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

4
votes
1answer
43 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
29 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
33 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
24 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 ...
-1
votes
0answers
10 views

How to find saturation point of a four parameter S curve?

I want to find the saturation point from where the S-curve values remain constant for a four parameter logistic equation i.e Alpha, Beta, Inflection and Slope are provided.
0
votes
3answers
18 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
27 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
54 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
45 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
41 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
28 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
20 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
23 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
45 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
125 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
56 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
133 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
35 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
39 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
61 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 ...
2
votes
1answer
35 views

Help with proof that that affine plane curves in $\mathbb{C}^2$ are not compact

This is a problem from Kirwan's Complex Algebraic Curves that I'm stuck on. She gives a hint suggesting that for $C = \{(x,y)\in\mathbb{C}^2: P(x,y) = 0\}$, show that at all but finitely many points ...
3
votes
2answers
135 views

Completion of the proof of theorem 3.3 in Dale Husemoller: Elliptic Curves

I want to read the proof of the following theorem: This is from p.35. But it is not complete there. There is written that: Can someone tell me where I can find the rest of the proof? Any other ...
4
votes
4answers
129 views

Why $y=e^x$ is not an algebraic curve?

Why $y=e^x$ is not an algebraic curve over $\mathbb R$? I can say that is not a algebraic curve over $\mathbb C$ because $e^x$ is a periodic function, but what about $\mathbb R$? EDIT: I don't want ...
0
votes
0answers
31 views

Definition of trigonal curves

I'm reading Fulton's book and I'm trying to understand the concept of trigonal in Hartshorne's book (page 345): On the other hand, Fulton's book define the $g_d^n$ in the following manner: So Can ...
2
votes
1answer
89 views

Finding a stronger version of Cayley-Bacharach Theorem that applies in the case that the intersection multiplicities are not equal to $1$

Cayley–Bacharach theorem: Assume that two cubics $C_1$ and $C_2$ in the projective plane $\mathbb{P}^2$ meet in nine (different) points. Then every cubic that passes through any eight of the points ...
1
vote
0answers
38 views

Help with this correspondence in Fulton's book

I'm trying to understand this question in Fulton's book (page 110): I couldn't prove $div(\sum \lambda_if_i)+D\mapsto (\lambda_1,\ldots,\lambda_{r+1})$ is indeed a correspondence. I only could ...
0
votes
1answer
23 views

A simple example of a base point of a linear series

I'm reading Fulton's algebraic curves book and he make the following definition of linear series (page 110): Let $D$ be a divisor, and let $V$ be a subspace of $L(D)$ (as a vector space). The ...
3
votes
0answers
43 views

Are trivial vector bundles on curves semistable?

Let $C$ be an irreducible projective curve with at worst nodal singularities. Let $E$ be the trivial locally free sheaf of rank $r$ i.e., $E$ is the direct sum of $r$ copies of the trivial line ...
2
votes
2answers
42 views

Is it true that for every $n\in\mathbb{N}$ there exists an algebraic curve $C$ and a point $p$ in that curve whose tangent plane has dimension $n$?

The title is very self-explanatory, I was thinking of finding a curve $k^n$ (k is the field with a non-singular point p, such that its tangent plane is V(0) and thus would be $k^{n}$ whose dimension ...
2
votes
2answers
54 views

Finding some rational points on elliptic curves

If I am considering an elliptic curve, for example $$y^2=x^3-2$$ $$\text{Edit: and } y^2=x^3+2$$ over $\mathbb Q$, how to find rational points? What possibilities do we have to calculate ...
0
votes
2answers
37 views

Wich kind of splines are the 3DS Max graph editor splines?

I'm trying to reproduce the splines from the program , and I have the correct point data, but my representation of Bezier Splines using the same anchor point data fails to give me a correct curvature. ...
1
vote
1answer
57 views

Questions on branch points on elliptic curve

So let $(E,p)$ be an elliptic curve over a field $k$ with a choice of $k$-valued point $p$. Then by Riemann-Roch, there are two global sections of $\mathcal{O}_{E}(2p)$ which gives a double cover of ...
0
votes
1answer
35 views

Help in this exercise in Fulton's algebraic curves book

I'm trying to solve the exercise 8.37 (page 111) in Fulton's algebraic curves book: I've already solved almost every item, it miss just the equivalence: The curve $X$ has a hyperelliptic ...
1
vote
0answers
19 views

Basis of $L(D)$

Let $L(D)=\{f\in k(C)\mid \text{ord}_P(f)\ge -n_p,\ \text{for all $P$ in C}\}$ be the vector space defined on page 99 of Fulton's algebraic curves book. I would like to know how to find a basis ...
4
votes
2answers
57 views

Geometric Intuition Behind Blowing Up a Cusp on a Plane Curve?

I'm reading Hartshorne AG V.3 on monoidal transformations and embedded resolutions. I understand one sort of intuition behind blowing up a point on a surface (or more generally a subvariety of a ...
0
votes
1answer
40 views

Equivalence definitions of hyperelliptic curves

I'm reading Fulton's algebraic curves book and on page 111, he defines hyperelliptic curves. For him an hyperelliptic curve $C$ is a curve which has a hyperelliptic weierstrass point $P$, i.e., $2$ is ...
2
votes
0answers
56 views

Help in a proof in Fulton's algebraic curves book

I'm reading Fulton's algebraic curves book and I didn't understand this proof of proposition 7 (page 106) very well: So I have the following doubts: I didn't understand why $\text{ord}_P(f')\ge ...
0
votes
1answer
35 views

If a curve is hyperelliptic, we have an equality in Clifford's Theorem

I'm studying Fulton's algebraic curves book and I have the following question: Clifford's theorem says that if $D$ is a divisor and $W$ is a canonical divisor with $l(D)\gt 0$ and $l(W-D)\gt 0$, then ...
2
votes
1answer
57 views

Blowing up families of singular curves

I am stuck with a simple example, but I guess the more general question would be whether blow ups commute with restrictions to subsets (points) of the blow-up locus. Over $\mathbb{C}$, suppose that ...
1
vote
1answer
54 views

Clifford Theorem as an easy corollary of Riemann-Roch Theorem

I'm studying Fulton's algebraic curves book and on page 109 he proves the Clifford's theorem: I have these doubts: 1.Why does he consider only the divisors $D\ge 0$ and $W-D\ge 0$? 2.What ...
4
votes
0answers
48 views

Is there some relationship between algebraic curves and partial differential equations that goes beyond classifying different PDE's

I ask primarily because despite not having taken that many math classes (up to two semesters of a PDE class in college), it would be very interesting if maybe we could gain intuition regarding ...
2
votes
1answer
35 views

A curve of genus $g\geq 2$ has a closed point of degree at most $2g-2$ over base field.

I am working on the following problem [R. Vakil] Exercise 19.8.B: Suppose $C$ is a curve of genus $g>1$ over a field $k$ that is not algebraically closed. Show that $C$ has a closed point ...
0
votes
1answer
30 views

$div(z)=0\Leftrightarrow z\in k$

I'm reading algebraic curves book from Fulton and I didn't understand this corollary on page 98: Why $\deg(div(z-\lambda_0))\gt 0$? and why is this a contradiction? Thanks a lot
1
vote
1answer
44 views

Automorphisms of non-hyperelliptic curve of genus 3 in $\mathbb{P}^{2}$

I have a question from R. Vakil's exercise 19.7.C which goes as follows: Suppose $C'\subset\mathbb{P}^{2}$ is a smooth plane quartic curve. Show that there is bijection between automorphisms of $C'$ ...
6
votes
0answers
57 views

Finding equations for projective curves, low genus, Riemann-Roch.

Let $C \subset \mathbb{CP}^n$ be a nonsingular projective curve, and let $L \subset \mathbb{CP}^n$ be a hyperplane. We have that $L \cdot C$ is a divisor $H$ on $C$ if $C \subset L$. Let $R = ...
8
votes
0answers
98 views

Intuitive/geometric way of thinking about effective divisors?

What is the motivation/intuition/geometric way of thinking about an effective divisor? I know that a divisor is effective if all its coefficients are non-negative. We write $D \ge 0$ for ...