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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14 views

Of $n^2$ points of intersection, $np$ lie on curve of deg. $p < n$, then remaining $n(n - p)$ lie on a curve of deg. $n - p$

Let $C$, $C'$ be two plane curves of degree $n$. Is the following statement true or not? Suppose that of the $n^2$ points of intersection, $np$ lie on a curve of degree $p < n$, then the ...
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33 views

Show that the set $\{ P \in E(\mathbb{Q})\ |\ h(P) \le M \}$ is finite, for any constant $M$.

Show that the set $\{ P \in E(\mathbb{Q})\ |\ h(P) \le M \}$ is finite, for any constant $M$. Here $h(P)$ is logarithmic height of $P$, that is, $h(P):=\log H(P)$ and $H(P)=H(x)$, for $P=(x,y) \in E(...
2
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0answers
38 views

Geometric equivalent of the degree zero divisor class group of an algebraic function field (in the singular case)

In an algebraic function field $F/k$ we have the degree zero divisor class group $\text{Cl}^0(F/k)$. Now since any such function field corresponds to the function field of a normal projective curve ...
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1answer
29 views

How to linearlize level curves at a saddle point

Let $f(x,y)$ be a real-valued function on a domain $D$ in $\mathbb{R}^2$, and let $(x_s, y_s)$ be a saddle point of $f(x,y)$ in $D$. That is to say, \begin{align} \frac{\partial f}{\partial x}(x_s, ...
3
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1answer
74 views

Action of $\mathbb{Z}/3\mathbb{Z}$ on $P^{1}$

I am reading from the book Topics in Galois theory by Serre. I have the following question , take $G=\mathbb{Z}/3\mathbb{Z}$. The group $G$ acts on $P^1$ by $$\sigma x\;=\;1/(1-x)$$ where $\sigma$ ...
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2answers
73 views

What is the difference between surface and algebraic curve in general?

The question may seem dumb at first glance. But I couldn't figure out a satisfying answer after some research. A friend of mine told me that in an interview, she was asked to explain the sliding mode ...
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35 views

branched cover of projective line over rationals

I was reading something when I came across the following phrase "branched cover of projective line over rational " . To understand what does author mean , I started reading , Now I know about ...
3
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0answers
27 views

Add $P$ to itself $N$ times on elliptic curve $y^2 = f(x)$, end up with expression in denominator of $x$ vanishing iff $NP$ is point at infinity?

See the second to last paragraph from page 39 of Koblitz's Introduction to Elliptic Curves and Modular Forms. Why is it that when we add a point $P$ to itself $N$ times on an elliptic curve $y^2 = ...
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2answers
12 views

Find the values of p for which there is no turning points in the curve

The question: So, I have done almost everything. I am in the last part of the question. This is how I did it: I differentiated the curve $y = x^3 + px^2 + px$ and got: $dy/dx = 3x^2 + 2px + p$ I ...
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1answer
40 views

Algebraic Curve

I know that a curve of the type $$\vec{\sigma}(t)=\cos(mt)\hat e_1+\cos(nt)\hat e_2$$ with $m,n\in\mathbb{Z}$ is algebraic. My question is: what is the polynomial that define this curve?
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2answers
105 views

Proof verification affine curve not isomorphic to plane curve

I'm trying to prove that the affine curve $X\subset\mathbb{A}^3$ given by $\alpha:\mathbb{A}^1\to\mathbb{A}^3$, $t\mapsto(t^3,t^4,t^5)$, is not isomorphic to a plane curve. Here is what I've done: it ...
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0answers
29 views

Finding valuations/uniformizers for the branches of the blow up of a singular curve

I understand that for a nonsingular curve $C(x,y)$, the uniformizer at a point $(a,b)$ is either $x-a$ or $y-b$, since the partial derivatives with respect to $x$ and $y$ are not both 0. However, if ...
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1answer
104 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 ...
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1answer
46 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)...
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1answer
46 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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1answer
51 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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0answers
13 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
65 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 $x=0$? ...
3
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0answers
104 views

Is this true that, any algebraic curve has finitely many singularities?

Can we say that any algebraic curve has finitely many singularities?
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66 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 $$...
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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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15 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
20 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 ...
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0answers
42 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
votes
1answer
16 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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32 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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15 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
30 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
37 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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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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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
18 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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1answer
77 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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11 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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0answers
29 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
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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 ...
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47 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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0answers
29 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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1answer
54 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
132 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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vote
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 ...
6
votes
1answer
130 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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vote
1answer
71 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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votes
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 ...
0
votes
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 ...
4
votes
1answer
46 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}...
1
vote
0answers
25 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 ...