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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1answer
98 views

An question on effective divisor (Clifford 'S theorem)

For an effective divisor $D\ge 0$ on a curve $Y$, define $$\lvert D\rvert =\{ D' \in \mathrm{Div}(Y) \mid D'\ge 0 \;\text{ and }\; D' \sim D \}$$ where $D\sim D$ means $\exists$ a rational ...
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1answer
44 views

get the length of a curve with integral

I need to get the length of a curve which equation is : $$y= (4-x^\frac{2}{3})^\frac{3}{2}$$ I need to find the length using the method : $$L=\int_a^b \sqrt{ 1 + \left(\frac{dy}{dx}\right)^2}$$ So ...
1
vote
1answer
60 views

A question on the morphism of projective varieties

The continuation of this, my question I want to show that $X$ and $Y$ are smooth and irreducible curves then $f(X)$ is either $Y$ or a point. Note that I know the proof of this ...
2
votes
1answer
82 views

the diagonal $\Delta (Y) =\{(y,y)\in Y \times Y\}$ is closed in $\Bbb P^m \times \Bbb P^m $

Asumme that $\Pi : \Bbb P^n \times \Bbb P^m \rightarrow \Bbb P^m $is a closed map. $X\subset \Bbb P^n$ And $Y\subset P^m$ $f: X\rightarrow Y$ be a morphism of projective varieties. the ...
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0answers
109 views

Can a “negative degree” line bundle on a reducible curve have global section?

Suppose $A,B$ are curves on smooth projective surface, having no common components and intersect, so $(A.B)>0$,do we have $H^0(O_A(-B|_A))=0?$ (here $A,B$ are effective divisors, may be irreduced ...
2
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0answers
74 views

Étale cohomology and Picard group of curves

Say we have $X$ a smooth projective curves over $\mathbb{Q}$, then I know there is an isomorphism $H_{ét}^1(X\times_\mathbb{Q}\overline{\mathbb{Q}},\mathbb{G}_m)\cong Pic(X\times_\mathbb{Q}\overline{\...
2
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1answer
142 views

intersection multiplicity from Shafarevich

In Basic Algebraic Geometry, Shafaravich proves the following theorem: Theorem. If $X$ is an irreducible affine curve, and $P \in X$ is a nonsingular point, then there is a function $t$, regular at $...
3
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1answer
120 views

local parameter on an irreducible affine algebraic curve

On page 14 of Shafarevich's Basic Algebraic Geometry 1, it is stated that for an irreducible affine algebraic curve $X: f(x,y) = 0$, and a nonsingular point $P \in X$, there is a regular function $t$ (...
2
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1answer
118 views

Cubic curve in projective space

Is it true that every cubic curve in $\mathbb{P}^3$, which is not contained in a plane, can be parametrized by polynomials? $\\\\\\\\$
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1answer
124 views

Why is the projection map proper?

It might be a silly question. I got stuck there. For the context, see S.K.Donaldson's Riemann Surfaces, chapter 4, section 4.2.3. Suppose $P(z,w)=a_0(z)+a_1(z)w+\dotsb+a_n(z)w^n\in\mathbb C[z,w]$ is ...
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51 views

Automorphisms of rational curves

Let $X$ be a non-empty open subscheme of $\mathbb P^1_{\mathbb C}$. What is the automorphism group of $X$ in terms of PGL$_n(\mathbb C)$ and the points on the boundary?
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1answer
113 views

Defined matrix in Catmull Spline Curve

I am trying to use Catmull spline curve in my program , I am trying to understand it but why we only use below given Matrix , because the examples I saw I only found the below one In Catmull spline ...
2
votes
1answer
72 views

covering of projective curve by affine parts

For $\mathbb{P}^n$ we can let $U_i = \{(x_1:\cdots:x_i:\cdots:x_{n+1}) : x_i \neq 0\}$. Then let $C \subset \mathbb{P}^n$ be a projective plane curve. We can decompose $C$ into a union of affine plane ...
2
votes
2answers
188 views

Eliminating a parameter when intersecting a manifold with a hyperplane

In the Euclidean space $\mathbb R^4$ we look at the intersection of the equations$$x^2 + y^2 = 1 \\ z^2 + w^2 = 1$$ sometimes known as the Clifford torus. This is known to be a 2-dimensional manifold, ...
2
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0answers
79 views

Stereographic projection for a link/knot

I've been trying to understand the topological "link" between algebraic varieties and their associated knots/links and to this end I've been reading F. Kirwan's book, "Complex algebraic Curves". The ...
0
votes
1answer
126 views

definition of affine plane curve

Let $k$ be a field. On page 5 of Milne's Elliptic Curves, the author defines an algebraic curves to be defined by polynomials $f \in k[x,y]$ with no repeated irreducible factors in $\overline{k}[x,y]$....
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1answer
219 views

multiplicity of intersection of projective curves

This example is taken from Silverman and Tate's Rational Points on Elliptic Curves. Suppose we have the line $C_1: x+y=2$ and the circle $C_2: x^2 + y^2 = 2$. We see that these intersect at the ...
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0answers
139 views

Surjective morphism of complete non-singular curves is normalization

My syllabus on algebraic geometry states the following: ''Let $\phi: X \to Y$ be a surjective morphism of complete non-singular curves. Then $X$ is the normalization of $Y$ in the function field of $...
2
votes
1answer
83 views

Question about divisors

Let $\Lambda=<2\omega_1,2\omega_2>$ be a lattice in $\mathbb{C}$, $C=\mathbb{C}/\Lambda$ an elliptic curve. Let $O(0,0)$ (neutral element of the lattice), and $A \in \mathbb{C}/\Lambda$ point of ...
3
votes
1answer
56 views

branched covering factors through a primitive one

I'm struggling with an assertion I found in an article I'm reading. A projective complex curve $X$ is rationally uniformized by radicals if there exists a branched covering $X\to \mathbb{P}^1$ such ...
4
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2answers
113 views

Isomorphic Elliptic Curves

I want to solve the following exercise: Show that the two elliptic curves $E/ \mathbb{Q}$ and $E'/ \mathbb{Q}$ are isomorphic. $E: y^2 = x^3+x-2$ and $E': y'^2 = x'^3-\frac{1}{3}x' - \frac{52}{27}$. ...
3
votes
0answers
61 views

Inequality involving multiplicities of points introduced via Quadratic Transformations of a Plane Curve

I've been learning about the resolution of singularities for plane curves, and have become stuck at exercise 7.15 of Fulton's Algebraic Curves (page 91 of the PDF). The question is: Let $F=F_1, \...
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2answers
149 views

Looking for an introductory Algebraic Geometry book

I am looking for recommendations on an AG text to work through this summer, possibly with the help of a mentor. I would want this book to have some introduction to categories, and then develop the ...
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0answers
151 views

Computing the genus in positive characteristic

That's an exercise but I'm not so sure how to approach this. Let $k$ be a field of characteristic $p$ and let $f(t)$ be a polynomial in $k[t]$ of degree $d$. Let $C$ be the curve that corresponds to ...
3
votes
2answers
135 views

What's the relation between prime spectrum and affine space?

Let $A$ be a ring ,$X$ be the set of all prime ideal of $A$.For each subset $E$ of $A$,let $V(E)$ denoted the set of all prime ideals of $A$ which contain $E$. we have: $V(0)=X,V(1)=\emptyset$...
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1answer
53 views

Intersection Multiplicites

I have the following problem; Let $C = \{Q:=x_0x_2^2 -x_1(x_1-x_0)(x_1+x_0)=0\}$ and $L = \{ax_0 + bx_1 = 0\}$ be two projective curves with $(a,b) \ne (0,0)$. Let $p=[0,0,1]$, then I am asked to ...
0
votes
1answer
97 views

Bézier curve limits

Can be any curve of any shape (without sharp edges) described by Bézier curve with unlimited (but finite) number of control points? The answer to the question above would probably be no, because I ...
2
votes
1answer
66 views

$L(D)$ is Vector Space

Given a divisor $D$ on a curve $X$, define $L(D)=\{0\}\cup \{f \in k(X),f\ne 0 \, | (f)+D \ge 0\}$. where $(f)=\sum \nu_P(f)P$ and $ \upsilon_{P}(f)= |zeros| − |poles| $ of $f$ at $P$. I want to ...
2
votes
1answer
258 views

Calculating the projective closure with more than one generator

I am given a variety $X = Z(f_1,f_2)$ in affine 3-space (in $x,y,z$), and I would like to compute its projective closure $Y = Z(g_1,\dots,g_n)$ in projective 3-space (in $x,y,z,w$). I have seen this ...
2
votes
1answer
51 views

Proving that a map is a birational equivalence

I am trying to prove that the map $\phi:P^1\to X = Z(x^2y^3-z^5)$, given by $[r:s]\mapsto [u^5:v^5:u^2v^3]$ is a birational equivalence, i.e. that there exists some map $\psi:X\to P^1$ such that $(\...
4
votes
2answers
402 views

Coordinate ring of the unit circle is never a UFD?

I'm reading some notes about coordinate rings. On the third example on the second page, the author notes that the coordinate ring $K[\mathcal{C}]$ is not a UFD. If $f=X^2+Y^2-1$, then in $K[\mathcal{...
2
votes
1answer
94 views

Existence of finite morphism to projective line

As we all know, Belyi's theorem says: A complex curve $X$ is defined over a number field, if and only if there exists a finite morphism $t:X\to \mathbb{P}^1_\mathbb{C}$ of varieties over $\mathbb{C}$ ...
2
votes
1answer
68 views

Existence of $g_{d}^{r}$ implies existence of which $g_{d'}^{r'}$'s?

Suppose I have a Riemann surface with a $g_{d}^{r}$. I am wondering what $g_{d'}^{r'}$'s exist for sure. For instance, since $h^{0}\left(L\left(-p \right)\right)$ is either $h^{0}\left(L\right)$ in ...
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votes
2answers
68 views

5th order Polynomial not accurate enough?

I have a data plot XY that goes from (X 0-127, Y -70.0 - 6.0 db) Im using the 5th order polynomial function from plotting this data on this site [http://www.zizhujy.com/en-us/Plotter][1] However, ...
2
votes
1answer
156 views

Intersection Multiplicity and Multiplicity of Zeros in Polynomial

I study coding theory and we use the textbook Fundamentals of Error-Correcting Codes . In the section related to Algebraic Geometry Code, we need to compute Intersection Multiplicity of two curve in ...
3
votes
1answer
116 views

Proof of the Belyi's theorem: where it is really used the hypothesis?

Consider the Belyi's theorem: If a smooth projective curve $X$ is defined over $\overline{\mathbb Q}$, then there exists a finite morphism $X\longrightarrow\mathbb P^1(\mathbb C)$ with at most $3$ ...
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vote
1answer
125 views

When branch points lie in $\mathbb P^1\left(\overline{\mathbb Q}\right)$.

I'm working on the following problem for several days without finding any solution: Let $X$ be a complex smooth projective curve and suppose moreover that $X$ is defined over $\overline{\mathbb Q}$....
2
votes
2answers
73 views

Strange property of parametrization of a class of plane curves

My studies lead me to the following parametrization of perhaps a new class of plane curves ( which are similar in shape to the classical sinusoidal spirals but not identical ). If the curves are not ...
10
votes
3answers
219 views

Is there a better way to find the polynomial equation for this curve?

Consider the curve in $\mathbb{R}^2$ defined by the equation $$ x^{1/3} + y^{1/3} + (xy)^{1/3} = 1, $$ where $x^{1/3}$ denotes the real cube root of $x$, etc. Since the equation above involves only ...
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votes
1answer
203 views

Find all the intersection points of a vector parabola (in R3) and a sphere

Given that I have a vector in R3 (7t, 10t - 2t^2, 5t) | (These numbers are arbitrary for the sake of the process) A sphere centered at the point ( 15, 25, 10) with a radius of 20 There is a ...
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0answers
59 views

Divisors of Fermat Curves.

I've been studying Algebraic Geometry (in coding theory), and my book has been very ambiguous about what some of these ideas actually look like. So I was curious as to what some of the Divisors of ...
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votes
1answer
264 views

Exactness of the pullback of the Euler sequence

Let $(L,V)$ be a base point free $g^r_d$ on a curve $C$. Then we have an exact sequence $0\rightarrow M_{L,V}\rightarrow\mathcal{O}_C^{r+1}\xrightarrow{\theta} L\rightarrow 0$, where we just let $...
2
votes
1answer
119 views

Line Meeting a Plane Curve at One Point

Given a curve (smooth, projective, irreducible) $X$ in $\mathbb{CP}^2$, this curve meets all other curves in the same space. Generically, it will meet a line (a copy of $\mathbb{CP}^1$ in $\mathbb{CP}^...
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votes
4answers
322 views

What are some applications of the Weil conjectures for algebraic curves?

I have been interested in the Weil conjectures for some time, and the easiest place to start has been in studying them for elliptic curves. I've been able to see some of their applications and ...
4
votes
1answer
124 views

Degree of ramification divisor and number of fixed points under a group action.

Let $C$ be a projective plane nonsingular complex curve and a finite group $G$ acts on $C$. Consider the quotient $f: C\rightarrow C/G=:C'$. Then, by Riemann–Hurwitz $2g_C-2=|G|(2g_{C'}-2)+\deg R,$ ...
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0answers
109 views

Irreducible Linear Subspace

Let k be an infinite field. Prove that any linear subspace of $A_k^n$ is irreducible. My first question is, what would a linear subspace be? Is is a variety that is generated by linear equations? ...
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0answers
110 views

On the bidegree of a curve in $\mathbb{P}^1 \times \mathbb{P}^1$

I was reading Beauville's Complex algebraic surfaces, at page 5 there is an example in which curves in $\mathbb{P}^1 \times \mathbb{P}^1$ are classified by the bidegree up to linear equivalence. I'm ...
3
votes
1answer
73 views

Elliptic points on modular curves and a uniformising parameter

I'm working with the modular curves $X_0(p)$ for $p=2,5$, with $$f_2=q\prod_{n\geq1}(1+q^n)^{24}$$ and $$f_5=q\prod_{n\geq1}(1+q^n+q^{2n}+q^{3n}+q^{4n})^6$$ being inverses of the Hauptmodul and ...
3
votes
1answer
125 views

$d$-branched coverings and finite morphisms of degree $d$

Consider a smooth projective curve $X$ over $\mathbb C$ (so $X$ is a projective $\mathbb C$-scheme, integral, of finite type....), and let $t: X\longrightarrow\mathbb P^1_{\mathbb C}=\operatorname{...
7
votes
2answers
218 views

Branch points lie in $\mathbb P^1\left(\overline{\mathbb Q}\right)$

Let $X$ be a complex smooth projective curve and suppose moreover that $X$ is defined over $\overline{\mathbb Q}$. Now consider a finite map $f:X\longrightarrow\mathbb P^1(\mathbb C)$ of degree $d$ ...