# 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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### 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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### 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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### 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 ...
1 vote
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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,...
1 vote
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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 $X$, ...
1 vote
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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 ...
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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 ...