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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11 views

Hartshorne IV.6.4 - no curve of degree 9 and genus 11 in P^3

I'm working on this exercise in Hartshorne: there are no curves of degree 9 and genus 11 in $\mathbb{P}^3$. The hint says to show that it would have to lie on a quadric surface. This is the part I'm ...
1
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1answer
23 views

Proof that Epicycloids are Algebraic Curves?

Epicycloids are most commonly described by the parametric equations, $x(t) = (R + a)\cos(t) – a \cos \left(\frac{R + a}{a} t \right),$ $y(t) = (R + a)\sin(t) – a \sin \left(\frac{R + a}{a} t \right)...
5
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1answer
127 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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1answer
44 views

Definition of singular points on an algebraic curve

From what I understood, given a point $p$ on a scheme $X$ over a field $k$, we have \begin{equation} \dim \mathcal{O}_{X,p} \leq \dim_{\mathcal{O}_{X,p}/\mathfrak{m} }\mathfrak{m}/\mathfrak{m}^2 \end{...
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2answers
807 views

The genus of the Fermat curve $x^d+y^d+z^d=0$

I need to calculate the genus of the Fermat Curve, and I'd like to be reviewed on what I have done so far; I'm not secure of my argumentation. Such curve is defined as the zero locus $$X=\{[x:y:z]\...
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0answers
36 views
0
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1answer
50 views

Why $\frac{{\partial D}}{{\partial x}}$ and $\frac{{\partial D}}{{\partial y}}$ don't have any common factor?

Let ${A_j} \in {\mathbb{C}^{n \times n}},0<{w_j}\in \mathbb{R} (j = 0,1,2....m)$ and $\lambda $ is a complex variable such that $\lambda=x+iy$ and $x,y\in \mathbb{R}$. ${\rm{P(}}\lambda {\rm{) = ...
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1answer
129 views

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 $C_n=\{(x,y): y^2 = (x-1)(x-2)\cdots(x-2n)\} \subset \mathbb{C}^2$. He claims that the topology of this curve is ...
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0answers
10 views

Conics not contained in any plane in $\mathbb{P}^3$

The cubic twisted curve is the most common example of a curve in $\mathbb{P}^3$ which is not contained in any plane. I was wondering if it is possible to find a conic in $\mathbb{P}^3$ that is not ...
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3answers
64 views

What is the power series expansion at $x=0$ of the algebraic function defined by $(27x-4)y^3 + 3y + 1 = 0$?

Let $y$ denote the complex-valued algebraic function defined implicitly near $x=0$ by $(27x - 4)y^3 + 3y + 1=0$ and such that $y(0)=1$. What is the power series expansion of this function at $x=0$? ...
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0answers
64 views

Riemann-Roch and quartic

I know very little in algebraic geometry, but I want to learn!! So I know the Riemann-Roch theorem as follow: let $$L(D)=\{\text{ meromorphic functions, s.t. }\operatorname{div}(f)\geq D \}$$ and $$...
3
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0answers
100 views
0
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0answers
46 views

Exterior Algebra VS Torsion

Let $C$ be an irreducible and reduced rational curve, and $f: \mathbb P^1\rightarrow C$ be the normalization. If $\mathcal F$ is a coherent sheaf of rank $r$ over $C$, then I was wondering if we can ...
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0answers
14 views

intersection multiplicity from kirwan’s book

I'm working on F. Kirwan's Complex Algebraic Curves. To define intersection multiplicity, Kirwan choose some special projective coordinates and calculate the resultant. She claims before the ...
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0answers
17 views

Vanishing polynomial in complex projective space

Assume we are working in $n$-dimensional complex projective space. Why does a (homogeneous) polynomial of degree less than or equal to $d - 1$ which equals $0$ on $d$ points on a line $L$ in ...
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0answers
24 views

Help with corollary 4.6 in Griffiths

Corollary 4.6 (P.72) in Griffith's 'Introduction to Algebraic Curves' proves that $\mathcal{O}=\mathbb{C}\{x,y\}=$set of all holomorphic functions in $x,y$ is a UFD, using the Weierstrass preparation ...
2
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0answers
39 views

Riemann-Roch Space for Quotient Curve

Let $C$ be a curve defined over a finite field $\mathbb{F}_q$. Let $\{f_1,..f_m\}$ be a basis for the riemann-roch space of functions, L(D), for the divisor $D= t\infty$. Suppose you have a subgroup ...
2
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1answer
15 views

Bounding the number of points at infinity of a curve of degree $n$.

I'm trying to prove the following statement. Let $C$ be a curve of degree $n$. Give a bound for the number of points at infinity. I tried it for $C$ defined by a polynomial in two variables only. ...
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0answers
30 views

How do I show the tangent to an elliptic curve over the complex numbers meets the elliptic curve at another point?

If $E(\mathbb{C})$ is an elliptic curve given by $y^2=ax^3+bx+c$ for $a,b,c\in \mathbb{C}$, and $\ell$ is a line tangent to $E(\mathbb{C})$ at some point $p$, then why does $\ell$ meet $E(\mathbb{C})$ ...
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0answers
14 views

In a DVR, why does $u=f(t,u)$, with $f$ a homogeneous polynomial of degree $3$ and $t$ a uniformizer, imply $\nu(u)=3$?

In this answer by Georges Elencwajg, it is stated that $$u=t^3-\dots-e_1e_2e_3u^3=\text{a homogeneous polynomial of degree $3$ in t,u}\quad(\ast)$$ [...] Now in the local ring $\mathcal O_{...
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0answers
27 views

Reference request on general Hurwitz schemes

Let $G$ be a fixed finite group. I'm interested in the structure of the set $\mathcal{H}_{r,g,h,G}$ of tuples $(C,f,\delta)$, where $C$ is a smooth projective genus $g\geq 2$ curve, $\delta:G\to\mbox{...
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1answer
36 views

Group of $\mathfrak a$-torsion points

Silverman defines the Group of $\mathfrak a$-torsion points of an elliptic curve $E/\mathbb C$ (with $\mathfrak a$ an ideal in $\mathrm{End}(E)$) in Advanced topics of elliptic curves as $$E[\mathfrak ...
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1answer
42 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
159 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} Ax²+Bx+C=u²,\...
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0answers
26 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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0answers
10 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 H^1(C,\wedge^{i+1}...
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1answer
69 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 $x\...
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1answer
26 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 $c:\mathbb{R}\...
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0answers
17 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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0answers
46 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 ...
4
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1answer
73 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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8 views

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 non-...
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1answer
87 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 $g'(x)=a'x^{4}+b'x^{3}+c'x^{2}+d'x+e'\in{k[...
2
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0answers
27 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 ...
2
votes
1answer
31 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 ...
4
votes
4answers
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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0answers
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 ...
1
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1answer
27 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 ...
2
votes
2answers
49 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 ...
3
votes
1answer
52 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 ...
3
votes
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 I've ...
4
votes
1answer
127 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 ...
1
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1answer
95 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
63 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 ...
2
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0answers
78 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 $X(\mathbb C)$...
4
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1answer
45 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 \mathbb{P}...
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0answers
24 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 ...
0
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1answer
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 ...
2
votes
0answers
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 ...
3
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0answers
77 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 $\...