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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
35 views

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. $$sin(2n + 1)\alpha = sin^{2n + 1}(\alpha)\sum_{k=0}^n(-1)^k{2n + 1\choose 2k + 1}[cot^2(\alpha)]^{n - k}$$
user avatar
0 votes
0 answers
13 views

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 ...
user avatar
  • 113
0 votes
1 answer
56 views

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 $ \...
user avatar
0 votes
0 answers
50 views

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 $...
user avatar
1 vote
0 answers
32 views

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 ...
user avatar
0 votes
1 answer
28 views

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{...
user avatar
2 votes
1 answer
53 views

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 ...
user avatar
  • 6,738
0 votes
0 answers
35 views

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 ...
user avatar
1 vote
1 answer
37 views

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(\...
user avatar
0 votes
2 answers
74 views

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}[...
user avatar
0 votes
0 answers
83 views

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(...
user avatar
  • 367
1 vote
1 answer
55 views

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 ...
user avatar
0 votes
0 answers
15 views

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,...
user avatar
2 votes
0 answers
37 views

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 ...
user avatar
  • 512
0 votes
0 answers
21 views

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 ...
user avatar
0 votes
0 answers
49 views

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....
user avatar
1 vote
1 answer
53 views

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 ...
user avatar
3 votes
0 answers
29 views

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 $H^0(E\otimes L)=0$ ?
user avatar
0 votes
0 answers
14 views

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 ...
user avatar
  • 1,121
0 votes
0 answers
45 views

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 ...
user avatar
  • 1,719
0 votes
0 answers
36 views

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 ...
user avatar
  • 1,012
0 votes
1 answer
50 views

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?
user avatar
  • 93
0 votes
1 answer
55 views

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) ...
user avatar
2 votes
0 answers
51 views

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, ...
user avatar
  • 1,482
1 vote
0 answers
35 views

$K[t^2,t^3]_{(t^2,t^3)}$ 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]...
user avatar
0 votes
1 answer
42 views

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 ...
user avatar
0 votes
0 answers
19 views

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,...
user avatar
  • 131
2 votes
1 answer
81 views

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,...
user avatar
0 votes
0 answers
48 views

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 ...
user avatar
1 vote
2 answers
58 views

$(f_1,\dots,f_r)$ 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 ...
user avatar
  • 177
0 votes
1 answer
80 views

$V(y^2-x^3)$ 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 ...
user avatar
1 vote
1 answer
57 views

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 ...
user avatar
0 votes
1 answer
44 views

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:...
user avatar
  • 455
1 vote
0 answers
35 views

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 $g>1$. ...
user avatar
  • 2,159
1 vote
0 answers
42 views

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!...
user avatar
0 votes
0 answers
42 views

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 ...
user avatar
1 vote
1 answer
37 views

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,...
user avatar
  • 455
1 vote
1 answer
61 views

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 \...
user avatar
2 votes
1 answer
40 views

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 ...
user avatar
  • 23
2 votes
0 answers
42 views

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 = \...
user avatar
0 votes
0 answers
35 views

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 ...
user avatar
  • 87
2 votes
0 answers
34 views

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 ...
user avatar
  • 512
3 votes
1 answer
70 views

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 ...
user avatar
  • 33
0 votes
1 answer
81 views

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 $X$, ...
user avatar
  • 1,482
1 vote
0 answers
29 views

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 ...
user avatar
0 votes
0 answers
27 views

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 ...
user avatar
  • 61
2 votes
1 answer
66 views

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 ...
user avatar
  • 1,482
1 vote
0 answers
34 views

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 ...
user avatar
  • 1,143
3 votes
2 answers
130 views

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)...
user avatar
  • 2,887
1 vote
0 answers
42 views

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 ...
user avatar
  • 21

1
2 3 4 5
45