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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Linearly normal embedding and varietes lying on quadrics

Let $X\subset\Bbb{P}^N$ be a smooth algebraic variety and assume that $X$ is not contained in a hyperplane. One says that the embedding $i\colon X\hookrightarrow\Bbb{P}^N$ is linearly normal if the ...
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15 views

Finding and analyizing the singularities in Affine and Projective space

Hi guys I am working this $F(x,y,z)=xy^4+yx^4+xz^4$ I need to find the singularities in affine and projective space and find the multiplicity of them.I would really appreciate some help tips. So ...
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1answer
11 views

Finding a curve given only its basic form and its tangent line

The basic form is $f(x)=k\sqrt{x}$ and the tangent line is $4x+36$. I've spent over half and hour with wolfram alpha and my notes on this problem: I've tried $(k\sqrt{x})'=4x+36$ and got ...
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18 views

There exist cuspidal cubic sections in a nonsingular cubic surface in $\mathbb{P}^3$

This is part of Exercise 7.3 in Undergraduate Algebraic Geometry by Reid. Let $S: (f=0) \subset \mathbb{P}^3$ be a nonsingular cubic surface. For $P\in S$ prove that if $P$ is not on a line of $S$ ...
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1answer
19 views

Isomorphism on cubics group law

Let $C$ a non singular cubic projective plane curve with a fix point $O$, we use que chord-tangent method to make $G = (C,+,O)$ an abelian group, if I choose another fix point $O'$ and construct the ...
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1answer
11 views

Set of roots of quadratic form $B(x,y,z,t)$ on the line $z=t=0$ is nonempty.

This is a proof from Section 7.1 of Undergraduate Algebraic Geometry by Reid. Suppose $S\subset \mathbb{P}^3$ is a nonsingular cubic surface, given by a homogeneous cubic $f=f(x,y,z,t)$. Consider ...
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24 views

Integers characterizing singularities of algebraic curves

The problem in the following : given an algebraic curve $C$, it's well-known that a smooth projective model of $C$ can be construct as the set of discrete valuations $v$ on it's function field ...
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1answer
30 views

Proving a projective quadric is nonsingular

Let $K$ be an algebraically closed field of characteristic $\neq 2$. Let $C$ be an irreducible quadric curve in $\mathbb{P}^2$, i.e. $C = Z(F)$ where $F$ is an irreducible degree 2 form. I think we ...
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1answer
29 views

Finding length of curve $y^2 = 64(x+3)^3$ for $0 \le x \le 3$

Not getting the right answer for this, can someone point me to where I'm going wrong?
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22 views

Abstract regular curve over non-algebraically closed field

In Hartshorne chapter I.6 is discussed the construction of the abstract nonsingular curve as part of the proof for the well known correspondence between complete regular irreducible algrebaic curves ...
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1answer
63 views

Image of a line or conic on Veronese surface.

This is part of Exercise 5.13 from Undergraduate Algebraic Geometry by Reid: Consider the Veronese surface $S$ defined by the map: $$\phi: \mathbb{P}^2\rightarrow \mathbb{P}^5$$ where ...
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1answer
26 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
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1answer
53 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 ...
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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 ...
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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 ...
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1answer
28 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 $U$ ...
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32 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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29 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 ...
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
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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 $\phi(t)=(t^2,t^3)$ ...
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2answers
50 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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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 ...
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2answers
46 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 ...
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1answer
94 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 ...
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2answers
35 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 ...
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342 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 ...
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39 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 ...
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1answer
31 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!
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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 ...
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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
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1answer
46 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
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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
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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 ...
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vote
1answer
28 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 $k$ ...
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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 ...
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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 ...
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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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0answers
44 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 ...
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36 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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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 ...
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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 ...
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1answer
60 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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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 ...
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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 ...
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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 ...
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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 ...
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1answer
55 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
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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 ...
8
votes
1answer
144 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 $$ ...