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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2
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1answer
16 views

Affine variety over a field which is not algebraically closed

I am now trying to prove the following statement. If the field $K$ is not algebraically closed, then any $K$-variety $V\subset\mathbb{A}$ can be written as the zero set of a single polynomial in ...
2
votes
1answer
51 views

Is the map $\phi: \mathbb{P^2\rightarrow \mathbb{P^1}}$ defined by $\phi(x,y,z)=(x,y)$ birational?

This is Exercise 5.3 (a) in Undergraduate Algebraic Geometry by Reid. Does the map $\phi: \mathbb{P^2\rightarrow \mathbb{P^1}}$ defined by $\phi(x,y,z)=(x,y)$ define a rational map? Determine ...
0
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0answers
14 views

Fit a curve to Data in R [on hold]

I'm fairly new to R, and attempting to fit a curve to a super simple set of data I have. I've tried nls but can't seem to get it correct. My data is as follows: ...
1
vote
0answers
24 views

Does this base change yield another dominant morphism?

Here's something that seems to be true, or at least I hope it to be true, but I'm unable to prove it: Let $S$ be a $k$-rational surface and $B$ a curve, both projective, smooth and geometrically ...
0
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0answers
30 views

What is and how to find a dual form?

In an article I was reading the notion of dual form came up, which I wrote down below, and I was interested in learning about this. I would greatly appreciate an explanation of how we can find such a ...
0
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1answer
26 views

Irreducibility of an affine variety in an affince space vs in a projective space.

Proposition 5.5 in Undergraduate Algebraic Geometry by Reid says (I only write down a brief idea since the proposition is long and involves some other notations to define): The affine variety $U$ ...
1
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0answers
29 views

$f\in k(\mathbb{A^2})$ not regular at the origin implies it is not regular at points of a curve passing through the origin.

This is Exercise 4.12 (a) in Undergraduate Algebraic Geometry by Reid. Prove that any $f \in k(\mathbb{A}^2)$ which is not regular at the origin $(0, 0)$ also fails to be regular at points of a ...
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0answers
28 views

Singularities in affine and projective space.

Sorry to bother you guys I am trying to read a text that is a bit out of my league. I am doing some of the problems in the book to understand it better. Specifically the singularities and the tangent ...
0
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0answers
16 views

Question Regarding Variaties finding the coresponding polynomial [closed]

Hi guys I have a general question. Say we have a set and we suspect it is a variety. How does one find the polynomial corresponding to it. I am thinking say the set ${(a^2,a^3+1)}$ where a is a ...
2
votes
1answer
39 views

What's so special about hyperbolic curves?

This is really a two-part question, but I would be happy to get an answer for either bit. By a hyperbolic curve as defined by e.g. Szamuely in Galois Groups and Fundamental Groups (p.137) I mean an ...
3
votes
1answer
40 views

The map $\phi: k\rightarrow C: (y^2=x^3) \subset k^2$ over a finite field

On page 76 of Reid's book Undergraduate Algebraic Geometry, he says that Over an infinite field $k$, the polynomial map $\phi: k\rightarrow C: (y^2=x^3) \subset k^2$ given by $\phi(t)=(t^2,t^3)$ ...
0
votes
1answer
17 views

How to find the perimeter of the region formed by two different curves [closed]

How to find perimeter of the region bounded by $x^2 + y^2 = 100$ and $x^2 + y^2 - 10x - 10(2-3^{\frac{1}{2}})y = 0$?
1
vote
2answers
49 views

Function field on a non-singular algebraic curve as the field of meromorphic functions

Let $k$ be an algebraically closed field and let $X$ be a non-singular algebraic projective curve over $k$. Very often, when most books present the function field $k(X)$, they say that "it is the ...
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0answers
29 views

Any solutions to problems in Harris book moduli of curves

Is there any place where I can find solutions to problems of the book Moduli of Curves. I am learning the subject, and want to do some of the problems. Also if there is any good source for problems ...
3
votes
2answers
43 views

Normalization of the projective closure of affine plane curve over $\mathbb{C}$

I am trying to understand how to do explicit calculations for finding the normalization of a plane curve. The intuition is somewhat clear to me: "separate" the singularities or smooth them out (for ...
4
votes
1answer
92 views

Noether normalization in algebraically closed field

The Noether normalization lemma states that if $k$ is a field, and $A$ a finitely generated $k$-algebra, then there exist elements $y_1,...,y_m\in A$ such that $y_1,...,y_m$ are algebraically ...
0
votes
2answers
34 views

How to differentiate $ y=\sin^2(2x)\cos(x) $?

I was solving some A Level past papers and I came across this question. We have the equation of the line $ y=\sin^2(2x)\cos(x) $ for $ 0\leq x \leq \frac{\pi}{2} $ and there is a maximum point M. We ...
17
votes
0answers
334 views

Intersection of twisted cubics in $\mathbb{P}^3$

Suppose we have two twisted cubics $C_1$, $C_2$ in $\mathbb{P}^3$ such that both of them lie in some cubic surface, which means that $h^0(\mathbb{P}^3, I_{C_1\cup C_2}(3))>0$. I want to show that ...
2
votes
0answers
38 views

Classify the set $\{(x,y,z):f(x,y,z)=0, \nabla f=0\}$, where $f$ is a polynomial of degree at most 3.

Suppose that $f(x,y)$ is a nonzero polynomial of degree at most 2. Observe the following set: $$S=\{(x,y) : f(x,y)=0 ,\; \partial_{x}f(x,y)=0,\; \partial_{y}f(x,y)=0 \}.$$ Note that this set is the ...
0
votes
1answer
30 views

How to convert $x^3+y^3-x^2+y-1=0$ to homogenous form using the variables $X,Y,Z$

I'm to figure out how to convert algebraic curve such as $x^3+y^3-x^2+y-1=0$ to homogenous form using the variables $X,Y,Z$. Any help will be appreciated!
3
votes
1answer
55 views

Smooth affine plane curve with non-trivial cotangent sheaf?

Question: Let $A = \mathbb C[x,y]/(f)$ be a non-singular plane curve. Under what conditions is the module of Kahler differentials $\Omega_A^1$ (over $\mathbb C$) a free module? I am not sure what ...
3
votes
2answers
56 views

Proving that set of points $(x,y)$ in $\mathbb{R}^2$ satisfying $y - \cos(x)= 0$

How can one prove that set of points $(x,y)$ in $\mathbb{R}^2$ satisfying $y-\cos(x)=0$ is not a algebraic curve. That is there does not exist a polynomial $f(x,y)$ in two variables $x$ and $y$ and ...
3
votes
1answer
45 views

If the $m-1$ first derivatives of a rational function vanish at a point, does the function have a zero of order $m$ at that point?

Let $C\subseteq\mathbb{P}^{2}$ be a projective smooth algebraic curve, and let $$ \alpha:K(C)\rightarrow K(C) $$ be a derivation, i.e. $\alpha$ is a $K$-linear map such that $$ ...
2
votes
1answer
33 views

Computing the restriction $T_{\mathbb{P}^3|X}$ for twisted cubic in $\mathbb{P}^3$

Let $i:X=\mathbb{P}^1\to\mathbb{P}^3$ be a twisted cubic given by the embedding $(u:v)\mapsto(u^3: u^2v: uv^2: v^3)$. How to compute $T_{\mathbb{P}^3|X}$?
3
votes
1answer
37 views

Are all rationally parametrized plane curves algebraic? How does one find their degree?

Suppose a plane curve is given parametrically by $x=p(t),y=q(t)$, where $p,q$ are rational functions. I originally assumed that this means that the parametrized curve is algebraic, i.e. that it is the ...
1
vote
1answer
27 views

Schaum's Differential Geometry exercise on curvature

Page 72 exercise 4.5, there is the following situation: There is a curve $\underline{x}(t)$ with $t$ not a natural parameter. I have to find the curvature vector $\underline{k}$ and the curvature $k$ ...
2
votes
2answers
65 views

Generator for Kahler differentials of an affine elliptic curve

Consider the affine (nonsingular) elliptic curve $A = \mathbb C[x,y]/(y^2-x^3+x)$. Since the cotangent bundle is trivial, $\Omega_A^1 = A\,dx\oplus A\,dy /(2y\,dy - (3x^2-1)\,dx)$ is a free ...
1
vote
2answers
66 views

Two polynomials $f,g \in K[x,y]$ ring. Prove that $K[x,y]/(f,g)$ is finite dimensional vector space

Let $f,g \in K[x,y]$ be polynomials with no common factor. Prove that $K[x,y]/(f,g)$ is a finite dimensional vector space. I know there are non-zero (this word is correct?) $r(x)$ and $s(x)$ in ...
0
votes
2answers
18 views

Area using definite integrals with a straight line

I'm really stuck on this. Say you have a curve $y = 3x - x^2$ that cuts the x-axis at points $O$ and $A$, and meets the line $y = -3x$ at the point $B$. How would you find the area of this shaded ...
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votes
0answers
43 views

The normal bundle of conic

Let $C\subset\mathbb{P}^2\subset\mathbb{P}^n$ be a smooth conic (everything is over the field $\mathbb{C}$). I want to compute $T_{\mathbb{P}^n|C}$ and $N_{C/\mathbb{P}^n}$. Let $z_0,z_1,...,z_n$ be ...
0
votes
0answers
35 views

Computing the ramification index of a morphism of curves

Definition: Let $f: C_1 \to C_2$ be a nonconstant map of smooth curves and let $P \in C_1$. $$e_f (P) = \textrm{ord}_P (f^* t_{f(P)})$$ where $t_{f(P)} \in K(C_2)$ is a uniformizer at $f(P)$ ...
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0answers
40 views

The divisor of a nonconstant function on a smooth curve

Let $C/K$ be a smooth curve and $f \in K(C)$ be a function. Then by identifying $f$ with a rational map, we can get a 1-1 correspondence with maps $C \to \mathbf{P}^1$, with one direction being given ...
2
votes
0answers
28 views

Subgroups of $\text{PSL}(2, \mathbb{R})$ Closed under Transposition

I am wondering, does anyone know if there is a classification of transposition-closed (Fuchsian) subgroups of $\text{PSL}(2, \mathbb{R})$? I can't read French, so for all I know it's sitting in the ...
0
votes
1answer
59 views

Show that a infinite discrete subset of $\mathbb{R}^n$ is not an algebraic set

I want to prove that a set which is discrete in $\mathbb{R}^n$ (with the euclidean topology) and infinite cannot be an algebraic set. How could I do it?
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votes
2answers
30 views

Quartic in $\mathbb{P}^2_k$ are not hyperelliptic

Let fix an algebrically closed field $k$. It is easy to show that a curve of genus $3$ over $k$ is hyperelliptic or a quartic in $\mathbb{P}^2_k$. I have some difficulties to prove that there not ...
0
votes
1answer
49 views

Surjectivity of morphisms of smooth projective varieties

I have a question regarding a proof of the "surjectivity of morphisms of projective varieties" (a whole mouthfull). Though there are proofs using completeness of varieties, I am interested in an ...
2
votes
0answers
25 views

Explicit form of certain polynomials and intersection of curves

Let $X$ be a smooth degree $d$ hypersurface in $\mathbb{P}^3$ and $C, D$ two effective divisors on $X$ intersecting at finitely many points. Is it true that if $C$ and $D$ intersect in ''low'' number ...
3
votes
0answers
24 views

Reference to an atlas of curves and surfaces?

I remember at more than one university math department there being a set of glass cabinets with a number of physical models of surfaces. They were all algebraic varieties on the reals (of limited ...
4
votes
1answer
54 views

Divisors of degree $2g-2$ on a hyperelliptic curve of genus $g$

Suppose I have a divisor $D$ of degree $2g-2$ on a hyperelliptic curve of genus $g$. Then I can prove that either a) $K_C\otimes\mathcal{O}(-D)=\mathcal{O}_C$, that is $K_C\cong \mathcal{O}(D)$, or ...
4
votes
1answer
48 views

Prove that a set in $\mathbb R^3$ is not an algebraic set

I want to prove that the set $\{(\cos(t),\sin(t),t)\in A^3(\mathbb R); t\in \mathbb R \}$ is not an algebraic set. I already proved that the set $\{(\sin(t),t)\in A^2(\mathbb R);t\in \mathbb R \}$ ...
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vote
0answers
33 views

Continuous maps from an absolute Galois group

Let $\xi$ be a continuous homomorphism from an absolute Galois group $G_{\bar{K}/K}$ (Krull topology) to a finite abelian group $M$(discrete topology), where $K$ is a number field and $\bar{K}$ is its ...
7
votes
1answer
142 views

Characterization of the $m$-torsion points of an elliptic curve.

Let $(E,\mathcal{O})$ be the elliptic curve of equation $$ f=Y^{2}+a_{1}XY+a_{3}Y-X^{3}-a_{2}X^{2}-a_{4}X-a_{6}, $$ $\alpha:K(E)\rightarrow K(E)$ the derivation such that $$ ...
2
votes
0answers
37 views

Equation to Draw Curves with Saturation and Peak

I am looking for an equation to draw a graph like this: The curve should have a peak and saturation. Would you please let me know what is the equation that can generate similar curve ? Here is ...
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vote
0answers
32 views

Global sections of symmetric product of curves

Let $C$ be a irreducible, smooth, projective curve over $\mathbb{C}$. Let $L$ be a globally generated line bundle over $C$. Let $h^0(C,L)=m.$ Consider the product $C \times C$. If $p_i:C \times C ...
5
votes
2answers
125 views

Normal bundle of twisted cubic.

Let $C$ be a twisted cubic in $\mathbb P^3$. I'd like to compute the splitting type of normal bundle $N_{C/\mathbb P^3}$? I understood that $T_{\mathbb P^3}|_C=\mathcal O(4)^{\oplus 3}.$ So we have an ...
3
votes
2answers
50 views

Computing $l(D)$ for certain divisor.

Let $C$ be a smooth projective curve of genus $g=2$. I want to prove that there exist $P,Q\in C$ such that $$ l(P+Q)=2. $$ I know that if $D\in Div(C)$, and $x\in C$, then $$ l(D)\leq l(D+x)\leq ...
3
votes
0answers
48 views

The Picard group of an Elliptic Curve

Let $(E,O)$ be an elliptic curve. Let $\operatorname{Pic}^0(E)$ stand for the divisors that have degree $0$ where : $$D = \sum_{p\in E}n_p(P) \text{ and } \deg D = \sum_{p\in E}n_p.$$ I understand ...
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vote
0answers
48 views

Genus-Degree formula gives the wrong answer: ordinary points?

I'm trying to compute the genus of the normalization of the curve: $y^5=x(x-1)(x-2)$ Now I calculate the ramification points of the projection x: they are $(0,0),(1,0),(2,0)$ and they are of ...
0
votes
0answers
20 views

Is this the correct way to compute the blow up of a curve

I'm trying to calculate the blowup of the curve $y^5=z^2-3z^3+2z^4$ at $(0,0)$ We have the relation $Ay=Bz$, now I split it into two charts: The first chart$(y,a=A/B)$: ...
1
vote
0answers
39 views

Endomorphism ring of Drinfeld modules.

Let $\mathcal{X}$ be a smooth geometrically irreducible projective curve over $\mathbb{F}_q$. Fix a closed point $\infty\in \mathcal{X}(\bar{\mathbb{F}_q})$. Let $K$ be the function field of ...