intersection multiplicity

I am reading the properties of the intersection multiplicity and in my lecture notes there is the following: We have $f(x, y) \in \mathbb{C}[x, y], g(x, y) \in \mathbb{C}[x, y]$ and $P=(a, b) \in ...
1
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2answers
23 views

How is the resultant defined?

In my lecture notes we have the following: We have that $f(x, y), g(x, y) \in \mathbb{C}[x, y]$ $$f(x,y)=a_0(y)+a_1(y)x+ \dots +a_n(y)x^n \\ g(x, y)=b_0(y)+b_1(y)x+ \dots +b_m(y)x^m$$ The ...
2
votes
1answer
23 views

Find the singular point

Let $f(x, y)=(x-y)^2$. We want to find the singular points. We do the following: Let $P=(a, b)$ be the singular point. $$f(a,b)=0 \Rightarrow (a-b)^2=0 \Rightarrow a=b \\ ...
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0answers
29 views

Irreducible components of the curve-Algebraic set

In my lecture notes I have the following: $f, g \in \mathbb{C}[x,y]$ $V(f)=V(g)$ if $f=p_1^{a_1} \cdot p_2^{a_2} \cdot \dots \cdot p_s^{a_s} , g=p_1^{b_1} \cdot p_2^{b_2} \cdot \dots \cdot ...
0
votes
1answer
38 views

Counting parameters for the intersection of a quadric and a cubic surface in $\mathbb{P}^3$

I have to count parameters for the intersection of a quadric and a cubic surface in $\mathbb{P}^3$, up to linear automorphisms of $\mathbb{P}^3$. I take account of the theorem according to which a not ...
3
votes
1answer
59 views

A simple lemma on divisors…

Let $D$ be a strictly positive divisor defined on a compact Riemann Surface such that $\operatorname{dim} \mathfrak{L}(D)=1+\operatorname{deg} D$. There exists a point $p \in X$ such that ...
3
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1answer
38 views

Bertini's theorem for surfaces: informations about singular fibers.

Let $S$ be a complex non-singular projective surface embedded in some $\mathbb P^n$. Thanks to the Bertini's theorem (Hartshorne theorem II.8.18) there exists a hyperplane $H\subseteq\mathbb P^n$ ...
6
votes
3answers
129 views

Fact check: global geometry / topology of moduli space of curves

Question: Is the moduli space of smooth complex curves of genus $g\geq2$ isomorphic to the affine space $\mathbb A_{\mathbb C}^{3g-3}$? (Note: I am not asking about the compactification of this ...
1
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1answer
27 views

Why does $L(0)=k$?

Definition 1: For every divisor $D=\sum_{P\in C}n_PP$ over a curve $C$, we define the vectorial space: $L(D)\doteqdot\{f\in k(C);\text{ord}_P(f)\ge -n_P,\forall P\in C\}$ Furthermore, $L(D)$ is a ...
6
votes
4answers
230 views

Is there a name for the curve $t \mapsto (t,t^2,t^3)$?

Is there a name for the curve given by the parametrization $\{(t,t^2,t^3); t\in\mathbb R\}$? Here is a plot from WA. An another plot for $t$ from $0$ to $1$. This curve is an example of a ...
1
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1answer
31 views

Degree of min distance function between two algebraic curves

Suppose I have two algebraic curves $C_1$ and $C_2$ in the plane. I would like to find the minimum distance between the two curves. If the two curves have degrees $n_1$ and $n_2$, what is ...
4
votes
1answer
62 views

What's the connection between exceptional divisor and projectivized tangent space?

This is one homework problem and hence I want some hint but not a whole answer. Let $P$ be a projective space and $X\subset P$ be a non-singular variety. Prove that the collection $L_p$ of lines ...
3
votes
1answer
43 views

Basic computation for the degree of an isogeny

I am trying to compute the degree of the isogeny $\phi:E_{1} \to E_{2}$ where $\phi(x,y)=(\frac{y^2}{x^2},\frac{y(b-x^{2})}{x^2})$ with $E_{1} : y^{2} = x^{3} + ax^{2} + bx$, $E_{2} : Y^{2} = X^{3} - ...
2
votes
0answers
42 views

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 $X$ ...
2
votes
1answer
93 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 ...
1
vote
1answer
75 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 ...
2
votes
3answers
160 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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0answers
51 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 ...
2
votes
1answer
57 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 ...
4
votes
1answer
85 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 ...
2
votes
0answers
51 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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0answers
69 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 ...
1
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1answer
47 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 ...
5
votes
2answers
83 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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9 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 $$f(x,y) = y^n+a_1y^{n-1}+\ldots+ a_iy^{n-i}+\ldots+a_n \in k[x][y]$$ irreducible in $y$, and ...
2
votes
1answer
43 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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votes
0answers
65 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 ...
2
votes
1answer
37 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 ...
4
votes
1answer
61 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 ...
0
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1answer
34 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 ...
2
votes
2answers
124 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? ...
1
vote
1answer
151 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 ...
1
vote
2answers
161 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 ...
1
vote
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 ...
3
votes
0answers
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 ...
5
votes
1answer
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 ...
4
votes
1answer
66 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 ...