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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38 views

Why is the morphism induced by this linear system birational?

I have seen 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 point of an ...
0
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2answers
29 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
36 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
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0answers
52 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
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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 ...
2
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1answer
56 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
121 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 ...
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0answers
25 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
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1answer
84 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 ...
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0answers
36 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
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1answer
22 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
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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 ...
1
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1answer
51 views

Is the finite union of algebraic curves an algebraic curve? [closed]

Is the finite union of algebraic curves an algebraic curve? I'm kind of new to the study of algebraic curves and I believe this is intuitive.
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
52 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
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2answers
32 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
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1answer
48 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
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1answer
32 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
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0answers
17 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
50 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
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1answer
38 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
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0answers
54 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
34 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
52 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 ...
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1answer
52 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
40 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 ...
1
vote
1answer
32 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 ...
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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
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1answer
40 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
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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
91 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 ...
0
votes
0answers
27 views

What is the definition of Osculating plane in algebraic geometry?

I'm studying Fulton's algebraic curves book and in order to understand this paper in Algebraic Curves I need the definition of the d-dimensional osculation plane. Can I understand properly this ...
3
votes
0answers
39 views

Kähler differentials, define valuation?

See my previous question for a definition of the $K$-module of Kähler differential $\Omega_{K/k}$. This question is sort of a follow up on it. Suppose $k$ is a field of characteristic $0$, $R$ is a ...
2
votes
0answers
57 views

Kähler differentials, define valuation?

See here for a definition of the $R$-module of Kähler differential $\Omega_{R/k}$. Suppose $k$ is a field of characteristic $0$, $R$ is a $k$-algebra, and let $K$ be a finite extension of $k(x)$. If ...
0
votes
0answers
18 views

Result showing that a certain valuation ring in some function field has to be a DVR?

I know that if $R$ is a valuation ring such that $0 \to \mathbb{C} \to R$ is a left-split exact sequence, then there exists a discrete valuation ring $C$ with $R \subset C$ so that $0 \to \mathbb{C} ...
2
votes
1answer
37 views

Isomorphic function fields of projective curves, bijection of points.

Suppose curves $C$, $D \in \mathbb{CP}^2$ are nonsingular. If their function fields are isomorphic, i.e. $K_C \cong K_D$, then do we necessarily have a bijection of points on $C$ and $D$? Can we do ...
1
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0answers
152 views

“Necessary” condition for Power Diophantine Equation.

Motivation: Brocard’s problem $n!+1$ being a perfect square Observations: Given a power Diophantine equation of $k$ variables with a “general solution” (provides infinite integer solutions) to ...
0
votes
1answer
69 views

Intersection counting without Bézout

I am trying to solve the following problem: Let $C$ be a non-degenerate line (resp. conic) in $\mathbb{C}\mathbb{P}^2$ and $D$ a projective curve in $\mathbb{C}\mathbb{P}^2$ of degree $d$ such ...
1
vote
1answer
58 views

If $\mathcal O_P(C)$ is a DVR, then $P$ is non-singular

Let $C$ be an irreducible curve over $\mathbb A^2$ and $P\in C$. I would like to prove if $$\mathcal O_P(C)=\{f\in k(C)\mid f=a/b, b(P)\neq 0\}$$ is a DVR, then $P$ is non-singular, i.e., the ...
1
vote
1answer
24 views

How to calculate the intersection points of the same implicit curve in parametric form?

Take the following parametric equation of an implicit curve as an example: $$ \left\{\quad \begin{array}{rl} x=& \dfrac{27}{14} \sin 2 t+\dfrac{15}{14} \sin 3 t \\ y=& \dfrac{27}{14} \cos 2 ...
-1
votes
0answers
40 views

Length of an algebraic curve

Good day! A problem has been posted looking for the expected number of real roots of random algebraic polynomials $$a_nx^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + ... + a_0$$ whose coefficients followed a ...
3
votes
0answers
26 views

$D$ is divisor of both $d(x/z)$ and $y/z$. [closed]

Let $C \subset \mathbb{CP}^2$ 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 = ...
7
votes
0answers
65 views

Relationship between twisting sheaves and divisor sheaves

I'm not really entirely sure how to think about Serre's twisting sheaves $\mathscr{O}(i)$ - on any $\text{Proj}$ construction, really, but let's stick to something like $\mathbb{P}_2$ for now for ...
1
vote
1answer
92 views

How to prove $\mathcal O_P(C)$ is a DVR for $P$ non-singular?

Let $C$ be an irreducible curve over $\mathbb A^2$ and $P\in C$ a non-singular point. I want to prove that $\mathcal O_P(C)=\{f\in k(C)\mid f=a/b, b(P)\neq 0\}$ is a DVR. I've already proved that ...
2
votes
0answers
56 views

Directional derivative expression

We have $n=\sqrt{{\mathbf N}\cdot{\mathbf N}}$, where ${\mathbf N}$ is the normal vector to a curve, let's accept ${\mathbf N}=\ddot{{\mathbf r}}$, say the curve is unit-speed. We also have a scalar ...
9
votes
2answers
81 views

Nonsingular curve $C$ of degree 4, exists rational function $f: C \to \mathbb{CP}^1$ of degree 2?

Suppose $C \subset \mathbb{CP}^2$ is a nonsingular curve of degree $4$. Does there exist a rational function $f: C \to \mathbb{CP}^1$ of degree $2$?
3
votes
1answer
61 views

Exists rational function on curve in $\mathbb{CP}^2$ such that pole of order $2g + 2$?

Let $C \subset \mathbb{CP}^2$ be a nonsingular curve of degree $d$, and $p_1$, $p_2$, $q$ distinct points in $C$. For any $a_1$, $a_2 \in \mathbb{C}$, does there necessarily exist a rational function ...
1
vote
0answers
21 views

Computing the order of a divisor in the Jacobian of a hyperelliptic curve.

Given a hyperelliptic curve of genus $g$, of equation $H: y^{2}+h(x)y=f(x)$ and defined over the finite field $\mathbb{K}$, how does one compute the order of a (reduced) divisor defined over ...
1
vote
1answer
27 views

Normal vector in curvilinear coordinates

Is it true that the normal vector, or, $\ddot{\mathbf r}$ always vanishes for: a helix in cylindrical coordinates a loxodrome in spherical coordinates a torus knot in toroidal coordinates When ...
5
votes
2answers
77 views

Any curve of genus three is either hyperelliptic or trigonal?

A curve $C$ is said to be trigonal if it admits a rational function $f: C \to \mathbb{CP}^1$ of degree $3$. Is any nonsingular plane curve of degree four trigonal? Can the map be chosen so that it has ...