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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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 ...
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41 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 ...
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42 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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59 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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69 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
113 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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36 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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366 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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42 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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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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59 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
57 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 ...
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1answer
49 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 $$ ...
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1answer
36 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 $T_{\mathbb{P}^3|X}$?
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42 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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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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74 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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70 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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20 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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49 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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53 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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42 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 ...
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31 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
62 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
34 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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64 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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26 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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26 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
71 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 ...
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1answer
57 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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34 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 ...
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1answer
149 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 $$ ...
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64 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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37 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 ...
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155 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
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2answers
53 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 ...
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0answers
57 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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62 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 ...
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24 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$(y,a=A/B)$: ...
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1answer
87 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 ...
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1answer
39 views

Smoothing transverse self-intersection

Let $S$ be a complex surface, and let $C \subset S$ be an immersed complex curve with a transverse self-intersection at point $P$. Let $\tilde{C}$ be a curve obtained from $C$ by smoothing the ...
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40 views

Which curves have reflexive structure sheaf?

Let $C$ be a curve, i.e. a purely one-dimensional scheme, embedded in a smooth projective threefold $X$. For a coherent sheaf $E$ of codimension $c$ on $X$, let $E^D=\mathscr Ext_X^c(E,\omega_X)$ be ...
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1answer
51 views

Cohomology of rational quartic in $\mathbb{P}^3$

I have to do this exercise. Let $X\subseteq\mathbb{P}^3$ a rational curve of degree $4$. Show that $$H^1(\mathcal{O}_X(1))=0=H^1(\mathcal{I}_X(2))$$ I tried to look at $X$ as closed immersion by ...
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1answer
89 views

Is there something similar to $\mathbb{R}^2$ for elliptic curve point representation?

Let $E$ be an elliptic curve over a finite field $\mathbb{F}_p$ and denote with $E(\mathbb{F}_p)$ its set of points over $\mathbb{F}_p$. Consider a coordinate system in $\mathbb{R}^2$. Every point is ...
2
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2answers
107 views

The equation of a jelly bean curve?

What is the equation of a curve with jelly bean shape? I have found a quartic equation for bean shaped curves, but nothing for jelly beans. If somebody doesn't know the shape, here is a link: jelly ...
3
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1answer
60 views

Computing cohomology over projective curve in $\mathbf{P}^3$

Let be $k$ an algebraically closed field and Let be $X\subseteq \mathbf{P}^3:=\mathbf{P}_k^3$ a smooth, irreducible curve that is not contained in any hyperplane. Let's call $d=\deg(X)$. A well known ...
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1answer
13 views

Prove that the tangent at $P$ intersects $C$ twice at $P$ and once at $4P$; the tangent at $5P$ intersects $C$ twice at $5P$ and once at $2P$.

This is a problem from Conics and Cubics by Bix. Please help me answer this one. Let $C$ be a nonsingular, irreducible cubic with a flex $O$. Add points (commutative) of $C$ with respect to $O$ as ...
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0answers
15 views

An application of the Max Noether's theorem

I'm reading chapter IV of Robert J Walker's book 'algebraic curves'. The last section of this chapter is about Max noether's AG+BF theorem. I am stuck on an exercise in this section. The exercise ...
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1answer
18 views

Prove that $2P$ is a flex of $C$ collinear with $P$ and $3P$ and $4P$ is a flex of $C$ collinear with $5P$ and $3P$.

I encountered this problem from Conics and Cubics by Bix. Please help me answer this. Let $C$ be a nonsingular, irreducible cubic with a flex $O$. Add points (commutative) of $C$ with respect to $O$ ...
1
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1answer
40 views

Show that the irreducible quartic projective curve is rational

How can I show that the irreducible quartic curve $\Gamma=V_+((x^2-z^2)^2-y^2(2yz+3z^2))$ on $P^2(\mathbb{C})$ is rational by considering the family of conics through the double points $(1:0:1), ...