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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.

If $K$ is a field, then an algebraic curve is an equation $f(X,Y)=0$ where $f(X,Y)$ is a polynomial with coefficients in the field. In other words, it is the set of points on the Euclidean plane whose coordinates are zeros of some polynomial in two variables.

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Divisors of Fermat Curves.

I've been studying Algebraic Geometry (in coding theory), and my book has been very ambiguous about what some of these ideas actually look like. So I was curious as to what some of the Divisors of degree 2, degree 3, and degree n Fermat Curves are.
KaneZ
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Bitangents corresponds to nodal points in the dual space

I'm beginning to study algebraic curves and I couldn't prove if we have $L$ a line bitangent to $F$, i.e, there are points $p_1, p_2\in F$, such that $L=T_{P_1}F=T_{P_2}F$, then $P_L\in F^\vee$ is a nodal point. I'm starting to prove this point is…
user122342
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Bijection between the projective plane and its dual

I didn't understand why we can't identify the projective plane with its dual. Let's take for example a line $L=aX+bY+cZ$ with $(a,b,c)\neq (0,0,0)$ in the projective plane $\mathbb P^2$. The dual projective plane $\mathbb P^{2v}$ is these lines,…
user42912
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Definition of holomorphic differential

I am studying algebraic curves through the book "Intr. to Algebraic Curves - Griffiths". The following definition puzzled too: Definition: Suppose $C$ is a Riemann surface. Then a holomorphic differential $\omega$ is by definition a family ${(U_i,…
user8186
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kernel of the norm map of jacobians

Given an étale double covering of curves $f: C\to C_0$, there is an induced norm map $Nm: J(C) \to J(C_0)$, which sends $\sum_i p_i$ to $\sum_i f(p_i)$. On page 285 of the book Geometry of algebraic curves, question 18 and question 19 are concerned…
user40153
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How to construct an example in which exactly 12 irreducible curves of degree 3 and genus 0 pass through 8 common points in the projective plane

The Gromov-Witten number for degree 3 curves of genus zero is 12. I would like to see an example in which exactly 12 irreducible curves of degree 3 and genus 0 pass through 8 common points in the projective plane I do understand that the 8 points…
Simon M
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Every curve has a finite number of multiple points?

I've encountered this assertion and I'm wondering how it is proved. (Here, a multiple point is defined as a point whose local ring is not a DVR, [EDIT] and a curve is a variety whose function field has transcendence degree 1 over the base field).
Tony
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Why is $K_C^3$ very ample for smooth curves?

Here is a screenshot from J. Alper. I notice a fact about algebriac curves, that for a smooth curve $C$ of genus $g\geq2$, the line bundle $\Omega_C^{\otimes 3}$ is very ample. Where can I find a reference for this statement?
Display Name
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Inflection points of a real planar singular cubic curve

Consider a real planar singular cubic curve. After an affine change of coordinates we may assume that the singularity is at the origin $(0,0)$, and that the equation of the curve is $$y^2 - e x^2 + a y^3 + b x y^2 + c x^2 y + d x^3=0$$ where…
orangeskid
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number of integral points on an ellipse

Let $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ be an ellipse. How can the number of integral points lying on such an ellipse be calculated ($A,B,C,D,E,F$ are, of course, integers) ?
Peter
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Fulton exercise 4.2

Let $F\in k[x_1,...,x_{n+1}]$. Write $F=\sum_{i} F_i$, where $F_i$ is homogeneous. Let $P\in \mathbb{P}^n(k)$, and suppose $F(x_1,...,x_{n+1})=0$ for every choice of homogeneous coordinates $(x_1,...,x_{n+1})$ for $P$. Show each…
Jun Xu
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Degree of rational map $\phi$ between curves: $[\overline K(E_1):\phi^*\overline K(E_2)]=[K(E_1):\phi^* K(E_2)]$?

Given a non-constant rational map $\phi\colon E_1\to E_2$ between projective (irreducible) curves $E_1$ and $E_2$, we can define the pull-back $\phi^*\colon K(E_2)\to K(E_1)$. The degree of $\phi$ is the degree of the field extension $K(E_1)/\phi^*…
Sha Vuklia
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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 section to F. My question is that I don't see how the…
User20354
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Why smooth projective curves are connected?

Why smooth projective curves are connected? I actually want to show that they are connected, compact, 2-manifolds. (projective algebraic curves are zeros of a homogenous polynomial of three variables with complex coefficients in CP2. We will call a…
Ahmadreza Khazaeipoul
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How to find the dimension of linear system of curves of degree $d$

Consider two curves $C_1$ and $C_2$ in $\mathbb P^2 (\mathbb C)$ . How can i find the expected and real dimension of the linear system of cuves of degree $d$ passing through points lying on the both curves . Expected dimension can be found using…
Theorem
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