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 ...

learn more… | top users | synonyms

-2
votes
1answer
34 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) ...
0
votes
0answers
22 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
vote
2answers
26 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
27 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 \\ ...
1
vote
0answers
32 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
48 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
60 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
votes
1answer
44 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
137 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
vote
1answer
28 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 ...
7
votes
4answers
251 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
vote
1answer
39 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
71 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
54 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
55 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
100 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
77 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
231 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 ...
1
vote
0answers
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 ...
2
votes
1answer
62 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
90 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
57 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 ...
1
vote
0answers
76 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
vote
1answer
90 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
94 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 ...
0
votes
0answers
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 ...
2
votes
1answer
52 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 ...
0
votes
0answers
95 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
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 ...
5
votes
1answer
83 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
votes
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
125 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
168 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
188 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
35 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
47 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
96 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
71 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 ...
2
votes
1answer
56 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 ...
1
vote
0answers
81 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 ...
2
votes
0answers
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., ...
5
votes
1answer
77 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 ...
5
votes
1answer
75 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 ...
1
vote
0answers
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 ...
0
votes
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 ...
1
vote
2answers
84 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 ...
0
votes
0answers
46 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 $\Omega^{n-1}(F-D)\omega_{g-1}+\Omega^{n-1}(F-D)\omega_g$ ...
1
vote
0answers
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!
1
vote
0answers
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 ...
0
votes
0answers
44 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 ...