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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Degree 1 divisor on a curve

Let $C$ be a smooth projective curve of genus $g$ over complex numbers. Is it true that any degree one divisor is linearly equivalent to $1.p$ for some point $p$? I know it's true for $\mathbb P^1$. ...
Angry_Math_Person's user avatar
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the local behavior of a function at a place and its relation to the concept of divisors

I am studying functions in the Riemann-Roch space $L(sP)$, with $s \in \mathbb{Z}^+$ and $P$ a place of degree $r > 1$. My focus is on the local behavior of these functions at higher-degree places. ...
Researcher in Academia's user avatar
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Problem 7-5 Algebraic Curves William Fulton

I have been trying problem number 5 of chapter 7 of the Fulton, I attach the statement in the following link; Problem 7-5 Fulton QUESTION: Let P be an ordinary multiple point on C, $f^{-1}(P) = \{P_1,...
Killua83's user avatar
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"Untwisting" line bundle on nodal curve

Let $L$ be a line bundle on a nodal curve $C$. Under which assumptions on $C$ and $L$ does there exist a "positive" Cartier divisor $D$ that "twists away" zeros of $L$ in the sense ...
Matthias's user avatar
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How do I translate the intersection of two affine curves in a plane into a statement about ideals in $k[X, Y]$?

Let $E : y^2 = x^3 + Ax + B$ be an elliptic curve over a field $k$ and let $x = x_0$ be a line that intersects $E$ in $(x_0, \pm y_0)$. According to Lemma 10 of this paper, this is expressed ...
Fred Akalin's user avatar
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Blow up and hyperelliptic curves.

Let $C'$ be a non singular affine curve $y^2=x^5+3$ over $\mathbb{C}$. $C'^\#$ be its projective closure : $Y^2Z^3=X^5+3Z^5$. It has singular point at $\mathcal{O}'=(0:1:0)$. On the other hand, let $C$...
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Explicitly calculating the tangents to a projective curve

In Fulton's algebraic curves, the definition of a line tangent to a projective curve in $\mathbb{P}^2$ is following: We can define a line $L$ to be tangent to a (projective) curve $F$ at $P$ if $I(P,...
Haoqing Yu's user avatar
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Relative dualizing sheaf of the family $\varphi: C\rightarrow \mathbb{A}^1$ where $C$ is the vanishing locus of $xy=t^k$ in $\mathbb{A}^3$.

This is an exercise from the book Moduli of Curves. Let $C$ be the vanishing locus of $xy= t^k$ in $\mathbb{A}^3$. Define a morphism $$\varphi:C\rightarrow \mathbb{A}^1, (x,y,t)\mapsto t.$$ Then $C$ ...
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Rank of elliptic curves does not decrease under field extension?

Let $E/\Bbb{Q}$ be an elliptic curve. Rank of elliptic curve over quadratic extension $L=K(\sqrt{D})/K$ is calculated by a formula $rank(E/L)=rank(E/K)+rank(E_D/K)$(this is easy to prove). In ...
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Smooth part of an Affine Plane Curve defined by irreducible polynomial is a Riemann Surface

Miranda states the following. Given an irreducible polynomial in $\mathbb{C}[x,y]$, the singular points of its locus of roots $X$ forms a finite set. If we delete these points of $X$ the the ...
Jeff's user avatar
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Quotient curve of Fermat curve over a finite field (Ireland -Rosen exercise)

I'm thinking a exercise 12 of Ireland-Rosen "A Classical Introduction to Modern Number Theory" p.226 as follows: But I have no idea, could you give some hints starting from (a) ?
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Can a polynomial with even exponents be expressed as a multiple of $x^3+y^3-1$

Given a polynomial in two variables $P(x,y)$ with even exponents, can there be another polynomial $Q(x,y)$ such that $P(x,y)=(x^3+y^3-1)Q(x,y)$? The original problem I am trying to solve asks: Is ...
Numeral's user avatar
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Descartes folium

The geometry of Descartes' folium, $x^3+y^3=3axy$ has been well studied. Can someone tell me which geometric property characterizes the following cubic curve: $$bx^3+y^3=3axy$$ The previous curve is a ...
Felipe 's user avatar
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Necessary and sufficient conditions for the existence of a bitangent

For a 2D curve defined over a limited range, is the existence of a region over which the sign of the curvature is the opposite from the rest of the range both necessary and sufficient to prove the ...
Ted Lorance's user avatar
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Exercise V.1.5. Hartshorne

Let $X$ be a surface of degree $d$ in $\mathbb P^3_K$, we compute $K_X^2$. We have $S\sim dH$ and $K_{\mathbb P^3}\sim -4H$ for a hyper plane $H$. We have by Hartshorne II.8.20 that $$\omega_X\cong \...
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Checking that a differential form is holomorphic on an algebraic curve and that it extends to a holomorphic form on its compactification

Consider the following algebraic curve: $$C=\left\{ (x,y) \in \mathbb{C}^2 | x^3+y^3+3\lambda xy + 1=0 \right\} $$ …where $\lambda^3 \neq -1$. I think this is called the Hesse pencil, and it’s a ...
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Infinite Ramification Places of Finite Extension

Let $F/K$ be an algebraic function field, with constant field $K=K_{F}$, and $L/F$ be an finite extension. Is it possible that infinitely many places of $L$ ramifiy? Somehow I feel that it is not ...
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degree, genus, and birational map

I'm confused on the following: Suppose we have a polynomial $f(T)$, e.g. $T^d$, with degree $d>2$, then I consider the curve defined by $y-f(x)$. Then there seems an obvious birational map given by ...
S.Gau at Math's user avatar
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Linearly equivariant prime divisors on a curve [duplicate]

Let $C$ be an algebraic curve, and $P_1,P_2$ be prime divisors on $C$. I would like to prove : $$P_1\sim P_2\Rightarrow C \text{ is rational.}$$ I tried to use Riemann-Roch theorem to prove that the ...
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Two definitions of places on a curve

In the literature on algebraic curves, I have found two definitions of places on a curve. Let $K$ be an algebraically closed field. Let $\mathcal{C}$ be an irreducible curve defined by $f(x,y) \in K[x,...
grove's user avatar
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Vector Bundles on a Curve as an Adelic Double Quotient

I'm looking for an explanation or a reference on why rank $n$ vector bundles on a curve over a field can be described as $\mathrm{GL}_n(\mathcal O)\backslash \mathrm{GL}_n(\Bbb A)/\mathrm{GL}_n(K)$, ...
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Visualisation of the cubic curve $C: y^2 z − x^3 − x^2z − xz^2 − z^3$

To get some insight on the zero locus of the cubic curve, I've tried a couple of online visualisers on Google and mostly failed to generate the plot (run time errors...) except Wolfram Alpha which ...
Rowing0914's user avatar
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Differentials on a hyper elliptic curve that are preserved by involution can be pulled back from $\mathbb{P}^1$ from $\pi$, hence must be 0.

This is the exercise 21.3.C from FOAG. I did the first two, but got stuck at the third one. I'm not entirely sure what it's asking. Given a double cover $\pi: C\rightarrow \mathbb{P}^1$ branched over $...
pcZhang's user avatar
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Use of Vertical bar notation in Sec 2.8 of Reid's UG Algebraic curve book

(At least) In 2.8 Group law on a plane cubic of the book, he repreatedly used the notation | inbetween curves, e.g., $F | L$ where F is a curbic curve and L is a ...
Rowing0914's user avatar
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Computation of Intersection points of an algebraic curve and a line

Sorry for long post... In Example 9.3 of the lecture notes of MA40188 Algebraic Curves (2016) by Ziyu Zhang from ShanghaiTech (Lecture notes), he goes like ...
Rowing0914's user avatar
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A question in the definition of the function field of a projective variety

Remark: A “projective variety” means an irreducible algebraic set of $\mathbb{P}^n(k)$, where k is an algebraic closed field throughout the question. Recently, I've run into to some problems in ...
user1059356's user avatar
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Projective curve without regular points?

Let $C$ be some integral projective curve on some field $k$. Can $C$ consist only singular points (except the generic point)? From Is this true that, any algebraic curve has finitely many ...
onRiv's user avatar
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A generator which is algebraic

Let $F\subset L$ be a field extension, and $x,\alpha,\beta\in L$. My question is: Assume that $F(\alpha)\subset L$ is an algebraic extension. If $\alpha$ is algebraic over $F(\beta)$, then is $x$ ...
sola's user avatar
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Group action of $\text{Aut}(X)$ on ramification points (Hartshorne, exercise IV.2.5)

I'm having some hard times trying to prove this point of the exercise. We have $X$ a curve of genus $2$ over a field with characteristic $0$. $G=\text{Aut}(X)$ (we know it is a finite group, let's ...
Fedeladrin's user avatar
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Dedekind Domain gives rise to "smooth" curve.

I have already seen multiple times (e.g. in Neukirch ANT, Chapter 1.13) that if we have a one-dimensional noetherian ring $R$, then the spectrum Spec$(R)$ defines a smooth (i.e. non-singular) curve if ...
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Parameterisation of Curve in projective space

Background: I've started reading Miles' "Undergraduate Algebraic Geometry" (Link) recently though struggling a lot. I'm stuck at processing the following paragraph... Sec. 1.7 ...
Rowing0914's user avatar
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1 answer
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What must be true for thirteen points to impose independent conditions on quartics?

Recently I've been trying to generalize the Cayley-Bacharach theorem to the case of quartics. Here's one version of what the theorem says for cubics: Let $P_1,\dots,P_8$ be eight (distinct, closed) ...
Hank Scorpio's user avatar
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Silverman's proof of III.6.2(c) (dual of sum of two isogenies is the sum of the duals)

I am teaching a course on Silverman's nice book "The Arithmetic of Elliptic Curves", and I am puzzled by the proof of III.6.2(c), which asserts $\widehat{\phi+\psi}=\hat{\phi}+\hat{\psi}$ as ...
Cheng-Chiang Tsai's user avatar
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Why is $K_C^3$ very ample for smooth curves?

Here is a screenshot from J. Alper. I notice a fact about algebriac curves, that for a smooth curve $C$ of genus $g\geq2$, the line bundle $\Omega_C^{\otimes 3}$ is very ample. Where can I find a ...
Display Name's user avatar
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3 votes
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On Siegel's Theorem in integer points.

Siegel’s theorem on integer points: Let $P \in {\mathbb{Q}}[x,y]$ be an irreducible polynomial of two variables, such that the affine plane curve $C = \{ (x,y): P(x,y)=0\}$ either has genus at least ...
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On the definition of singular points for algebraic curves

For simplicity, let us consider curves in the Euclidean plane ${\mathbb{R}}^2$. They may be defined as the set of points $(x_1, x_2)$ satisfying an equation of the form $P(x_1,x_2)=0$ where $P(x_1, ...
ric.san's user avatar
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smooth proper connected curve over a field is projective

I have the following question: In this answer is remarked that to show that a smooth proper connected curve $X$ over a field $k$ is projective we may assume to work over $\overline{k}$. Why can one ...
JackYo's user avatar
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Resolving a Node of a Plane Curve

In the book Algebraic Curves and Riemann Surfaces, Miranda explains how to resolve the node of a plane curve by plugging the hole at the node using hole charts. The idea is at a node $p$, the surface $...
The Special One's user avatar
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Zeta functions of hyperelliptic curves

I have been wondering recently about the geometric information encoded in the zeta function of a (smooth, projective) variety over a finite field - or in its étale cohomology (i.e. l-adic cohomology) ...
Basil's user avatar
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Find, if it exists, a quintic curve in $A^{2}$ over complex numbers, with the following conditions:

The curve has two asymptotes, $x + 3y - 2 = 0$ and $x + 3 y - 3 = 0$ The curve has a two degree inflation point in $(0,0)$, with inflational tangent $x + 3 y = 0$ (Sorry if the terminology is not ...
Goffredo Valenza's user avatar
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Properties of the symmetric square of a curve

Under what conditions on a curve $C$, defined over a ring $R$, is its symmetric square, $C^{(2)}$ smooth/proper? Is it enough for $C$ itself to be smooth/proper over $\text{Spec}R$? How does one see ...
kindasorta's user avatar
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Confusion about a sequence of subspaces of $\mathbb{P}^n$

Let $C$ be a smooth projective algebraic curve over an algebraically closed field $k$ of characteristic zero. Let $K$ be the function field of $C$. We take rational functions $f_0,...,f_n \in K$, not ...
oleout's user avatar
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Can a quartic polynomial curve in the two dimensional affine space over complex numbers have 4 parallel asymptotes?

I'm studying plane curves in affine and projective spaces over real and complex numbers. The question is: can a quartic polynomial curve in the affine space over the complex numbers have 4 different ...
Marco Andreoli's user avatar
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Automorphism group of the Klein quartic over field of characteristic 2,3,7(Hartshorne exercise IV.5.7)

Let $k$ be an algebraically closed field of characteristic $p$, and $X$ be the plane quartic curve defined by $$x^3y+y^3z+z^3x=0,$$ which is the so-called Klein quartic. Hartshorne claims in exercise ...
ZCC's user avatar
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Maximal number of multiple points for an irreducible quartic

I was working on this problem, but I don't see how I can solve it. I was given a hint, but I don't know how to use it. Can anyone help me? Thanks in advance! Let $f \in \Bbb C[x_0, x_1, x_2]$ be an ...
Oopsilon's user avatar
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Proof of Belyi's theorem with additional requiremt that ramification indices above 1 are equal to 2

I'm trying to prove an exercise problem from Szamuely's "Galois Groups and Fundamental Groups", which is a version of belyi's theorem. The exercise in question The normal version without the ...
358's user avatar
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Polynomial parametrization and degree nominator of parametrization of a rational curve

Let $\mathcal{C}\subset\mathbb{R}^2$ given by $f(x,y)=0$ be a rational curve. Then, there exists a rational parametrization $\phi(t)\,:\mathbb{R}\rightarrow \mathbb{R}²$. Under which circumstances is ...
user382144's user avatar
1 vote
1 answer
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Understanding a curve system on the boundary of Teichmuller space

I am reading the paper: https://arxiv.org/pdf/1511.07635.pdf and I try to understand the behavior on the boundary. I have some problems understanding this definition. On page 26, the authors give a ...
Framate's user avatar
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1 answer
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$j$-invariant of a genus one curve defined over $k$

The theorem and proof are from these notes : https://ocw.mit.edu/courses/18-782-introduction-to-arithmetic-geometry-fall-2013/a60e5c7aad91910265ee1fc686308413_MIT18_782F13_lec26.pdf Theorem 26.7.Let $...
user631874's user avatar
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Where is the hypothesis that $f_{n-1}(x, y)+f_n(x, y)$ is irreducible used in my solution?

I'm working through Shafarevich's Basic Algebraic Geometry, and one of the problems is: Prove that the curve given by the equation $f_{n-1}(x,y)+f_n(x,y)=0$ is rational if it is irreducible. Here $...
littleman's user avatar
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