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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Is there something similar to a discriminant for algebraic curves of the form $y^2 = f(x)$ (similar to the discriminant of elliptic curves)?

Consider an algebraic curve $X$ defined by an equation $y^2 = f(x)$ where $f$ is a polynomial of degree $d \geq 3$ with coefficients in a field $k$. For $m = 2$ and $d = 3$, we can get an equation of ...
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33 views

Is there a rational parameter for the cylindrical curve?

Is there a rational parameter for the cylindrical curve? \begin{align*} \left\{ \begin{split} x^2-y^2=2\\ y^2-z^2=3 \end{split} \right. \end{align*} It seems to have something to do with the topic of ...
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Galois covers of curves of arbitrary degree.

For smooth projective algebraic curves over finite fields. Do they admit a finite Galois cover of any degree, that is not induced by just base extension?
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What happens to point at infinity when you complete the square of an elliptic curve?

Suppose we have some elliptic curve (just suppose we are working over $\mathbb{R}$) defined by the equation $$Y^2Z + a_1XYZ + a_3YZ^2 = X^3 + a_2X^2Z + a_4XZ^2 + a_6Z^3 $$ with a point at infinity $[0,...
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Algebraic proof of equivalence of Virasoro relations with OPE relation (vertex operator algebras)

In Ben-Zvi, Frenkel's book on Vertex Algebras and Algebraic Curves, we find the identity (page 43, Lemma 2.5.4, second edition): $[T(z),T(w)]=\dfrac{c}{12}\partial_{w}^{3}\delta(z-w)+2T(w)\partial_{w}\...
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Fulton's Algebraic Curves Exercise 7.2.3

I have a (hopefully) quick question about this. Fulton's Algebraic Curve, exercise 7.2.3 on page 85 says: Let $F$ be any plane curve with no multiple components. Generalize the results of this ...
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$p$ is an inflection point if and only if the Hessian $\mathcal{H}_{P}$ vanishes at $p$

I'm trying to comprehend the proof of the following proposition. Let $C$ be a projective plane curve over $\mathbb{C}$ and let $p$ be a smooth point of $C .$ Let $P$ be a homogeneous polynomial in $\...
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40 views

Is a rational and projectively normal curve a rational normal curve?

I am confused about the notion of "rational normal curve". First of all, if a projective curve is rational and normal, then using the Zariski's Main Theorem, we can conclude that it is ...
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Dualizing sheaf on singular curves

I am trying to understand two points brought up in Harris and Morrison's moduli of Curves, but I would like to have a more general understanding than simply considering nodal curves. So let $C$ be a ...
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Proving uniqueness of intersection multiplicities

I'm trying to understand a proof from my Algebraic curves course. Additional context is provided Geometric Interpretation of Intersection Multiplicities, but I'll copy the relevant details here. ...
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Using intersection multiplicities for computing the Resultant

Let $P(x_0,x_1,x_2) \in \mathbb{C}[x_0,x_1,x_2]$ be a homogenous polynomial of degree $d$, and $Q(x_0,x_1,x_2) = d_0 x_0 + d_1 x_1 + x_2$. I'm trying to prove that $R_{P,Q} = P(x_0,x_1, -d_0 x_0 - d_1 ...
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Kodaira dimension of an elliptic curve is zero

Let $C$ be an elliptic cure, i.e. a projective curve over $\Bbb{C}$ having genus $1$. I'm trying to understand why the Kodaira dimension of $C$ is zero (according to this article). I know that the ...
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33 views

Multiplicity of curve compatible with restriction to charts

I was trying to understand a proof from an Algebraic Curves course. Let $P$ be a non-zero homogeneous polynomial in $\mathbb{C}\left[x_{0}, \ldots, x_{n}\right]$ and let $a=\left(a_{0}, a_{1}, \ldots,...
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34 views

Geometric Interpretation of Intersection Multiplicities

I'm taking a course on Algebraic curves, and this is excerpted from a portion of the lecture notes. Theorem: Let $p$ be a point of $\mathbb{P}_{\mathbb{C}}^{2}$. There is a unique way to associate ...
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An analogue of the Jacobian variety for affine curves

Given a nonsingular complete irreducible algebraic curve $C$ of genus $g$ over an algebraically closed field $k$, we can associate to it its Jacobian variety $J_C$, which is an abelian variety of ...
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Holomorphic differential on the algebraic curve $y^3=(x^2+1)^2(x^3-1)$

I'm trying to find three holomorphic differential non-constant on the algebraic curve $y^3=(x^2+1)^2(x^3-1)$ in order to obtain a basis for $\mathcal{L}(K)$. I have found the canonical divisor $$div\...
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William Fulton 4.23 Describe all subvarieties of $\mathbb{P}^1$ and $\mathbb{P}^2$

For $\mathbb{P}^1$, I know subvarieties are of the form $V(aX+bY)$ where $a,b\in k$ as forms of degree $n$ in $2$ variables in a algebraically closed field $k$ will be factored into product of linear ...
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Definition of very ample divisor

Sorry for my bad English. Let $X$ be scheme over $Y$, and $\mathscr{L}$ be invertible sheaf on $X$. By definition of Hartshorne's algebraic geometry (p120), $\mathscr{L}$ is very ample relative to $Y$...
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Multiplicities of Polynomials Multiplied

This is a follow-up to one of my past questions, Multiplicities of Polynomials. I thought it was more appropriate to ask in a new question. This question is self-contained, I have written the relevant ...
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Multiplicities of Polynomials

I'm trying to understand the following explanation. This comes up as a first step into defining intersection multiplicities for Bezout's theorem. Let $P$ be a non-zero polynomial in $\mathbb{C}\left[...
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Hyperelliptic iff there exist points $P,Q$ such that $\dim L(P+Q)=2$?

I want to prove that a curve $C$ of genus $\geq3$ is hyperelliptic iff there exist points $P,Q\in C$ such that $\dim L(P+Q)=2$. My definition of hyperelliptic curve is that it has genus $\geq2$ and it ...
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Non-isomorphic curves with the same genus

I need to show that the curves $y^7=x^2(x-1)$ and $xy^3+zx^3+yz^3=0$ are not isomorphic even if the are both of genus 3. I have tried to show that they are a quartic and a hyperelliptic curve in order ...
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Shimura curves over totally real number fields

Suppose that $F$ is a totally real number field such that $[F:{\Bbb Q}]$ is odd. Then we shall choose the quaternion algebra $D$ everywhere unramified at finite places and at all but one infinite ...
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Non-degenerate conic is projectively equivalent to smooth conic

I'm trying to understand the proof of the proposition: Every non-degenerate conic $C$ in $\mathbb{P}_{\mathbb{C}}^{2}$ is projectively equivalent to the smooth conic $$ C_{0}=\left\{\left[x_{0}, x_{1}...
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How do I compute the action of the Frobenius on the torsion of the Jacobian in Sage?

I have a curve $C/\mathbb F_q$ and I would like to compute the eigenvalues of the Frobenius action on $H^1(C,\mathbb Z/\ell)$. One way would be to compute the integral Frobenius action and reduce mod $...
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Sketches of stable 6-pointed curves

I was wondering if I could ask about stable $n$-pointed curves (essentially intersecting copies of $\mathbb{P}^1$ with at least $3$ special points on each copy - these may be either a point of ...
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2answers
153 views

Liu's “Algebraic Geometry and Arithmetic Curves” - Proposition 4.4. (Ch. 7) and Exercises 5.1.29 and 5.1.30

In Liu's book, Chapter 7, Proposition 4.4, the question is about closed embeddings (or: closed immersions) coming from very ample divisors. The theorems are quite well-known when the ground field $k$ ...
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36 views

Computing basis for $\mathscr{L}(D)$, $D$ a divisor on a algebraic curve

In reading Hartshorne IV.1.3 on Riemann-Roch, I wonder if there is a general method (an algorithm) for finding a basis for $\mathscr{L}(D)$, where $D$ is a divisor on a (complete, nonsingular corve ...
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1answer
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Understanding blow up of curves

I am trying to desingularize the curve $V=V(y^n-x^d)$, where $n>d$ and $gcd(n,d)=g$. It is singular at $P(0,0)$ so I am trying a blow-up. My intuition is that a blow-up is locally like $(x,y)\...
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28 views

Semistable bundle over a curve of a sufficiently high degree is globally generated

This is an exercise in Newstead's notes "Vector bundles on algebraic curves". Let $E$ be a semistable bundle over a genus $g$ complex curve, i.e. for any proper subbundle $F \subset E$ we ...
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Restricting to affine charts

this is a fairly basic question that I'm struggling with. I'm trying to solve this problem: Let $C$ be an affine or projective plane curve over $\mathbb{C}$, and let $p$ be a smooth point of $C$. ...
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What are $A,B,C,D$ in the proof of Chasles' theorem?

I read A Treatise on Algebraic Plane Curves by Julian Lowell Coolidge and I have a question about the proof of Chasles' theorem: I do not really understand where $A,B,C,D$ are first defined? What are ...
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Basis for $L(D)$ over the hyperelliptic curve $y^2=x^6-1$

Let $X$ be the Riemann Surface associated to the curve $y^2=x^6-1$ and let $\pi$ be the x-projection. Let $p+q=\pi^{-1}(\infty)$ and $r_{k}=(e^{ki\pi/3},0)$. I need to find a basis for the Riemann-...
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A torsion free sheaf of rank $1$ on a nodal curve

Let $X$ be a irreducible curve which has a unique node, $\pi \colon X^{\prime}\rightarrow X$ be the normalization, and $\mathcal{F}$ be a torsion free sheaf of rank $1$ on $X$. Then,$\pi^{*}\mathcal{F}...
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Surjectivity of the degree homomorphism

Regarding the following question and discussion: Is the degree homomorphism $\text{deg}: \text{Pic}(X)\to \mathbb{Z}$ surjective? We agree that if $X$ is a curve over an algebraically closed field the ...
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Common zeros of all elements in $H^0(X,D)$ for Riemann surface or algebraic curve

Let $X$ be a Riemann surface and $D=D^+-D^-$ is divisor(e.g. $P-2Q+3R=(P+3R)-(2Q)$), then I want to ask if the common zeros of $f\in H^0(X,D)$ is exactly the support of $D^-$. By definition, $f\in H^0(...
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Normalization of $y^3=(x^3-1)(x^2+1)^2$, blow up or something else?

I need to normalize the curve $\mathcal{C}:y^3=(x^2+1)^2(x^3-1)$ which is singular at $(\pm i,0)$. I have moved the x-positive one to the origin getting $y^3=x^2(x-2i)^2((x-i)^3-1)$ and I have blown ...
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Definition of Hecke operators for Shimura curve

I am following Wiles' paper On class groups of imaginary quadratic fields, and it's my first time learning about Shimura curves. There is part of the setup that I don't understand, concerning the ...
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Blowing up of the curve $x^4+y^4-xyz^2$ in $\mathbb{P}^2$

[Problem] (Fulton's Algebraic curves Problems 7.9) Let $C=V(x^4+y^4-xyz^2)$. Write down equations for a nonsingular curve $X$ in some $\mathbb{P}^N$ that is birationally equivalent to $C$. (Use the ...
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Galois cohomology of projective linear group

I am currently trying to compute Galois cohomology $H^{1}(\overline{k}/k, PGL_2(\overline{k}))$. As far as I know these cocycles correspond to isomorphism classes of smooth genus-$0$ curves over $k$. ...
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Linear system the algebraic curve $y^2=x^6-1$

Let $\mathcal{C}$ be the Riemann surface associated to the curve $y^2=x^6-1$ realized as a branched covering of $\mathbb{P}^1$. Let $r_{\alpha}=(e^{\alpha\pi i/3},0)$ and $p+q=\pi^{-1}(\infty)$. I ...
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Resolution of an ordinary multiple point on an irreducible plane algebraic curve

Let $C \subset \mathbb{A}^2$ be an irreducible plane algebraic curve and $P \in C$ be its ordinary point of multiplicity $m$, i.e., there are exactly $m$ tangents of $C$ at $P$ and they are pairwise ...
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If $\psi$ the projection away from $R\in X$, then why does '$\psi$ inseperable' imply that $R$ lies on every tangent line of $X$?

Im reading section 3 in chapter IV of Hartshorne, on embeddings of curves in projective space. More precisely, I'm at Proposition 3.8: Let $X$ be a curve (complete, nonsingular, over an algebraically ...
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46 views

Number of connected components of complement of plane curve

Let $$P(x,y)=\sum_{k=0}^m\sum_{l=0}^n a_{k,l}x^k y^l$$ be a polynomial over $\mathbb R$ in $x$ and $y$ of degree $m+n$. The zero locus $$\mathcal C=\{(x,y)\in \mathbb R^2\,|\, P(x,y)=0\}$$ is a plane ...
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Connection between quadratic characters and quadratic twists of elliptic curves

So apparently there is some kind of a connection between quadratic characters and twists of elliptic curves that I can't understand. I am well aware of quadratic twisting of an elliptic curve $E$, we ...
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1answer
26 views

Dimension of the linear system of conics passing through four points.

Let $P_1$,$P_2$,$P_3$,$P_4$ $\in \mathbb{P}^2$. Let $V$ be the linear system of conics passing through these points. Then show that if $P_1$,$P_2$,$P_3$,$P_4$ lie on a line, then $\dim(V)=2$ and if ...
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17 views

Morphism on Elliptic Curve to Intersection Point

I am working on a problem found in the Gathmann Algebraic Geometry notes (2019 edition) that I cannot figure out. I would appreciate any help that I could get on this. Let $X \subset \mathbb{P}^2$ be ...
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1answer
74 views

uniformizer on elliptic curve in Silverman's book

I'd like to ask remark 1.1 in chapter 2 of Silverman's "Arithmetic of elliptic curves". Let $K$ be a field and $C$ be a curve with a smooth point $P$, after proving $\bar{K}[C]_P$ is a ...
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16 views

$x^iy^j\frac{dx}{\partial f/\partial y}$ is a basis of holomorphic forms on algebraic curves

Let $f(x,y)=0$ be a smooth algebraic curve of degree $d$ over a field $k$, then I want to prove that $x^iy^j\frac{dx}{\partial f/\partial y}$ is a basis of the space of holomorphic forms, where $0\leq ...
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64 views

Genus of $y^3=(x^2+1)^2(x^3-1)$

I'm trying to proving that the genus of the normalization of $C:y^3=(x^2+1)^2(x^3-1)$ is 4 by using the Riemann-Hurwitz formula on the projection map $\pi:C\rightarrow\mathbb{P}^1$ which sends $[x:y:z]...

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