# Tagged Questions

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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### 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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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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 $t$ (...
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### 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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### 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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### 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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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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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### 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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### 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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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### $L(D)$ 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 ...
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### 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 ...
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### 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 $\mathbb{C}$ ...
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### 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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### 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, ...
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### 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 ...
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### 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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### 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 $\overline{\mathbb Q}$....
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### 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 ...
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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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 ...
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 ...