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

0
votes
0answers
25 views

example of computing ramification index

I am trying to understand example 2.2.9 of Silverman's "Arithmetic of Elliptic Curves". In this example, Silverman considers a map $$ \phi:\mathbb{P}^1\to \mathbb{P}^1; [X,Y]\mapsto [X^3(X-Y)^2, ...
3
votes
0answers
61 views

Fulton Algebraic Curves, Exercise 2.32(a).

Let $R$ be a DVR satisfying the conditions of Problem 2.30. Any $z \in R$ then determines a power series $\lambda_i X^i$. If $\lambda_0, \lambda_1, \dots$ are determined as in Problem 2.30(b). Show ...
1
vote
0answers
35 views

Automorphisms of cubic nodal curve

How to calculate the automorphism group of the nodal cubic curve $y^2=x^3+x^2$ ? Should I use the rationality of this cubic curve ?
0
votes
2answers
43 views

Sketch the graph $x=e^{-t}\sin t$,$ t\ge 0$

My graph is always negative though, and that doesn't make sense cause $t$ is supposed to be positive. I substituted $x$ as $y$ and $t$ as $x$.
2
votes
1answer
44 views

Is the quotient morphism from product of curves to to their symmetric product flat?

Suppose $C$ is a smooth curve, is the morphism $C^2=C\times C\to C^{(2)}=C\times C/S_2$ flat? What about the general case?
0
votes
0answers
9 views

Tweaking function to reduce the rate of decay of a logarithmic based curve

Im not even sure if this is possible or perhaps I may need to use a different function altogether but I currently have one that looks like this: $$y = a\log(x+b)+c$$ That produces the red curve ...
1
vote
1answer
64 views

Dimension of $\mathfrak{m}^k/\mathfrak{m}^{k+1}$?

Let $C \subset \mathbb{P}_\mathbb{C}^2$ be a projective curve, and $p \in C$ a point of multiplicity $m$. If $\mathcal{O}_p(C)$ is the local ring of $C$ at $p$, and $\mathfrak{m} \subset ...
3
votes
1answer
29 views

Nonsingular cubic curve, quotient of $d(x/z)$ and $y/z$ is differential which is regular everywhere.

Let $C \subset \mathbb{P}_2$ be the cubic curve defined by$$y^2z = x(x-z)(x-\lambda z)$$with $\lambda \in \mathbb{C} - \{0,1\}$. Let $p = [0, 0, 1]$, $q = [1, 0, 1]$, $r = [\lambda, 0, 1]$, and $s = ...
5
votes
1answer
41 views

Riemann-Roch analysis of point divisor ring on smooth genus 3 Riemann surface

Let $C=C_4\subset\mathbb{P}^2$ be the smooth genus 3 Riemann surface given by a quartic curve. Let $P\in C$ be a point, and $D=P$ the divisor given by the point $P$. Let ...
0
votes
0answers
20 views

Compute outer and inner outlines of graph of curves

Let's have some cubic Bezier curves and straight lines. Some of the Bezier curves and straight lines might have a shared start or end point, some might intersect. Input: A list of cubic Bezier ...
1
vote
1answer
33 views

Surface area of circle extracted from a tube wall

I have made a hollow tube (thickness $1$mm) having inner radius $89$ mm and outer radius $90$ mm (length $400$ mm, can be higher). then I made a circular (circle radius $25$ mm) cut perpendicular to ...
1
vote
0answers
30 views

Subsheaves of locally free sheaves on a rational curve

Let $k$ be an algebraically closed field, $\mathcal{E}$ a locally free sheaf on $\mathbb{P}^1_k$ and $\mathcal{L} \subset \mathcal{E}$ a subsheaf of rank $1$. By Grothendieck's theorem, we know that ...
6
votes
0answers
87 views
+50

Siegel's Theorem for Genus 0 Curves over Function Fields

Let $V$ be a quasi-projective algebraic curve over a field $k$. Are there finitely many morphisms from $S$ to the triply-punctured sphere $\mathbb{P}^1 -\{0,1,\infty\}$? This is true if $k = ...
0
votes
1answer
13 views

Tangent from points on a curve meeting the curve again and again

tangent at a point C1 on the curve y=x^3 meets the curve again at C2 .the tangent at C2 meets the curve at C3, and soo on, so that the abscissa of c1,c2,c3.....,Cn form a G.P. find the ratio of area ...
1
vote
1answer
40 views

Proving $x^2 - x = y^5 - y$ is a hyperelliptic curve

Greetings to one an all! How can we prove the curve "$x^2 -x = y^5-y$" is a hyperelliptic curve? Is a hyperelliptic curve the same as a hyperbolic elliptic curve or are there any differences?
1
vote
0answers
20 views

Recovering curve's equation from a given Kummer surface?

Assuming I'm not in even characteristic and that the ground field is algebraically closed for simplicity, it is known that every genus 2 curve is associated to a Kummer surface in $\mathbb P^3$. If ...
2
votes
0answers
24 views

Arakelov Theory and Arakelov curves

There exists a definition of Arakelov Curve in Arakelov theory? My question is because Neukirch (Algebraic Number Theory, Chapter III) defined Arakelov divisors in the set $X=Spec(\mathcal ...
1
vote
1answer
21 views

Why are all non-singular curves absolutely irreducible?

I eas reading Judy Walker's book Codes and Curves, and one of the exercise's in the book (ex. 4.6) was proving that every non-singular curve is abaolutely irreducible. I'm not so familiar with ...
1
vote
0answers
27 views

Riemann-Roch for nodal curves

Let $X$ be an irreducible, nodal curve and $E$ a coherent subsheaf of a free sheaf $\oplus_{i=1}^r \mathcal{O}_X$ on $X$ of rank strictly less than $r$. Assume that $r \ge 2$. It follows that $H^0(E)$ ...
2
votes
0answers
37 views

Detail regarding tangent spaces and dual varieties from Harris's Algebraic Geometry: A First Course

In Harris's Algebraic Geometry: A First Course, Example 16.20, the author shows that the dual of the dual variety $X^{*}$ is the original variety $X$. I think in chapter 15, Harris mentions that he'll ...
2
votes
1answer
33 views

height of formal group of an elliptic curves

I have an elliptic curve $E$ defined over a complete discrete-valued field $K$ of characteristc $0$. the residue field $k$ is of positive characteristic $p$. Then $E[p]=\mathbb{Z}/p\mathbb{Z} \times ...
-5
votes
2answers
83 views

How can I use left inverse to f(x)=3x format equation? [closed]

I want to solve linear equations as following. $$f(x)= 3x^3 -4x^2 +3x -7$$ $$f(x)= 2x^3 -3x^2 +2x -1$$ $$f(x)= 1x^3 -7x^2 +1x -2$$ But these seem that there is no $y$. How can I solve by using ...
9
votes
3answers
121 views

When is a Morphism between Curves a Galois Extension of Function Fields

My apologies if this question has already been answered somewhere on this site: when I searched, I could only find specific examples rather than the general question. It is known that the category of ...
4
votes
2answers
54 views

Reduction map on torsion points of an elliptic curve and their valuation

Let $K$ be a field of characteristic zero, complete with respect a discrete valuation $v$. Assume that the residue field $k$ is of positive characteristic $p$. Now take an elliptic curve $E$ defined ...
0
votes
1answer
30 views

How to approach to fitting curve?

I'd like to approximate fitting curve some kind of curves like below. (1, 3.5), (2, 4.3), (3, 7.2), (4, 8) which is having 4 points. and I heard that this solver is PINV() of matlab function. But ...
0
votes
2answers
59 views

How to find a equation from approximate curve? [closed]

I want to know a way to find equation from a curve. for example, if I have 4 point (1, 3.5), (2, 4.3), (3, 7.2), (4, 8) then how to find a good approximation equation ? What if I got above curve ...
0
votes
3answers
18 views

Gradient on curves

Please with a bit of explanation, what is the gradient on the curve $y = 16/x$ where $x = 8$. I'm finding it hard to solve problem like this.
4
votes
1answer
54 views

Can a bidegree $(3,4)$ curve be embedded in plane?

Suppose $C$ is a curve on $\mathbf{P}^1\times\mathbf{P}^1$ of bidegree $(3,4)$, why such a curve cannot be embedded as a curve in $\mathbf{P}^2$?
1
vote
0answers
30 views

How many $g_3^1$ does a smooth $(3,3)$ type curve on $\mathbf{P^1}\times\mathbf{P^1}$ has?

Suppose $C$ is a smooth curve of type $(3,3)$ type curve on $\mathbf{P^1}\times\mathbf{P^1}$. Does the two projections provide all the $g_3^1$s for $C$?
1
vote
1answer
37 views

Non hyperelliptic curves of genus 5 form a family of dimension 12

Suppose $C$ is a complete intersection of three quadrics in $\mathbb{P}^4$, how to count naively the dimension of its parameter spaces? One needs $|O_{\mathbb{P^4}}(2)|=14$ parameters to describe one ...
0
votes
0answers
25 views

How would one define polynomials over the projective line $P_K^1$

May $K$ be a field. If I set $\varXi=(X:Y)$ as a "projective variable" and "projective coefficients" $a_k=(x_k:y_k)\in P_K^1$ - may I then write a polynomial map $P_K^1\longrightarrow P_K^1$ in a form ...
0
votes
3answers
22 views

How to move the position of a curve $x,y$ coordinates?

I have some silly problem. I want to know how to move the curve in $x,y$ coordinates which I have some curve. For example, $f(x) = x^2$ and this is originally start at $(0,0)$. But I want to this ...
0
votes
1answer
36 views

How can I get a continuous piecewise polynomial curve, with a turning point (not differentiable)?

I would like to make a curve which has turning point(x,y). y= x^2*2 for x<= 0.5 y= 1-(1-x)^2*2 for x> 0.5 and still have a smooth S-shaped curve, where ...
4
votes
1answer
56 views

Identifying two points on an algebraic curve

Given a smooth algebraic curve $C$, say projective over an algebraically closed field $k$, is it always possible to identify two distinct closed points $x, y$ on $C$ to produce a curve with a single ...
1
vote
1answer
14 views

Question about r(t), movement along line

So I'm studying for an exam in calculus when i came across the concept of objects moving along a curve. I have a general idea of how to calculate speed, velocity and such when r(t)(position vector I ...
4
votes
2answers
50 views

Third point of intersection is also a point of inflection?

Let $C \subset \mathbb{P}_2$ be a nonsingular cubic. If $L$ is a line through two distinct points of inflection on $C$, how do I show that the third point of intersection is also a point of ...
0
votes
1answer
47 views

injective morphism between line bundles on curves

Let $X$ be a smooth projective, irreducible, curve, $\mathcal{L}$ be an invertible sheaf on $X$ and $\mathcal{L}' \subset \mathcal{L}$, an invertible subsheaf. Is $\deg(\mathcal{L}') \le ...
0
votes
0answers
29 views

Self-intersection number of fibered surface

Let $R$ be a complete discrete valuation ring with residue field $k$ (can assume algebraically closed), $f:X \to \mathrm{Spec}(R)$ a flat, proper family of projective curves (i.e., $f$ is also ...
0
votes
0answers
22 views

Locally free sheaves on reducible curves and their subsheaves

Let $X$ be a reducible (but reduced), connected, projective curve with at worst nodal singularities and $\mathcal{F}$ be a locally free sheaf of rank $r$ on $X$. Suppose that $\mathcal{F}'$ is a depth ...
0
votes
1answer
13 views

About the maximum number of ordinary points on algebraic surface

http://mathworld.wolfram.com/OrdinaryDoublePoint.html I'm trying to figure out the (3) statement ( $\mu(d)\leq \frac{1}{2}(d(d-1)-3) $ ) That can't be true if the table bellow it is correct (and it ...
2
votes
2answers
24 views

The simplest way to find a parametrization of the plane projective curve $XZ-Y^2=0$.

I have to explain to some first year math students that the projective algebraic set $\textbf{Z}(XZ-Y^2)\subset\mathbb P^2_k$ is $$V=\{(a^2_0:a_0a_1:a^2_1)\subset\mathbb P^2_k \,:\, \textrm{for}\; ...
2
votes
1answer
40 views

Characterization of linear system without base points

My question is really simple. Where can I find characterizations of linear system without base points? I searched on Hartshorne's book without success. Thanks
1
vote
1answer
47 views

linear systems and maps

Given a regular map $\varphi:C\to \mathbb P^n,P\mapsto \mathbb (f_0(P):f_1(P):\ldots:f_n(P))$, we can associate a linear system $|\varphi|$ in the following manner: let the divisor $D=-\min div(f_i)$ ...
0
votes
1answer
71 views

Example of a curve with this property

I'm reading Fulton's book and he defines the linear series $g_n^r$: So a curve $C$ is trigonal if it has a divisor which has a linear system $g_3^1$. I'm looking for a simple example of a trigonal ...
3
votes
1answer
128 views

Proof verification of a weak version of Bezout's Theorem

I'd like to make sure here that my reasoning seems sound. I am working from Kirwan's book on algebraic curves. I was not totally happy with her proof of this theorem, so I wanted to see if I could ...
1
vote
0answers
57 views

Proof of the formula for dimension of moduli of stable vector bundles on smooth curves

Let $C$ be a smooth curve of genus $g \ge 2$ over an algebraically closed field of positive characteristic. If I understand correctly, the dimension of the moduli space of vector bundles on $C$ of ...
4
votes
0answers
135 views

Why is the morphism induced by this linear system birational?

I have seen (what seems to be) the following statement used in a few places, but I am not sure why it is true. Any explanation as to why it is (or is not) true would be appreciated. Let $P$ be a ...
0
votes
2answers
37 views

How do I reverse the smooth-step equation?

I'm using the "smooth step" equation for an easing curve: $y = 3x^2 - 2x^3$ I would like to reverse this equation so that given y, I can find ...
1
vote
0answers
42 views

Difference between quadric and conic

What is the difference between a conic and a quadric? I'm guessing that this depends on your ambient space? I think that conics are just special quadrics and are a codimension 1 object and a quadric ...
0
votes
0answers
64 views

Computation of Riemann-Roch space L(kQ) to a specific Divisor D

I am trying to build a Reed-Solomon Code through a Goppa-Code Construction. I start with the projective line $\mathcal{X}$ $aX+bY+cZ=0$. The genus $g$ of this line is $0$. Futhermore, let ...