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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I want the proof (or the article where I can find the proof) of this vietas formula?
I need the proof of this vietas formula please.
$$sin(2n + 1)\alpha = sin^{2n + 1}(\alpha)\sum_{k=0}^n(-1)^k{2n + 1\choose 2k + 1}[cot^2(\alpha)]^{n - k}$$
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Basis of holomorphic 1-forms on smooth projective curve with affine equation $C\colon s^6 = t^{p+1}+1$
Let $k = \mathbb{F}_{p^2}$ be a finite field with $p \equiv -1 \mod 6 $. Via Riemann-Hurwitz one can calculate that the genus of this curve equals $5(p-1)/2$ and hence the the space of holomorphic ...
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The local ring at the origin of $ \Bbb{A}^2 $
Consider the curves $ F = y-x^3 $ and $ G = y^3-x^4 $ over $ K. $ Find a polynomial representative of $ \frac{1}{1+x} $ in $ \mathscr{O}_0/ \langle F,G \rangle. $ I am having trouble simplifying $ \...
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Non singular point of a projective plane curve
In problem 5.1 of Fulton's Algebraic curves, we're asked to show that a point $P\in\mathbb P^2$, $P=[P_1:P_2:P_3]$ is multiple iff $F(P)=F_X(P)=F_Y(P)=F_Z(P)=0$.
Here $P$ is said to be multiple if $...
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How to compute the ramification indices in the degree-genus formula
I've been study the proof of the genus formula and I am a little confused about the ramification index of one of the maps involved. Let $C$ be a projective algebraic curve in $\mathbb{C}P^2$ such that ...
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Order of Vanishing on Projective Variety
I know for a curve $C$ in affine space, we can define a the order of vanishing at a smooth point $p \in C $by noting that $\mathcal{O}_p$, the local ring at $p$, is a DVR with maximal ideal $\mathfrak{...
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Special definition of “algebraic curve” for Riemann surfaces?
On the book Algebraic curves and Riemann Surface by Rick Miranda, page 169, I see the following definition:
(Last part of definition 1.1) A complex Riemann surface $X$ is an algebraic curve if the ...
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Polynomial Curve Fit without floating point
big math dummy here hoping to get some advice. I'm working on a closed loop servo system that requires a curve fit on some feedback. The controller for this system is $16$-bit. With the help of excel ...
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Ring of regular functions
For $V\subseteq \mathbb A^n$ an affine variety set $$\Gamma (V)=\{f:V\to K| \exists F\in K[x_1,\dots,x_n]\text{ such that }f(c)=F(c)\forall c\in V\}=\{f:V\to K|f\text{ is regular}\}$$ $$K(V)=Frac(\...
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An equivalence condition for $\mathcal{I(V}(I)) = \sqrt{I}$ over real fields
Question: Does $\mathcal{I}_{\mathbb{R}}\mathcal{V}_{\mathbb{R}}(I) = \sqrt{I}$ imply that $\overline{\mathcal{V}_{\mathbb{R}}(I)} = \mathcal{V}_{\mathbb{C}}(I)?$, where $I$ is an ideal of $\mathbb{R}[...
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Fulton, Exercise 2.28: Show that $R$ is a DVR.
We define as order function on a field $K$ a function $\varphi$ from $K$ onto $\mathbb{Z}\cup \{\infty\}$ satisfying:
(I) $\varphi(a)=\infty$ if and only if $a=0$
(II) $\varphi(ab)=\varphi(a)+\varphi(...
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Open set of the cuspidal curve is not principal
I'm struggling with this problem:
Let $k$ a field and $X=\operatorname{Spec} k[x,y]\big/(x^3-y^2)$ and $U=X\setminus\{(x-1,y-1)\}$. Show that $U$ is not a principal open set in $X$.
I tried to show ...
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Separability of the Frobenius morphism
We are in an algebraically closed field $k$ with positive characteristic $p>0$. Let $X\subseteq \mathbb{P}^2$ be an irreducible algebraic curve and $\phi:X\to\phi(X),\ (x_0,x_1,x_2)\mapsto (x_0^p,...
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Uniqueness of model of curve over DVR
I understood that properness of $\overline{\mathcal{M}_{g,n}}$ tells us that for a fixed field $K$, a curve $C$ over $K$ and a DVR $V$ with fraction field $K$, there is a unique curve over $V$ with ...
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Equivalence of two notions of intersection multiplicity for curves.
Let $C,D\subset\mathbb{C}^{2}$ be algebraic curves with no common component, and suppose that $P=(0,0)$ is in $C,D$. Let $f,g$ be the equations of $C,D$ respectively and let $i_{P}(C,D)$ be any ...
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Rings of regular functions, affine and quasi affine [duplicate]
Edit: I don't know how you can think that linking something which doesn't answer my question would satisfy me. By the way I don't know what a sheaf is, maybe you could ask before taking such decisions....
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Harder Narasimhan polygon
We know for vector bundles on smooth projective curve we have a HN filtration. Let $0=E_0 \leq E_1 ... \leq E_n=E$ be such a filtration of $E$. Then we can construct a convex polygon with lattice ...
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Stable vector bundles on an algebraic curve
Let $E$ be a stable bundle on a smooth curve of genus $g$. Assume $\chi(E)\leq 0$ or equivalently, $E$ has slope $\leq g-1$. Is there a line bundle $L$ of degree 0 such that $H^0(E\otimes L)=0$ ?
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Intuition of semistable vector bundles over a curve.
Consider a characteristic zero algebraically closed field $k$.
Let $f(x)\in k[x]$ be a polynomial of degree $2g+2$ ($g\geq 2$) with no double roots and $f(0)\ne0$. Let $g(u)\in k[u]$ be the ...
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Why consider $dx/x$ on a complex curve?
In a paper I'm reading, the author considers a compact Riemann surface -- or smooth algebraic curve, you pick -- $X$ given by the equation $y^d=x^n-1$ for some natural numbers $n,d$. My understanding ...
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Degree of a map given by a rational function.
Let $X$ be a complete nonsingular curve and $f \in K(X)$ a rational function. By the general theory there is the map $g: X \to \mathbb{P}^1$ which usually people write as $x \to (f(x) :1)$.
I have two ...
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Is a degree zero divisor on a curve always basepoint-free?
Let $X$ be a smooth projective curve and $D$ a divisor on $X$ of degree zero.
Is it always the case that $D$ is basepoint-free?
If not, then is there always some (positive) power $mD$ which is?
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Confusion in showing [2] is a rational map between elliptic curves
Let $E:y^2=x^3+ax+b$ be an elliptic curve over $\mathbb{C}$ with the identity $\mathcal{O}=(0:1:0)$.
It is well known that if $P=(x,y)$
(affine coordinate), then $2P=(f(x,y),g(x,y))$ where
$$
f(x,y)
...
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A general pencil $\{t_0F+t_1G\}$ of plane quartics consists entirely of smooth and nodal curves
Let $F,G$ be general plane quartics. It states in Harris, Morrison, Moduli of Curves, pg. 182, that the pencil fixing a value of $t$ for $H(X,t)=t_0F(X)+t_1G(X)$ yields either a smooth or irreducible, ...
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$K[t^2,t^3]_{(t^2,t^3)}$ is not regular
I want to show that $K[t^2,t^3]_{(t^2,t^3)}$ is not regular, and to do so I want to show that $dim_F(m/m^2)=2$ for $m=(t^2,t^3).K[z^2,t^3]_{(t^2,t^3)}=(\frac{t^2}{1},\frac{t^3}{1})\subseteq K[t^2,t^3]...
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Semicubical parabola is not isomorphic to the affine line (module of differentials)
Here is exercise 1 from chapter 8.1 of Bosch - Algebraic Geometry and Commutative Algebra:
(Exercise:)
For a field $K$, consider the coordinate ring $A = K[t_1, t_2]/(t_2^2
− t_1^3)$ of Neile’s ...
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Dimension of a point in a polynomial ring.
Let $k$ be an algebraically closed field. Let $X \subset \Bbb A^n$ be a nonempty variety. Now suppose $X$ is a point then it easy to see that $\dim_k k[X] $ is finite. Here
$$k[X]:=\{f \in k[X_1,X_2,...
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Local rings of $V(y^2-x^3)$
I want to find the local rings of $V(y^2-x^3)$, and establish if it's isomorphic to $K[x]_{(x)}$, or maybe some other ring which I don't know.
We take $p=(t^2,t^3) \in V$ and we want to find $O_{(t^2,...
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On a particular exact sequence in cohomology
The setup for my question is as follows (from page 9 of Deschamps' expository notes on the Artin-Winters proof of semi-stable reduction here). We want to prove that for $X$ the special fiber of a ...
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$(f_1,\dots,f_r)$ coprime $\implies V(f_1,\dots , f_r)\subset \mathbb{A}^2_{\mathbb{K}}$ is finite
I'm trying to solve the following exercise.
Let $(f_1,\dots,f_r)\in \mathbb{K}[x,y]$ and suppose that $\gcd\{f_1,\dots,f_r\}=1$. Show that $V(f_1,\dots , f_r)\subset \mathbb{A}^2_{\mathbb{K}}$ is ...
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$V(y^2-x^3)$ not isomorphic to $\mathbb A^1_K$
We can find everywhere that $\phi :\mathbb A^1_K \to V(y^2-x^3)$ sending $t$ to $(t^2,t^3)$ defines a bijective map but not a morphism of affine varieties. To me it's not clear why it's even ...
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Ordinary Double Point Characterization
I'm reading Qing Liu's Algebraic Geometry and Arithmetic Curves in an attempt to understand Artin-Winters' proof of Deligne-Mumford's semi-stable reduction theorem. I'm having a bit of trouble ...
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Cubic curve with a cuspidal singularity equivalence
I want to show that an irreductible projective cubic $F$ with an cuspidal singularity $P$ is equivalent with the curve $G=Y^2Z-X^3$.
I know that exists an affine transformation that takes $P$ to $(0:0:...
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Hartshorne exercise V.2.5: What is $|D+K-E|+E$?
Hartshorne exercise V.2.5 is about possible values for $e$, the invariant of a ruled surface expressed as the relative proj of a rank-2 vector bundle $\mathcal{E}$ over a curve $C$ of genus $g>1$. ...
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Calculating polar form of a homogenous polynomial
I'm trying to figure out how to apply the formula given on this Wikipedia page for polarizing a homogenous polynomial $f(u)$ of $n$ variables $u=(u_1,u_2..,u_n)$
$$F(u^{(1)}, ...,u^{(d)}) = \frac{1}{d!...
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Intersection of all homogeneous ideals containing a given ideal [duplicate]
I am going through William Fulton's "Algebraic Curves" and when he started talking about projective spaces, I got a little confused as the idea I think is right is not mentioned anywhere in ...
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Find the dimension of an space of homogeneous polynomials
Let $K$ be a field and $S_2$ be the linear space of all homogeneous polynomials $F \in K[X,Y,Z]$ of degree $2$. Let $P_1,...,P_n$ be points of the projective space $\mathbb{P}^2$. Define as:
$$S_2(P_1,...
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Canonical effective divisors on a degree 4 curve
I am trying the following problem from Hartshorne:
Original Problem (IV.3.2.i): Let $X$ be a plane curve of degree 4. Show that the effective canonical divisors on $X$ are exactly the divisors $X \...
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For which (a, b) the subset is not a smooth curve in the plane R^2?
The full question is:
For which (a, b) the subset
C = {(x, y) ∈ $R^2 : y^2 = x^3 + ax + b$} ⊂ $R^2$
is not a smooth curve in the plane $R^2$?
I found out one similar question here:
How to check ...
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Why is this de Rham 1-cochain a cocycle on an elliptic curve?
I am trying to understand a proof about the de Rham cohomology of an elliptic curve in Kedlaya's notes, p7 Example 1.5.
The motivation for this question has overlap with my previous question
Let $X = \...
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Embedding of complex torus into $\mathbb{P}^3$
I'm dealing with Riemann Surfaces and I saw how a basis for the space $L(D)$ (where $D\in Div(X)$ is a divisor for the complex torus $T=\mathbb{C}/\Lambda$) of D-bounded meromorphic functions give ...
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Existence of covers of curves in characteristic $p$
For a given smooth curve over $\mathbb{C}$ and $n$ ramification profiles (which each sum up to a fixed degree $d$), by considering monodromy representations the existence problem of such covers can be ...
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What is a non-effective holomorphic line bundle $L$ of degree 0 such that $L \otimes L$ is effective?
I am trying to solve an exercise that asks for a non-effective holomorphic line bundle $L$ of degree 0 such that $L \otimes L$ is effective.
I think I have mostly solved it but since I am a bit shaky ...
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How to define the intersection product of a curve with itself: $(C\cdot C)$?
I'm reading through Positivity in Algebraic Geometry by Lazarsfeld, and it defines the intersection product for bundles $L$ and (Cartier) divisors $D$ on a complete, irreducible complex variety $X$, ...
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Projective curves under twistor fibration
Consider a holomorphic curve $f:\Sigma\to\mathbb{CP}^3$ of degree, say, $d$ from some Riemann surface to projective space. The Penrose twistor fibration $\pi:\mathbb{CP}^3\to\text{S}^4$ then allows us ...
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Real algebraic plane curves with triple point references
In Gibson book Singular Points of Smooth Mappings, at p.131 he gives a classification of real algebraic plane curves with double points: node, cusp, tacnode, etc... Is there a reference for a ...
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Possible degrees of nonconstant map $f:C\rightarrow \mathbb{P}^1$ for a plane curve $C$
I am looking for the possible degrees of nonconstant map $f:C\rightarrow \mathbb{P}^1$ for a plane curve $C$. By combining the Brill-Noether theorem with the equality $g={d-1\choose 2}$ for a plane ...
- 1,482
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Convergence of Puiseux series in an topological algebraically closed field
The classical Puiseux theorem states that the field of the Puiseux series over an algebraically closed field of characteristic $0$ $F$ is also an algebraically closed field. Hence for each point of a ...
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Map from X to $\mathbb P^1_k$ of degree $3$ with $X$ curve of genus $3$.
Let $X$ be a curve over $k$ algebraically closed field. Here $X$ is a proper, smooth, connected, $\text{dim}(X)=1$ scheme over $k$.
Suppose that $g(X) = 3$ with $g(X) = \text{dim}_k H^1(X,\mathcal O_X)...
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isomorphism classes over algebraic closure vs bigger extensions
I found a statement in "A first Course in Modular Forms" Diamond Shurman that essentially said(at the best of my understanding)
if two complex elliptic curves defined over $\mathbb{Q}$ are ...
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