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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37 views

Extrema and inflection points for $x^3 - 3x^2 + kx$

My girlfriend has a problem with her math task. I did all this stuff years ago when so I am pretty behind and clueless what to do. She has following function: $x^3 - 3x^2 +kx $$ Her tasks are ...
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14 views

Solving a curve of fifths

I have five questions (A to E) used in a scorecard, all are currently ranked 0 or 1 meaning if all are answered 1, the total score possible is 5. I want the total of all to be 100 where the increments ...
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34 views

Function fields for genus 2 curves

Let's say you were given a genus one algebraic curve by the equation $y^2 = (x-a)(x-b)(x-c)$ and you wanted to parametrize it. We could go ahead and convert it to Weierstrass form: $y^2 = 4t^3 - g_2t ...
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43 views

Finiteness of morphism of curves with fixed image

This question comes from the proof of "bend and break" lemma in "Higer-dimensional algebraic geometry" (p.59-60). I use the notations in compatible with the notation given there for convenience. Let ...
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63 views

Are $k$ points on a smooth algebraic plane curve ever in general position?

Let $C$ be a smooth plane curve of degree $d$ and genus $g=\frac{(d-1)(d-2)}{2}$. Let us choose $k\leq g+3d-1$ points on $C$. Is it true that the dimension of the space of plane curves of degree $d$ ...
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72 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
40 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 ...
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45 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 ...
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70 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. ...
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28 views

How many inflection points can a spatial parametric cubic curve have?

So in my knowledge, a planar cubic polynomial curve always has one and only one inflection point. My question is, for a spatial parametric cubic curve parameterised by t, which has the equation of: ...
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52 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 ...
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41 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 ...
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59 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 ...
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65 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 $t$ ...
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75 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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46 views

How to decide if an ideal in $\mathbf Q[X,Y]/(P)$ is principal?

Let $P(X,Y)$ be an irreducible polynomial in $\mathbf Q[X,Y]$. Given an ideal $I$ of the quotient ring $\mathbf Q[X,Y]/(P)$ (say given by a set of generators) how can I decide if $I$ is principal or ...
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103 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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44 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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54 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
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34 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 ...
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78 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, ...
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66 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 ...
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59 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 ...
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62 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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62 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 ...
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24 views

Cubic curves vs conics

What is the main difference between cubic curves and conics, i.e. why can cubic curves develop singularies while conics cannot? Is this in some way related to Poincare-Bendixon theorem of chaos ...
2
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1answer
77 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
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1answer
41 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 ...
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81 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 $E': y'^2 = x'^3-\frac{1}{3}x' - \frac{52}{27}$. ...
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53 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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112 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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127 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 ...
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2answers
74 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: ...
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1answer
25 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 ...
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60 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 ...
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1answer
61 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
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1answer
61 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 ...
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1answer
33 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 ...
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187 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 ...
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1answer
62 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 $\mathbb{C}$ ...
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24 views

Could a hyper elliptic curve of degree 5 admit only three ordinary double points?

The definition for hyper elliptic curves is those curves which admit a ramified double cover of $P^1$. The given degree five is for some homogeneous equations of degree five that defines the curve, ...
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61 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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65 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, ...
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1answer
80 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 ...
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72 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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1answer
107 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 ...
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54 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 ...
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203 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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14 views

Building a nonsingular curve

I have a quintic surface, defined by a homogeneous polynomial in the variables $x,y,w,z$ in $\mathbb{P}^3$. I know the polynomial to have the form $$ xP_1 + wP_2 + (y-z)^2(y^3 + z^3)=0 $$ where the ...
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1answer
67 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 ...