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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41 views

A basis for forms of degree $d$ (Fulton, 2.35)

I am trying to solve this exercise from Fulton's book: (2.35)(c) Let $L_1, L_2, \dots,$ and $M_1, M_2, \dots$ be sequences of nonzero linear forms in $k[X,Y]$ and assume no $L_i = \lambda M_j$ for ...
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1answer
158 views

How can functions disagree with the values of its expansions at some points on an algebraic curve

I found a curve, in which some function has at least two expressions, which differ infinitely much!! Is there any error in the thoughts? The curve is defined by "\begin{equation} ...
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22 views

Genus of Curves over finte fields

This may be a dumb question but is calculating the genus of a curve define over a finite field different than over $\mathbb{C}$. For example the following curve: $y^8 + y +x^{12} + x^5$ is genus 14 ...
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9 views

Reference needed for an exact sequence of an ACM curve with a homogeneous coordinate ring.

Let $C\subset\Bbb{P}^n$ be an ACM (arithmetically Cohen-Macaulay) curve with homogeneous coordinate ring $R$. Then there is an exact sequence $$0\to \text{Tor}_i(R,\Bbb{C})_k\to ...
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67 views

Intuition about formal brances of a curve at a point

Consider an algebraic surface $X$ and a curve $Y\subset X$. Here $X$ is a $K$-scheme integral of finite type of dimension $2$ and $Y$ is a closed subscheme of dimension $1$. Fix a closed point ...
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24 views

What curves have a closed-form formula for projecting a point onto them in multiple dimensions?

What curves have a closed-form formula for projecting a point onto them in multiple dimensions? For example, give a simple, straight line $$ c(t) = v t $$ where $v\in\mathbb{R}^m$ and ...
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86 views

Understanding the topology of $y^2=(x-1)(x-2)(x-3)(x-4)$

Andreas Gathmann's lecture notes on algebraic geometry start by considering the curve $C_n=\{(x,y): y^2 = (x-1)(x-2)...(x-2n)\} \subset \mathbb{C}^2$. He claims that the topology of this curve is the ...
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12 views

Finding Primitive Elements of Separable Function Field Extensions

Suppose you have a a curve $C$ defined by an equation in $x$ and $y$. There is a map from $C$ to $\mathbb{P}_1$ by projection onto $x$. This corresponds to a separable extension of function fields ...
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39 views

Question about the rational normal curve and different representations of it.

I know the rational normal curve as the image of a polynomial map \begin{gather} \phi:K\rightarrow K^n\\ \phi(t)=(t,t^2,\dots,t^n) \end{gather} My question is proving the variety defined by the set ...
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69 views

History and future of algebraic curves and the like?

Now that Fermat's last theorem has been proven, and also elliptic curves see widespread use in simple everyday applications, I would love to learn how the related theories came into beeing, how they ...
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Let $f = c{f^{r_1}}_1 … {f^{r_s}}_s$ be the unique factorization of the polynomial

$\ \ \space$ We are in the field of the theory of algebraic curves. Here $F$ is a projective curve. Choosing a plan affine $L$ whose equation is given by $ax + by + cz = d$ with coefficients of ...
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86 views

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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24 views

How many points at infinity in Artin-Schreier type curve

Let $Y$ be an affine curve over a perfect (yet not necessarily algebraically closed) field $k$ given by $$y^p+a(x)y=b(x)$$ (abs. irreducible) with $p$ a prime number. Now one can normalize $k[1/x]$ in ...
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29 views

$Z(y^2-x^3) \subset \mathbb{A}_{\mathbb{R}}^2$ is not isomorphic to $\mathbb{A}_{\mathbb{R}}^1$

Prove that the algebraic variety $Z(y^2-x^3) = \{(x, y)\in\mathbb{A}_{\mathbb{R}}^2\,\,|\,\,y^2-x^3=0\}$ is not isomorphic to the affine space $\mathbb{A}_{\mathbb{R}}^1$. [i.e., there are no ...
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209 views

How do I write down a curve with exactly one rational point

Let $g\geq 1$. I would like to write down (for all $g$) a smooth projective geometrically connected curve $X$ over $\mathbf{Q}$ of genus $g$ with precisely one rational point. Is this possible? For ...
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27 views

Are there curves of genus 2 and higher over number fields with everywhere good reduction?

a theorem of Fontaine states that there are no curves of genus $\geq 1$ over $\mathbb Q$ with everywhere good reduction. For curves of genus one over number fields, this is not true. There are number ...
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24 views

smooth affine algebraic curves and their subschemes

I am reading a lot about curves at the moment and I am a little confused: Let $X= Spec K[X]$ denote a smooth affine algebraic curve. Then, according to some sources, the ring $K[X]$ is a Dedekind ...
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46 views

Example of a dominating map

Unfortunately the book that i am reading (Algebraic curves by Fulton) has no examples, so i am trying to find an example of a dominating map that would be helpful for the understanding of the ...
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1answer
51 views

A version of Bezout's Theorem

I have read the following version of Bezout's Theorem, but I don't get to understand how it implies the classical version. Let $F,G\in K[X_{0},X_{1},X_{2}]$ be non-constant homogeneous polynomials ...
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1answer
40 views

Prove that a general monomial curve is smooth

Let $k$ be a field, $n_1<n_2<\cdots<n_r$ positive integers, and $C:=\{(t^{n_1},...,t^{n_r})\mid t\in k\}\subset \mathbb{A}^r$. Show that $C$ is a smooth curve iff $n_1=1$. This is what ...
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118 views

Defining the set $\{(t^3,t^4,t^5) : t \in \mathbb{C}\}\subset \mathbb{C}^3$ by two polynomial equations

What are two polynomials $f,g \in \mathbb{C}[x,y,z]$ such that $$\{(x,y,z): f(x,y,z)=g(x,y,z)=0\}\;=\;\{(t^3,t^4,t^5): t \in \mathbb{C}\}$$ holds as an equality of subset of $\mathbb{C}^2$? This ...
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93 views

$V=\mathcal{Z}(xy-z) \cong \mathbb{A}^2$.

This question is typically seen in the beginning of a commutative algebra course or algebraic geometry course. Let $V = \mathcal{Z}(xy-z) \subset \mathbb{A}^3$. Here $\mathcal{Z}$ is the zero locus. ...
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5answers
62 views

Solving equations.

How would you solve these equations and show that they do not intersect each other? $$x^2+y^2=2x-2y$$ $$x^2+y^2=4(x^2+y^2)^{1/2} +y$$ It's isolating a term which I am struggling with. General ...
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77 views

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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43 views

Smooth curve of genus $1$ in $\mathbb{P}_{\mathbb{C}}^1\times \mathbb{P}_{\mathbb{C}}^1$.

This question comes from Gathmann's notes of Algebraic Geometry: Show that $$\{((x_0:x_1),(y_0:y_1)): (x_0^2+x_1^2)(y_0^2+y_1^2)=x_0x_1y_0y_1\}\subset \mathbb{P}_{\mathbb{C}}^1\times ...
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20 views

Calculating the intersection multiplicities of algebraic curves using Gröbner Basis

In my class, the lecturer told me that in order to calculate the intersection multiplicities for multiple space curves, sometimes I may have to calculate the Gröbner Basis. I just can't see how ...
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46 views

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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18 views

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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75 views

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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73 views

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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72 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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53 views

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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1answer
24 views

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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32 views

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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34 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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0answers
43 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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1answer
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 $ ...
2
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1answer
56 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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64 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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0answers
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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46 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
35 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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1answer
32 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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105 views

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
72 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
59 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 ...
3
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1answer
55 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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1answer
46 views

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 ...
3
votes
1answer
58 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. ...
1
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0answers
45 views

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 ...