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Questions tagged [algebraic-curves]

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 of curves are extensively studied as a basic case in singularity theory. Via algebraic function fields and modular curves they have links to number theory.

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Non-singular algebraic curve with no rational points

It is known that if $C$ is a non-singular algebraic curve of genus $g = 0$, then the set of rational points on $C$ is either empty or it has infinitely many points. Under these circumstances, is there ...
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Fulton Exercise 3.4 how to get double partial derivatives?

I have a question: $P =(0,0)$ is a double point on a curve $f(x,y) \in k[x,y]$ where $k$ is algebraically closed field. If degree of $f$ is $n$ and we write $f = f_n + f_{n-1} +......f_m $ where $f_{...
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Support for Fulton reading (chap 3 and 5) [closed]

I am studying on my own the book of the Fulton (algebraic curves). My goal is to get to chapter 5. I happen to encounter difficulties in two chapters, 3 and 5. So I would like to know if there is ...
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Intersection of the strict transform of a curve and exceptional hypersurface in the blow up of the affine plane.

Consider the curve $Y$ to be $(\frac{t^3}{1-t},\frac{t^4}{1-t})$ with $t$ different from 1. I need to parametrize the curve, calling $x$ the first coordinate ad $y$ the second we find that $t^3=x(1-t)$...
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Does a differential with zero divisor have any special characterization?

I know that on a projective algebraic curve, a function with a zero divisor must be constant. Is there a similar characterization for a differential with a zero divisor? For example, on the ...
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How to show there are only two discrete valuation rings with quotient field $k(x)$?

I want to show that only discrete valuation rings with quotient field as $k(x)$ containing $k$ are: $\mathcal{O_{a} (\mathbb{A^{1}})}$ for each $a \in k$ and $\mathcal{O_{\infty}}$; the former is the ...
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Irreducible components of $V(y^2-x(x^2-1))$

Let $V=V(y^2-x(x^2-1)$. It's easy to know $y^2-x(x^2-1)$ is irreducible in $\mathbb{C}[x,y]$, then $V$ is an irreducible curve in $\mathbb{A}^2(\mathbb{C})$. If we consider it in $\mathbb{A}^2(\...
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computation of the genus of a linear system on a surface

I am interested in figuring out why there is a $\mathbb P^1$ on the symmetric product of a genus $2$ curve as the part (iii) : As there is no rational curves on an abelian variety, we see the Abel-...
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Fulton example section $2.4$, help needed in clarification

This is an example in Fulton, Algebraic Curves. Let $V= V(xw-yz) \subset \mathbb{A^{4}(k)}$, and $\Gamma(V)$, the co-ordinate ring of $V$ is $k[x,y,z,w]/ (xw-yz)$. and, $\bar{x}, \bar{y}, \bar{z}, \...
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Miranda - Classification of Curves of Genus 4

I'm trying to understand pag. 207 of Miranda's book "Algebraic Curves and Riemann Surfaces" where it is explained which homogenous polynomial equations define a Riemann Surface of Genus Four embedded ...
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Sufficient conditions for a projective curve $X$ over $k$ to satisfy $H^0(X,\mathcal{O}_X) = k$?

Let $X$ be a one-dimensional proper (resp. projective) scheme over a field $k$. By Tag 0BY5 it is necessary that $X$ is Cohen-Macaulay, connected and equidimensional. What are the weakest assumptions ...
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Families of genus zero curves vs Family of deformations of $\mathbb{P}_k^1$.

Suppose I am looking to classify isomorphism classes of connected, compact, projective curves of genus zero. We usually start this moduli problem by considering a scheme $S$ and families of genus ...
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Can Yoneda lemma for smooth projective varieties only use curves?

Let $X,Y$ be two projective smooth varieties over an algebraically closed field $k$, if we know functorial isomorphism $Hom(C,X) \cong Hom(C,Y)$ as sets for every smooth projective curve $C$, do we ...
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Computing the genus of the function field $y^l=(x-a_1)^{n_1}\cdots (x-a_m)^{n_m}$

The following is a question from Rosen's "Number Theory in Function Fields"- Let $l$ be a prime not equal to $char(F)$ and $K=F(x,y)$ a function field where $x$ and $y^l=(x-a_1)^{n_1}\cdots (x-a_m)...
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Are isomorphic plane curves projectively equivalent?

Let $C$ and $D$ be two projective plane curves (over $\mathbb{C}$) of degree $d>1$. Suppose that $C$ and $D$ are isomorphic. Are $C$ and $D$ projectively equivalent? For smooth curves this is a ...
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How to find this ring homomorphism and the required kernel?

I have been wasting my time since morning in this very 'easy looking' thing: Let k be algebraically closed field, $ V \subset \mathbb{A}^n$ be a non empty variety. $I(V)$ be the ideal of $V$, set of ...
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Show that these 3 projective lines intersect on the same point

I'm stuck with the following problem of projective geometry from an assignment: Let $P_1$, $P_2$, $P_3$ and $Q$ be four points in the projective plane (over an algebraically closed field) such that ...
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Parametrise $z_0^2 + \cdots + z_3^2 = 0$

Find a function $f:\mathbb{CP}^1 \times \mathbb{CP}^1 \longrightarrow \mathbb{CP}^3$ with image $$ \{[z_0:z_1:z_2:z_3] \in \mathbb{CP}^3 \mid z_0^2 + \cdots + z_3^2 = 0\}. $$ I have got no idea how ...
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Discrete valuation of a rational function composed with an automorphism

The answer to my question might be trivial, although I can not see it. Details: let $k = \mathbb{F}_q$, $\bar{k}$ the algebraic closure of $k$, $C \subset \mathbb{P}^n(\bar{k})$ a smooth projective ...
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Show that the following two lines intersect at a point with $z \neq 0$

Consider the following two lines in complex projective space $\mathbf{P^2}$: $a_{1}x + b_{1}y + c_{1}z = 0$ and $a_{2}x + b_{2}y + c_{2}z = 0$. We suppose further that $a_{1}b_{2} - b_{1}a_{2} \neq ...
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Interesection of two “homogenous lines”

Suppose we have two lines given by $a_{1}x + b_{1}y + c_{1}z = 0$ and $a_{2}x + b_{2}y + c_{3}z = 0$, where $(x,y,z) \in \mathbf{C^3}$. What I am confused about, is that clearly $(0,0,0)$ is a point ...
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Morphism of hyperelliptic curve to $P^1$

If $C$ is a hyperelliptic curve, then how many morphisms are there to $P^1$? For genus 0 and 1, there should be many. Is it unique for genus larger than 2?
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Existence of a line bundle of order 2

For a genus $g$ (nonsingular irreducible) curve, does there always exist a line bundle $L$ such that $L^{\otimes2}=O$ (besides the trivial one)? For genus 0, this is an easy question by every line ...
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Genus of a smooth projective curve

I was trying to prove that the genus of a smooth projective complex curve $F=0$ of degree $d$ is $(d-1)(d-2)/2$. My attempt was to take the standard projection $$\pi:\mathbb{P}^2 \to \mathbb{P}^1$$ $$...
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An affine transformation, and its effect on a curve and a polynomial.

Suppose $(u,v) = A(x,y)$ is affine transformation. Where $u = ax + by + e$, and $v = cx + dy + f$ , and the inverse transformation given by $x = a'u + b'v + e'$ and $y = c'u + d'v + f'$. Suppose ...
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Bombieri's Theorem (Weil Conjectures for Curves)

I am studying Weil's conjectures for curves. Bombieri's theorem says: Let $C$ be a smooth projective curve of genus g defined over $\mathbb{F}_q$. Assume $q> (1+g)^4$ and $q=p^a$ where $a$ is ...
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Verify Herwitz's formula for $z^3/(1-z^2)$

This is an exercise from Miranda's book "Algebraic curves and Riemann surfaces". Consider $f(z)=z^3/(1-z^2)$ as a holomorphic map from the Riemann sphere $\mathbb{C}_\infty$ to itself. Verify ...
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How to find the equation of an image under a central projection

Let $\pi:\mathbb{P}^3 \to V(x_2) \cong \mathbb{P}^2$ the linear projection with center $P =(0:1:0:0)$. Find the equation for the image of $C=\{(s^3:s^2t:st^2:t^3)|~(s:t) \in \mathbb{P}^1 \}$ under $\...
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Pushforward of a line bundle along a finite morphism of curves

Let $f:X\rightarrow Y$ be a finite morphism (a branched covering) of degree $n$ of smooth complex algebraic curves. It is a known result that for any line bundle $L$ on $X$, the pushforward $f_* L$ ...
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Desingularisation of curves (Lorenzini-Invitiation to Arithmetic Geometry, chap 6,ex 7)

Given a nonsingular complete curve over algebraically closed $\bar{k}$, which is interpreted as a field $\bar{k}(X)$ of transcendence degree 1 and its set of valuations trivial on $\bar{k}$, we may ...
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Divisor of two relatively prime

Let $X=\mathbb{P}^1$, let $f,g\in k[t]$ relatively prime, where $t=X_1/X_2$, two coordinates. Then what can we say $\operatorname{div}(f/g)$? I can just tell that $\operatorname{div}(f/g)=\...
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Do algebraic curves exclude the whole space? Prove that the locus $y = \sin x$ in $\mathbb R^2$ doesn't lie on any algebraic curve in $\mathbb C^2$

Artin Algebra Chapter 11 Here is the solution of Brian Bi: Here is the definition of algebraic curve: Why can't we have the zero polynomial?
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For a line $L$ and an algebraic curve $C$ of an irreducible polynomial, prove $C \cap L$ contains at most d points unless C = L.

Artin Algebra Chapter 11 This has been answered here. My questions are about the solution of Brian Bi: By stronger, does he mean that $C \ne L$ and $f$ is irreducible $\implies l \nmid f?$ If so, ...
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In proving an irreducible curve has only finitely many singular points, is $f_x \not \equiv 0$?

Artin Algebra Chapter 11 This has been answered on the site, but I want to ask about the solution given by Brian Bi: The constant polynomials are not considered irreducible, so f is not constant. ...
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Connected Smooth Projective curve $C$ is rational if unirational

Following question: Why and how to see that a connected, smooth, projective curve $C$ (so a so a $1$-dimensional, proper $k$-scheme) is rational if it is unirational. Remark: unirational means that ...
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Space of nonsingular cubics

The general nonsingular projective cubic is of the form $$F_\lambda = Y^2 Z - X (X - Z) (X - \lambda Z), \qquad \lambda \ne 0,1$$ where $F_\lambda$ and $F_\mu$ are isomorphic if they have the same ...
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Base point free for $g^1_2$ for hyperelliptic curve

Let $C$ be a curve of genus lager than 1. $C$ is called hyperelliptic if it contains a $g_2^1$ linear system, meaning that $D$ is of degree $2$ with $dim|D|=2$ if $D$ is such a divisor in this linear ...
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$p$-adic values of rational points on elliptic curves

The following question came up naturally whilst studying diophantine equations: given an elliptic curve $E$ of the form $Y^2 + aY = X^3 +bX^2 + cX + d$ defined over $\mathbb{Q}$, consider the subset $...
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Check injectivity of the map $\mathbb{P}^1_{\mathbb{K}} \rightarrow \mathbb{P}^2_{\mathbb{K}}$ defined by $(t_0:t_1)\mapsto(t_0^3:t_0t_1^2:t_1^3)$

The title of this question is explicit, I want to check the given map is injective, provided the field $\mathbb{K}$ is algebraically close; take for instance $\mathbb{K}=\mathbb{C}$ for clarification ...
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Simple Curve Integral using Parametrization

I know the answer but cannot for the love of everything figure out how the book got the answer and the solution is only the answer nothing step-by step at all. $$ F(D) = F*dr $$calculate curve ...
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Is there a substitution which transforms every Fermat curve into an elliptic curve?

A Fermat Curve of degree $n$ is the set of solutions to $x^n+y^n=z^n$, $x,y,z\in \mathbb R$. In this question, the OP provides a substitution which relates a Fermat Curve of degree $n=3,4$ to two ...
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Affine plane curves with constant curvature

Question I want to solve this differential equation for $P : \mathbb{R} \to \mathbb{A}^2$, a plane affine curve. $ P'''(t) = \frac{P'(t)}{t^2}$ Someone recognize this equation? Is a famous curve? ...
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61 views

When do two plane cubic curves have 9 real intersection?

What is the "minimal" condition I can have such that two plane cubic curve defined each by one implicit equation over the reals will have 9 distinct real intersections? Note that I do not want an ...
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Is this plane curve irreducible?

I want to define a plane curve in $\mathbb{A}^2(\mathbb{C})$ by the polynomial $f(x,y)=x(x-1)^2-(y-1)^2=0$ where $(x,y)\in\mathbb{A}^2(\mathbb{C})$, but my goal is for the plane curve to be ...
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Plot projective curves

Do you know any website, app or program that can plot curves in the projective plane? I know the projective plane $\mathbb{P}^2 \mathbb{R}$ can be visualized as a sphere with the antipodal points ...
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Using resultants to show extension of function fields of curves is algebraic.

We are given two irreducible nonsingular plane curves, the zero sets of $f,g\in \bar{k}[x,y]$, with $\bar{k}$ algebraically closed. We have an injective map given on algebras as $\phi^*:C_f\rightarrow ...
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Example for intersection, multiplicity and tangents for projective plane curves

We recently got introduced to projective plane curves in our class, however we just defined it and never really talked about examples. And when I am trying to come up with a just any curve, ...
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Why is this partial derivative zero? (Algebraic functions)

Why is $F'(a,z_i) \ne 0$? An algebraic function $y=f(x)$ is defined by the algebraic equation $$ F(x,y) := g_n(x)y^n + g_{n-1}y^{n-1} + \cdots + g_0(x) = 0 $$ where $g_j$ are polynomials. In ...
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Smooth curves of odd genus

Let $C$ be a smooth curve of genus $g$ and $J_C$ its intermediate Jacobian. Recall that $J_C$ is a ppav of dimension $g$. Fixing a point $p\in C$, one can define the Abel-Jacobi map $$a\colon C\...
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What is the relation between torsion elements of the class group and covering spaces of curves?

For a Dedekind domain $A$ we have the following result relating torsion elements of the class group to (mostly) unramified extensions: If $a\in A$ is such that there exists an ideal $I$ with $I^n=(a)$...