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 ...

learn more… | top users | synonyms

1
vote
0answers
45 views

Dimension of homogeneous polynomials passing through 4 points

Can anyone help me solve the following exercise? Let $p_1,p_2,p_3,p_4$ be distinct points in the projective plane $P^2$. What is the dimension of the vector space of homogeneous polynomials ...
2
votes
0answers
59 views

bases of a function field

I read this example Example 3.33: Consider the Hermitian curve with equation $x^{r+1}=y^r+y$ over the field $\mathbb{F}_{r^2}$. Here $Q = (0:1:0)$ and $X = x/z$, $Y=y/z$ is a monomial generating ...
1
vote
1answer
62 views

There exists a divisor linearly equivalent to $P$ not containing $P$

Let $X$ be a non-singular projective curve over $\mathbb C$ and let $P\in X$ be a closed point. Which is, in your opinion, the easiest way to prove that there exists a divisor $D$ of $X$ satisfying ...
1
vote
1answer
90 views

Every point lies on a unique secant through $C$

Let $C \subset \mathbb{P}^3$ be the twisted cubic (i.e., $C=\{(X_0^3:X_0^2 X_1:X_0 X_1^2:X_1 ^3) : (X_0,X_1) \in \mathbb{P}^1\}$). I need to show that every point $Q \in \mathbb{P}^3 \setminus C$ lies ...
1
vote
1answer
58 views

Generic point and pull back

Let's work over $\mathbb C$, Consider a finite morphism between two integral curves $f: X\rightarrow Y$, let $\eta_Y$ (resp $\eta_X$) be the generic points Questions: What is ...
0
votes
0answers
15 views

Can the Klein Quartic be parameterized by meromorphic upper-half plane functions?

It is known that Elliptic Curves in canonical form can be parameterized by the Weierstrass elliptic function and its derivative on a suitably chosen lattice: $$[\wp'(z)]^2 = ...
2
votes
0answers
35 views

Basepoint free line bundles in $Pic^d(C)$ is an open set?

Let $C$ be a smooth projective curve over $\mathbb{C}$. Let $Pic^d(C)$ denote line bundles on $C$ with degree $d$. Is the subset of basepoint free line bundles an open set of $Pic^d(C)$? Many thanks!
1
vote
1answer
51 views

Simple question from Kunz's book Introduction to Plane Algebraic Curves

I'd like to ask a question that arose when I was reading the book Introduction to Plane Algebraic Curves by Kunz. In the following passage, how can we conclude that ker$\alpha = (q)$? I can see that ...
0
votes
1answer
50 views

Flat families and section of family of curves

Let $X$ be a smooth projective surface. Let $L$ be an ample line bundle on $X$. Consider $Y=\{(x,C) : C\in |L|, x\in C\}\subset X\times |L|$. Then $p_2:Y\longrightarrow |L|$ is a family of curves over ...
0
votes
0answers
16 views

Compute $\ell(W+P)$ and $\ell(W-P)$ for a place $P\in\mathbb{P}(F/K)$ of degree $1$ and any $W$ canonical divisor.

Let $F/K$ be a function field of genus $g$. Let $P\in\mathbb{P}(F/K)$ be a place of degree $1$ and $W$ be any canonical divisor. Determine $\ell(W+P)$ and $\ell(W-P)$. If $\ell(P)=\deg(P)+1$, then we ...
0
votes
0answers
34 views

a coordinate change of a quadric in P^2 to get one of 3 curves

I need to show that for a quadric curve in P^2(k) of degree 2,if the characteristic of k is not 2 ,then up to a coordinate change any quadric is one of the following :1) non-singular quadric 2) a pair ...
0
votes
0answers
33 views

Hurwitz bound is not sharp for curves of genus 4.

I'm trying to prove that for any curve $C$ over complex numbers $\mathbb C$ of genus 4 the number of automorphisms is strickly less then $252$. I proved this inequality for hyperelliptic ones. As for ...
0
votes
1answer
75 views

The translation map between elliptic curves is a rational map

I want to see a reference or a prove that the following map is a rational map: Let E be an elliptic curve,$P\in E$ and $T_p$ defined as $T_p:E\rightarrow E,\text{ }T_P(Q)=P+Q$. It is important ...
0
votes
1answer
18 views

Reverse-engineering a parametrization

Let's say you have a polynomial depending on complex parameters $A,B$: $$p(x,y) = (y + Ax)(y + Bx) + (A-B)^2$$ One parametrization of zero points of this polynomial is given by $$ x(t) = -(t + ...
0
votes
1answer
94 views

polynomial curve fitting: higher order models' root mean square error does not decrease

I am trying to fit a curve for 15 data points. I started by creating a linear model and observing the root mean square difference, followed by quadratic, cubic and increasing the degree of polynomial ...
2
votes
1answer
79 views

Dimension of an affine cone without one variable is equal to the dimension of the projective algebraic set

Let $A:=V(F_1,...,F_k)\subset\mathbb{P}^n$ with $F_j\in k[X_0,...,X_n]$, a projective algebraic set. Let $C(A)\subset \mathbb{A}^{n+1}$ the affine cone over $X$. Show that $\dim A=\dim B$, where ...
2
votes
1answer
26 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 ...
0
votes
0answers
31 views

Irreducible components of a cone

I have the following definition for an affine cone: Let $Y\subset \mathbb{P}^n$ be a projective algebraic set, we define the affine cone associated to $Y$ as $C(Y)=\theta^{-1}(Y)\cup \{(0,...,0)\}$, ...
0
votes
1answer
34 views

Every irreducible component of an affine cone contains its vertex

Let $X=V(F_1,...,F_k)\subset \mathbb{P}^n$with $F_i\in k[X_0,...,X_n]$ an projective algebraic set. Let $C(X)\in \mathbb{A}^{n+1}$ the affine cone over $X$, that is $C(X)=\theta^{-1}(X)\cup ...
2
votes
1answer
59 views

How do I show that there is no irreducible algebraic set $Y \subsetneq \mathbb{A}^n$ such that $Y \supsetneq X$?

Let $f \in k[x_1, \dots, x_n]$ be irreducible. The variety $X = V(f)$ is called an irreducible hypersurface in $\mathbb{A}^n$. How do I show that there is no irreducible algebraic set $Y \subsetneq ...
2
votes
0answers
48 views

Use of Hilbert basis theorem to find a lone generator for $I$.

Let $I \subset \mathbb{C}[x]$ be the ideal generated by $\{x^n + n: n \in \mathbb{N}\}$. How do I use the proof of the Hilbert basis theorem to find a single generator for $I$?
2
votes
1answer
89 views

Global sections when we tensor by a degree zero line bundle

Let $C$ be a smooth projective curve over $\mathbb{C}$. Let $A$ be a degree $d$ line bundle on $C$, and $M$ be a degree 0 line bundle on $C$ such that $M^2=\mathcal{O}_C$, that is, it is a 2-torsion ...
3
votes
0answers
39 views

Is the given set an open subset of the $\mathcal{G}^r_d(|L|)$

Let $X$ be a smooth projective surface over $\mathbb{C}$. Let $L$ be a very ample line bundle on $X$. We have a variety $\mathcal{G}^r_d(|L|_s)$ associated to the linear system of curves $|L|$. The ...
0
votes
1answer
47 views

About the degree of a tensor power of a line bundle on a curve

Suppose that $H$ is a hyperplane in some $n$ dimensional complex projective space and $C$ is a smooth curve of positive genus. Can I say that $\deg((H|_C)^{\otimes 2}) = 2\deg(H|_C)$?
3
votes
2answers
48 views

Find polynomial equation for a cardioid in $\mathbb{R}^2$

We have the cardioid with equations: $$x(\theta)=\cos\theta+\frac{1}{2}\cos(2\theta)$$ $$y(\theta)=\sin\theta+\frac{1}{2}\sin(2\theta)$$ I have to show that you can define this cardioid with a ...
0
votes
1answer
72 views

Are elliptic curves algebraic varieties?

I got a short question. Are elliptic cubes also algebraic varietes? Say we have $E:y^2=x^3+5x=:f(x)$ Then we can $f(x)=x(x^2+5)$ So it can't be an algebraic variety.. I feel like I am totally ...
0
votes
2answers
42 views

Solving Integration problem, area of a curve?

I am working my way through a self study book of scientific and engineering principles. Although it covers a pretty broad range of subjects, there are obviously a few maths related topics in it as ...
0
votes
1answer
53 views

Rational maps between elliptic curves

I dont understand the definition of rational maps. Here is the definition: Let $E_1$ and $E_2$ be elliptic curves over a field $K$. (projectively written). A rational map $\Phi:E_1\rightarrow E_2$ ...
0
votes
0answers
32 views

Difference between $ X $ and $ X( \mathbb{C}) $ in a special case.

I would like to know why is, in the case of $ X $ is a smooth curve over $ \mathbb{C} $, $ X( \mathbb{C} ) $ a Riemann surface ? Thank you in advance for your help.
1
vote
1answer
40 views

Algebraic curves, intersection

Given the two plane curves, $$F=2X_0^2X_2-4X_0X_1^2+X_0X_1X_2+X_1^2X_2$$ $$G=4X_0^2X_2-4X_0X_1^2+X_0X_1X_2-X_1^2X_2$$ I want to calculate the multiplicity of the intersection at $(1:0:0)$, but I have ...
0
votes
0answers
60 views

Inflection points on cubic curves

Let's call $\Lambda$ the set of cubics generated by, $$F=2X_0^2X_2-4X_0X_1^2+X_0X_1X_2+X_1^2X_2$$ $$G=4X_0^2X_2-4X_0X_1^2+X_0X_1X_2-X_1^2X_2$$ I'm trying to find all the points that are inflection ...
2
votes
1answer
54 views

One point in intersection of conic and cubic, intersection multiplicity?

Let $C \subset \mathbb{P}^2$ be the conic defined by $P(x, y, z) = xz + y^2$,and $D \subset \mathbb{P}^2$ be the cubic defined by $Q(x, y, z) = y^2z - x^3 + xz^2$. Show that there exactly one point $p ...
2
votes
1answer
46 views

Nonsingular curve has 12 points of inflection. [closed]

Let $C \in \mathbb{P}^2$ be the curve defined by the polynomial$$P(x, y, z) = x^4 + y^4 + z^4.$$ Show that $C$ is nonsingular. Show that $C$ has exactly $12$ points of inflection.
0
votes
1answer
82 views

Pullback of globally generated line bundle is globally generated?

Let $f:C\longrightarrow C'$ be a finite morphism of curves. Let $A'$ be a line bundle on $C'$ and let $A$ be it's pullback to $C$. If $A'$ is globally generated, is it true that $A$ is globally ...
5
votes
1answer
100 views

Singular points of algebraic curve, multiplicity, ordinary?

Let $C \in \mathbb{P}_2$ be the curve defined by the polynomial$$P(x, y, z) = x^2z^2 + y^2z^2 + y^4.$$Find the singular points of $C$. For each one, calculate the multiplicity and say whether it is ...
1
vote
1answer
99 views

normalization of a curve with a node is not flat

Given the ring $$A = \frac{K[x,y]}{y^2-x^2(x+1)}$$ I know that its normalization is $K[t]$, where $$x\mapsto t^2-1\qquad y\mapsto t^3-t$$ I have to show that the normalization map is not flat. I know ...
2
votes
1answer
92 views

example of computing ramification index

I am trying to understand example 2.2.9 of Silverman's "Arithmetic of Elliptic Curves". In this example, Silverman considers a map $$ \phi:\mathbb{P}^1\to \mathbb{P}^1; [X,Y]\mapsto [X^3(X-Y)^2, ...
4
votes
1answer
106 views

Fulton Algebraic Curves, Exercise 2.32(a).

Let $R$ be a DVR satisfying the conditions of Problem 2.30. Any $z \in R$ then determines a power series $\lambda_i X^i$. If $\lambda_0, \lambda_1, \dots$ are determined as in Problem 2.30(b). Show ...
1
vote
0answers
68 views

Automorphisms of cubic nodal curve

How to calculate the automorphism group of the nodal cubic curve $y^2=x^3+x^2$ ? Should I use the rationality of this cubic curve ?
1
vote
2answers
127 views

Sketch the graph $x=e^{-t}\sin t$,$ t\ge 0$

My graph is always negative though, and that doesn't make sense cause $t$ is supposed to be positive. I substituted $x$ as $y$ and $t$ as $x$.
2
votes
1answer
53 views

Is the quotient morphism from product of curves to to their symmetric product flat?

Suppose $C$ is a smooth curve, is the morphism $C^2=C\times C\to C^{(2)}=C\times C/S_2$ flat? What about the general case?
0
votes
0answers
23 views

Tweaking function to reduce the rate of decay of a logarithmic based curve

Im not even sure if this is possible or perhaps I may need to use a different function altogether but I currently have one that looks like this: $$y = a\log(x+b)+c$$ That produces the red curve ...
1
vote
1answer
78 views

Dimension of $\mathfrak{m}^k/\mathfrak{m}^{k+1}$?

Let $C \subset \mathbb{P}_\mathbb{C}^2$ be a projective curve, and $p \in C$ a point of multiplicity $m$. If $\mathcal{O}_p(C)$ is the local ring of $C$ at $p$, and $\mathfrak{m} \subset ...
3
votes
1answer
42 views

Nonsingular cubic curve, quotient of $d(x/z)$ and $y/z$ is differential which is regular everywhere.

Let $C \subset \mathbb{P}_2$ be the cubic curve defined by$$y^2z = x(x-z)(x-\lambda z)$$with $\lambda \in \mathbb{C} - \{0,1\}$. Let $p = [0, 0, 1]$, $q = [1, 0, 1]$, $r = [\lambda, 0, 1]$, and $s = ...
5
votes
1answer
66 views

Riemann-Roch analysis of point divisor ring on smooth genus 3 Riemann surface

Let $C=C_4\subset\mathbb{P}^2$ be the smooth genus 3 Riemann surface given by a quartic curve. Let $P\in C$ be a point, and $D=P$ the divisor given by the point $P$. Let ...
1
vote
1answer
62 views

Surface area of circle extracted from a tube wall

I have made a hollow tube (thickness $1$mm) having inner radius $89$ mm and outer radius $90$ mm (length $400$ mm, can be higher). then I made a circular (circle radius $25$ mm) cut perpendicular to ...
1
vote
0answers
41 views

Subsheaves of locally free sheaves on a rational curve

Let $k$ be an algebraically closed field, $\mathcal{E}$ a locally free sheaf on $\mathbb{P}^1_k$ and $\mathcal{L} \subset \mathcal{E}$ a subsheaf of rank $1$. By Grothendieck's theorem, we know that ...
6
votes
0answers
108 views

Siegel's Theorem for Genus 0 Curves over Function Fields

Let $V$ be a quasi-projective algebraic curve over a field $k$. Are there finitely many morphisms from $S$ to the triply-punctured sphere $\mathbb{P}^1 -\{0,1,\infty\}$? This is true if $k = ...
0
votes
1answer
48 views

Tangent from points on a curve meeting the curve again and again

tangent at a point C1 on the curve y=x^3 meets the curve again at C2 .the tangent at C2 meets the curve at C3, and soo on, so that the abscissa of c1,c2,c3.....,Cn form a G.P. find the ratio of area ...
1
vote
1answer
46 views

Proving $x^2 - x = y^5 - y$ is a hyperelliptic curve

Greetings to one an all! How can we prove the curve "$x^2 -x = y^5-y$" is a hyperelliptic curve? Is a hyperelliptic curve the same as a hyperbolic elliptic curve or are there any differences?