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.
2,014
questions
0
votes
1answer
51 views
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 ...
2
votes
1answer
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 ...
2
votes
0answers
37 views
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?
0
votes
1answer
49 views
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,...
0
votes
0answers
7 views
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}\...
0
votes
0answers
20 views
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 ...
2
votes
0answers
29 views
$p$ 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 $\...
0
votes
1answer
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 ...
2
votes
0answers
37 views
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 ...
2
votes
1answer
37 views
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. ...
0
votes
0answers
32 views
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 ...
0
votes
1answer
32 views
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 ...
1
vote
0answers
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,...
1
vote
1answer
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 ...
0
votes
0answers
53 views
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 ...
3
votes
0answers
46 views
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\...
1
vote
0answers
34 views
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 ...
0
votes
0answers
32 views
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$...
2
votes
1answer
27 views
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 ...
3
votes
1answer
46 views
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[...
1
vote
1answer
32 views
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 ...
2
votes
1answer
75 views
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 ...
1
vote
0answers
66 views
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 ...
2
votes
1answer
50 views
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}...
1
vote
0answers
16 views
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 $...
1
vote
1answer
50 views
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 ...
1
vote
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 $k$ ...
0
votes
0answers
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 ...
1
vote
1answer
47 views
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)\...
0
votes
0answers
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 ...
2
votes
1answer
44 views
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 $C$.
...
0
votes
0answers
36 views
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 ...
2
votes
1answer
57 views
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-...
1
vote
0answers
63 views
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}...
2
votes
1answer
43 views
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 ...
0
votes
0answers
29 views
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(...
2
votes
1answer
71 views
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 ...
2
votes
0answers
47 views
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 ...
2
votes
1answer
71 views
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 ...
3
votes
1answer
124 views
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 $k$. ...
0
votes
0answers
56 views
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 ...
1
vote
0answers
25 views
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 ...
0
votes
0answers
19 views
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 ...
0
votes
0answers
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 ...
1
vote
1answer
38 views
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 ...
0
votes
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 ...
0
votes
0answers
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 ...
1
vote
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 ...
0
votes
0answers
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 ...
4
votes
0answers
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]...
