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 $\mathbb P^1$. ...
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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.
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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,...
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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 ...
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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 ...
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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,...
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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 ...
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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 ...
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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 ...
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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 ...
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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\} $$
…where $\lambda^3 \neq -1$. I think this is called the Hesse pencil, and it’s a ...
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Infinite Ramification Places of Finite Extension
Let $F/K$ be an algebraic function field, with constant field $K=K_{F}$, and $L/F$ be an finite extension.
Is it possible that infinitely many places of $L$ ramifiy?
Somehow I feel that it is not ...
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degree, genus, and birational map
I'm confused on the following:
Suppose we have a polynomial $f(T)$, e.g. $T^d$, with degree $d>2$, then I consider the curve defined by $y-f(x)$. Then there seems an obvious birational map given by
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Linearly equivariant prime divisors on a curve [duplicate]
Let $C$ be an algebraic curve, and $P_1,P_2$ be prime divisors on $C$. I would like to prove : $$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,...
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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 ...
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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 $...
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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 ...
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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 ...
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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 ...
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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 ...
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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$ ...
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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 ...
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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 ...
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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) ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 $...
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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) ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $...
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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 $...
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