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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Rational sections of invertible sheaves and hermitian inner products

Notations: Let $X$ be a $\mathbb C$-scheme of finite type, projective, integral and of dimension $1$ (i.e. an algebraic curve) and with function field $K(X)$. The set of closed points is $X(\mathbb ...
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Groups acting on Riemann Surfaces and Automorphic Function

Consider the following paragraph from a book of Magnus on Combinatorial Group Theory. ... the simple group $G_{168}$ of order $168$ acts on a genus $3$ surface, is important for the theory of ...
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Field of moduli relative to a different extension.

Let $X$ be a Riemann surface of genus $g\geq 3$ defined over $\mathbb{Q}(\sqrt{2})$. Let us consider the automorphism of $\mathbb{Q}(\sqrt{2})$, $$ ...
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About birational map between curves of the form $y^{2}=g(x)$

I'm trying to solve the following exercise but I'm stuck after trying for a long time. Suppose that $g(x)=ax^{4}+bx^{3}+cx^{2}+dx+e\in{k[x]}$ and similarly ...
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Proof of Chapter 2 Proposition 2.6a in Silverman Arithmetic of Elliptic Curves

The following is Proposition 2.6(a) in Silverman's AEC: "Let $\phi: C_1 \rightarrow C_2$ be a nonconstant rational map of smooth (projective) curves over an algebraically closed field $K$. Then ...
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Page 276 of Principles of Algebraic Geometry by Griffiths and Harris; wrong parameter count?

The following is taken from page $276$ of Principles of Algebraic Geometry by Griffiths and Harris: Now let $S$ be a Riemann surface of genus $g\ge 3$. By our last result, if $S$ has any ...
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Parametric Interpolation in the Plane

Given $i+j$ points in the plane, when can we find $x(t),y(t)$, polynomials of degree $i$ and $j$ respectively such that the parametric curve $(x(t),y(t))$ goes through each point? We can do this ...
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70 views

$L$ is algebraic curve. Why has $L$ finitely many singularities?($x,y\in \mathbb{R}$)

Let ${L_1} = \left\{ {x + iy:x,y \in \mathbb{R},{f_1}(x,y) = (\sqrt {{x^2} + {y^2}} )p{{(x,y)}^{}} + q{{(x,y)}^{}} = 0} \right\} \subseteq {L_2} = \left\{ {x + iy:x,y \in \mathbb{R},{f_2}(x,y) = ...
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Elliptic curves with trivial Mordell–Weil group over certain fields.

I am looking for elliptic curves $E,E'$ defined over $\mathbb{F}_{3}$ and $\mathbb{F}_{4}$ respectively and given by a Weierstrass equation such that their Mordell-Weil group is trivial, i.e. such ...
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The cardinal of the Mordell-Weil group is prime for certain elliptic curves over $\mathbb{F}_{q}$ for certain $q$.

Let $p\in\{2,3\}$ and $r\in\mathbb{Z}_{\geq 2}$. I would like to find if there exists an elliptic curve defined over $\mathbb{F}_{p}$ such that $|E(\mathbb{F}_{p^{r}})|$ is a prime number. If $p>3$ ...
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How are non-homogenous elliptic curves projective varieties?

So if I am given an elliptic curve such as $Y^2Z=X^3$ then I see how it can be realized as the projective variety $Proj(k[X,Y,Z]/(Y^2Z-X^3))$. But, given an elliptic curve like $Y^2 = X^3 + X$, then ...
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32 views

A question on Bézout's theorem

Bézout's theorem Bézout's theorem for curves states that, in general, two algebraic curves of degrees $m$ and $n$ intersect in m·n points and cannot meet in more than $m·n$ points unless they have a ...
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41 views

Dual curve of an algebraic curve in affine coordinates

$F$ is an irreductible algebraic curve and we consider the application that sends a non-singular point $(a : b : c)$ in homogenius coordinates, to $\phi(a, b, c) = \bigl(f_x(a, b, c), f_y(a, b, c), ...
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28 views

Characterization of the elements of a quotient ring

I'm in trouble with the following exercise: Consider the ideal $ I = (X^2-Y^3,Y^2-Z^3) $ in the polynomial ring $ k[X,Y,Z] $, where $k$ is any algebraically closed field. Show that every element of $ ...
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50 views

Image of divisors under étale cover

Let $f\colon X\to Y$ be an étale cover of degree $d$ between two smooth projective varieties. If $V\subset X$ is an effective reduced and irreducible divisor, does $f$ restrict to an isomorphism ...
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62 views

How can we prove, by Bézout's theorem, that $L$ has finitely many singularities?

Bézout's theorem Bézout's theorem for curves states that, in general, two algebraic curves of degrees $m$ and $n$ intersect in m·n points and cannot meet in more than $m·n$ points unless they have a ...
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8 views

Derivations on a plane curve with singularites.

I am looking for some general results on the Lie algebra of derivations $Der(R)$ of $R=\mathbb{C}[x,y]/(f)$. How to describe $Der(R)$ for $f=x^3-y^2$ or $f=xy$. Is it possible to characterise the ...
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45 views

A question on partial-derivative

Let $f(x,y) = \sqrt {{x^2} + {y^2}} p(x,y) + q(x,y)$, and $p$ and $q$ are two polynomials(non zero) Is it true that; $\frac{{\partial f}}{{\partial x}}$ and $\frac{{\partial f}}{{\partial y}}$ don't ...
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1answer
30 views

Order and residue of 1-form $x^{-1}dx$

Consider the rational 1-form $x^{−1}dx$ on $\mathbb{P}^1$. I am asked to compute its order and residue at all $P \in \mathbb{P}^1$. Could somebody help me with this? I do not really how to start ...
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101 views

Is it true that; $\frac{{\partial f}}{{\partial x}}$ and $\frac{{\partial f}}{{\partial y}}$ don't have a common factor?

Let $f(x,y) = ({x^2} + {y^2})p{(x,y)^2} - q{(x,y)^2}$ and where $p(x, y)$ and $q(x, y)$ are real polynomials. Is it true that; $\frac{{\partial f}}{{\partial x}}$ and $\frac{{\partial f}}{{\partial ...
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30 views

The set of commutative matrices is an irreducible algebraic variety

Let $A, B$ matrices $n \times n$. Let $X = \left\{(A, B) \in \mathbb{A}^{2n^2} \mid AB = BA \right\}$. Prove that $X$ is algebraic and irreducible variety.
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Intersecting two pencils of plane curves

In $\Bbb{P}^2$, let $D_1,D_2$ be two curves of degree $d_1,d_2$ respectively. Choose two pencils $|D_1(t)|\subset|D_1|$ and $|D_2(t)|\subset|D_2|$ (free of fixed components) parametrized by the same ...
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1answer
70 views

Some elements of the function field of the Fermat curve

For $n>0$, consider the Fermat curve: $$C(n): \{X^n+Y^n=Z^n\}\subset\mathbb P^2(\mathbb C)$$ the function field of $\mathbb C(n)$ can be explicitly described in the following way. It is the set of ...
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2answers
55 views

Degree of a morphism from a curve to $\mathbb P^1_\mathbb C$: explicit description

Let $f:X\to \mathbb P^1_{\mathbb C}$ be a non-constant (i.e. surjective) morphism (of $\mathbb C$-varieties/schemes) from a smooth complex projective curve to the projective line. The degree of the ...
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1answer
53 views

Computing the Picard group of Number Fields in analogy with genus $0$ curves

It is possible to compute the Picard group of (some? all?) genus $0$ curves in the following manner: For concreteness let, $A = k[x,y]/(x^2+y^2-1)$ and $X = \operatorname{Spec} A$. Let $Y$ denote ...
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Let $\gamma(t)$ be an integral curve of a vector field $X$ on $M$. $\dot{\gamma}(t)=0$ for some $t$. Prove that $\gamma$ is a constant map

Let $\gamma(t)$ be an integral curve of a vector field $X$ on $M$. Suppose that $\dot{\gamma}(t)=0$ for some $t$. Prove that $\gamma$ is a constant map, that is, its range consists of one point.
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Question about intersection multiplicity of a curve and it's tangent line

If we have a double point $a$ on some complex curve, call it $C$, defined by some polynomial $f$ and we have only one tangent line at $a$, call it $T_l$, then the intersection multiplicity $I(a,f \cap ...
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54 views

Differential forms on a projective curve: two constructions

Let $X$ be a projective curve on a perfect field $k$ ($k$-scheme integral, separated, of finite type) and let $K$ be the function field of $X$. Let's compare the following two constructions: 1. ...
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$L$ is algebraic curve. Why does $\nabla f({x_0},{y_0}) = 0$?

Let $L = \left\{ {(x,y):f(x,y) = 0} \right\}$ be algebraic curve, $f$ is polynomial and its coefficients are real, and we know differentiability in $(x_0,y_0)$ is lost, for $L$. Why does $\nabla ...
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Some Basic Properties of Tropical Amoebas

I am working through Sturmfels' new book on Tropical Geometry with another student, and we are stuck at a pretty important concept, namely the basic properties of amoebas. Let me reproduce a bit of ...
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Clarification in a theorem statement regarding intersection of Complex Algebraic Curves in $P_2$

I have a theorem in the book Complex Algebraic Curves- Frances Kirwan : *If two projective curves $C$ and $D$ of degrees $n$ and $m$ respectively in $P_2$ intersect at exactly $n^2$ points and if ...
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36 views

Compute the genus of a curve with a flex point

The genus of a smooth plane curve is $g=\frac{(d-1)(d-2)}{2}$ and I know that if the curve has $n$ nodes the genus decreases by $n$. What happens if the curve has singular (non ordinary) points? In ...
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1answer
109 views

Number of inflection points of an algebraic projective curve

I' m trying to prove that a curve in $\mathbb{P^{2}(\mathbb{C})}$ of degree $d$ has an infinite number of inflection points or it has at most $3d(d-2)$ inflection points. Let be $C$ the curve and ...
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71 views

Question on a “dual form” of $F$ that caprures the singularity of a plane section of $F$

In an article I was reading the notion of dual form came up, which I write down below. I was interested in learning more about this and I have two questions regarding it. This is how it came up: ...
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1answer
61 views

The independence of the degree of morphisms between two curves on field extension.

Suppose $C_1$ and $C_2$ are two regular projective geometrically irreducible curves over $k$ and $F$ is a surjective morphism between them, then the degree of $F$ is the degree of the field extension ...
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15 views

Non Vanishing Thetanulls

Let $C$ be a smooth curve over $\mathbb C$, a theta characteristic $\kappa$ is said to be non-vanishing if $h^0(\kappa)=0$. Does there exist always a non vanishing theta characteristic?
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58 views

Intersection of n hyperplanes in projective space of dimension n is not empty

I want to prove the following: Let $H_1,\dots,H_n$ be $n$ hyperplanes in $\mathbb{P}^n =\mathbb{P}^n \mathbb{C}$. Then $\cap_{i=1}^n H_i$ is not empty. Please be noted that this is an exercise ...
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If ${L_1} \subseteq {L_2}$ and and $L_1$ is union of continuous curves and $L_2$be algebraic curve. Can we say that $L_1$ is piecewise $C^∞$ curve?

Let $L_2=\{(x,y):x,y\in R,f(x,y)=0\}$ be algebraic curve, and $f(x,y)$ is polynomial. If ${L_1} \subseteq {L_2}$ and $L_1$ is union of continuous curves. Can we say that $L_1$ is piecewise $C^∞$ ...
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49 views

Maps to $\mathbb{P}^1$ induced by rational functions.

I am reading from Hartshorne, Corollary II.6.10 page 138. Given a nonsingular (is this necessary?) curve $X$ over a field $k$, let $f\in K(X)^*\setminus k$. Then the inclusion of fields ...
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35 views

Can we say that, every algebraic curve is a piecewise ${C^\infty }$ curve?

Let $L$ be algebraic curve. Can we say that, $L$ is a piecewise ${C^\infty }$ curve?
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Euler characteristic singular surface

The setting is the one of algebraic curves over the complex numbers. It is known that in an irreducible nodal curve each node reduces the arithmetic genus by one: if $\tilde{C} \rightarrow C$ is the ...
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33 views

Projective Mapping of a Sphere

Show that the projective completion of the curve $Y=X^2$ is topologically a sphere. Consider the parametrization $X=t, Y=t^2$, where t ranges the sphere $\mathbb C\cup {\infty}$. How do I prove this ...
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A question on algebraic curve

Let $A_j \in \mathbb{C}^{n \times n}$, ($j = 0,1,2,\ldots,m$) and $P(z) = A_m z^m + \cdots + A_1 z + A_0$ is a matrix polynomial, and $z $ is a complex variable. $s_1 \ge s_2 \ge \cdots \ge s_n$ ...
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Geometric interpretation of algebraic property

Let $X \subset \mathbb{C}^n$ be an irreducible affine algebraic curve with coordinate ring $$\mathbb{C}[X] = \mathbb{C}[z_1, \ldots, z_n] / (f_1, \ldots, f_m ) $$ with each $f_i \in \mathbb{Z}[z_1, ...
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1answer
24 views

Image of morphism between curves

I have this projective curve $C\dots y^2z=x(x+2z)(x-z)$ and I have function $f$ on $C$, ie $f\in k(C)$ given by $f(x:y:z)=(y:z)$. What would be image of this function? Can I see it in some way as ...
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1answer
47 views

Order of the pole of projective curve using uniformizer

I need to find divisor of functon $f=y$ i.e. $\frac{y}{z}$ for projective curve $y^2z=x^3-xz^2$ and I have some questions: For example, I have pole: $(0:1:0)$ and zeros $(0:0:1),(1:0:-i),(1:0:1)$. It ...
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1answer
56 views

When is a hypersurface rationally connected?

A projective variety $X$ is said to be "rationally connected" if any two points on it can be connected by a map $\mathbb P^1 \to X$. Let $X$ be a smooth hypersurface in $\mathbb P^n_k$ defined by a ...
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41 views

Smooth completion of algebraic curves

I am having trouble understanding the concept of smooth completions of algebraic curves. I know the definition (smooth, complete curves which contain the curve X as an open subset ) and according to ...
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1answer
31 views

local equation of a divisor, localization and local ring at a point

Let $X$ be a regular, proper surface (integral, separated, of fin. type) over a perfect field $k$ and let $Z\subset X$ be an integral irreducible subscheme. Moreover $\eta$ is the generic point of $X$ ...