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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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Prove polynomial map between two affine curves is bijection except origin

I wanna to solve this problem dealing with a polynomial map between two affine curves. First curve is $A \subset \mathbb{C}^2$ defined by equation $s(1+t^2)-1=0$ and the second curve is $B \subset \...
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Why does a pointed surface minus a countable set of points contain a curve?

Let $S$ be a surface over $\mathbb{C}$ and let $s_1,\ldots, s_n$ be closed points of $S$. We consider this data as fixed. It is not hard to see that there is a curve passing through $s_1,\ldots,s_n$....
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Smooth curves as iterated smooth hyperplane sections

Let $k$ be an algebraically closed field. Can every connected smooth projective one-dimensional $k$-scheme be obtained by taking iterated smooth hyperplane sections of the image of a projective ...
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Gluing functions from irreducible components of a reduced curve if they agree on the intersection points

The question is: What is the algebraic machinery/reasoning behind the following intuition? Given a reduced curve over some field $k$ and a regular function on each of its components such that those ...
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Tangent lines throught a point in an algebraic curve

In Fulton's Algebraic Curves, beginning of chapter 3, there is a quick discussion about multiple points and tangent lines. He gives many examples of affine curves, for instance: He then says that ...
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How can I compare two curves statistically?

I have points in a 2D space, and each point has X and Y coordinate. These points are stored in an array. There are two separate arrays each array has n points with X and Y coordinate. The points in ...
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Proving Pascal's theorem from a corollary to Max Noether's Fundamental theorem

A result in my book Fulton's Algebraic Curves states that: If $F$ and $G$ meet in $\deg(F\deg(G)$ distinct points, and $H$ passes through these points then there exists a curve $B$ such that $B•F = H•...
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Error in Algebraic Curves by Fulton?

The following lemma is from section 3.3 of Fulton's algebraic curves. Notation: $\psi$ is a map from $k[X,Y]/I^n×k[X,Y]/I^m$ to $k[X,Y]/I^{m+n}$ I'm having some difficulty understanding the ...
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The genus of a smooth algebraic curve in $(\mathbb{CP}^1)^2$

I came across this exercise in Riemann Surfaces by S. Donaldson: Let $Z$ be a smooth algebraic curve in $\mathbb{CP}^1\times\mathbb{CP}^1$. Let $d_1,d_2$ be the degrees of the projection maps ...
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How to find the equation of a hyperbola given the asymptote, equation of axis and a point

Given that a hyperbola has asymptote $y=0$, passes through the point $(1,1)$ and has axis $y=2x+2$, determine its equation. The answer arrived at is $\displaystyle{4xy+3y^2+4y-11=0}$. However, I have ...
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Belyi's Theorem for Proper Normal Curves

I have a question about an argument used in Szamuely's "Galois Groups and Fundamental Groups" in the excerpt below (or look up at page 127): In the proof of Belyi we beginn with the reduction step to ...
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Automorphisms of stable curves.

I am a bit confused by what Harris and Morrison write about the finiteness condition for stable curves in Moduli of Curves. First the defintions: Definition (2.12) A stable curve is a complete ...
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About the ramification divisor

I am reading about Hurwitz's theorem ``Let $\varphi : C' \to C$ be a surjective morphism of irreducible smooth projective curves over $k$ of genus $g(C')$ and $g(C)$. Then we have $$2g(C') - 2 = \...
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Curves in products of curves

Let $K$ be a (not necessarily algebraically closed) field of characteristic 0. Let $C$ be a smooth projective geometrically irreducible curve over $K$. Let $n$ be any positive integer and consider the ...
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Are tropical polynomials differentiable?

I know that for a function $f$ to be differentiable at $a$, the following equality must hold. $$f'(a)=\lim_{x\to a}{\frac{f(x)-f(a)}{x-a}}$$ I also know that the left hand limit differs from the ...
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Why is it sufficient to consider 'forms' instead of polynomials in the projective case?

So I am reading through William Fulton's Algebraic Curves and currently I am reading that bit about projective curves. Now, when the author discussed about the affine case (in the earlier chapters 1-...
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space of projective plane curves of degree d

This is basically from Hartshorne Exercise I.5.13 where he writes there is a correspondence from the set of plane projective curves of degree $d$ to points in a projective space $\mathbb P^N$ where $N=...
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How does one find a parameter representation for bounded region?

I need help with this question. I have been stuck at it for a few days. My main problem is how I use the curve $K_r$ to find the parametric representation. I have a curve $K_r$ in the $(x,y)$-plane ...
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Computing global monodromy of branched covering of Riemann sphere

Given an irreducible polynomial $F(X,Y)\in\mathbb{C}[X,Y]$, there is an associated Riemann surface $S$ obtained from the zero set of $F$. Projection onto the second coordinate gives a map $\pi:S\to \...
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Norm map on function fields of curves

For a curve $C$, $f\in \bar{K}(C)^*$ and a divisor $D = \sum n_P(P) \in \text{Div}(C)$ such that $D$ and $\text{div}(f)$ have disjoint supports (support of a divisor is the set of points with non-zero ...
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Reference request: proof about result of torsion in Jacobian varieties

I've seen the following result a few times, and I am trying to find a proof of the following fact in a book, paper, or notes. Let $\mathcal{C}$ be a genus $g$ curve defined over $\mathbb{F}_q$. ...
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Global sections of square root line bundle

Let $C$ be a smooth curve in $\mathbb{P}^2$ over field $\mathbb{C}$. Suppose that I have a very ample line bundle $L$ on $C$ of even degree. Then $L$ has $2^{2g}$ square roots in $Pic\ C$. These are ...
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Blow up of plane curve is Normalization of local ring?

I have a question concerning normalizations of plane curves, which I know little about. Consider the simple node $V(f = y^2 - x^3 - x^2)$. Then $t=y/x$ is integral over $k[x,y]$ so that $(k[x,y]/f) \...
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What is a proper curve?

In the language of Janos Kollar's ``Rational curves on algebraic varieties,'' what is the definition of a proper curve? Is it possible to understand this without using the language of schemes?
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Construction and properties of flat family of elliptic curves

I have the following situation: let $k$ be algebraically closed of characteristic 0 (one can assume $k=\mathbf{C}$ if this simplifies the discussion) and let $\varphi : \mathfrak{X}\longrightarrow \...
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Confusion regarding sheaf cohomology and singular cohomolgy

Let $X$ be a smooth, projective curve (in particular irreducible) of genus $g$ at least $1$. We know that $H^1_{\mbox{sing}}(X,\mathbb{Z})=2g$. But, since $X$ is irreducible, the locally constant ...
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Are superelliptic curves singular?

It is an easy corollary of the Riemann-Hurwitz formula that smooth double covers of $\mathbb{P}^1$ can only be branched over an even number of points. Let $F(x,z) \in \mathbb{C}[x,z]$ be a homogeneous ...
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Proving $y-x^2, z-xy$ generate the ideal of the twisted cubic

This question comes from Hartshorne's exercise 1.2. He defines $Y:=\{(t,t^2,t^3)\in \mathbb{A}^3\mid t\in k\}$ and asks us, among other things, to find generators for the ideal $I(Y):=\{f\in k[x,y,z]\...
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Computing A Basis of Global Differential Forms for a Given Algebraic Plane Curve

Consider a polynomial $P\left(x_{1},\ldots,x_{N}\right)\in\mathbb{C}\left[x_{1},\ldots,x_{N}\right]$ in $N$ variables, and let $C$ be the projective/affine/whatever curve/hypersurface/whatever ...
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Computing degree of hom sheaf of coherent sheaves.

Let $C$ be complex projective curve (Cohen-Macaulay at least). Let $\mathcal{F}$ and $\mathcal{G}$ be coherent sheaves on $C$. Is there any way to express the degree of $\underline{Hom}_{\mathcal{O}...
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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 ...