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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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Sum of multiplicities of a plane curve

Suppose $f$ is a bivariate polynomial that defines an irreducible plane curve of degree $d$ over some algebraically closed field $k$. We know that by Bézout's that summing over points on the curve: \begin{align*} \sum_P m_P(f)(m_P(f)-1) \leq…
apprentice8
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Scalar action on a projective curve

I have an homogeneous quartic in $(u,v,w)\:$: $Q(u,v,w)=u^4+au^2w^2+bw^4-v^2$ For $\lambda$ in the base field $K$, I wish to calculate: $\lambda^4Q(u,v,w)=Q(u',v',w')$. I 'm not sure but I find $u'=\lambda u$, $v'=\lambda^2v$, $w'=\lambda w$. Do you…
beginarray
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Change of coordinates for $X^3+Y^3+60Z^3=0$

I have to demonstrate that the curve $$ X^3+Y^3+60Z^3=0$$ is birationally equivalent to $$Y^2=X^3-2^43^360^2$$ or to $$Y^2=X^3-3^330^2.$$ I can't find a proper change of coordinates for this purpose. Does anyone know how can I do? Thanks!
robbis
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Space of nonsingular cubics

The general nonsingular projective cubic is of the form $$F_\lambda = Y^2 Z - X (X - Z) (X - \lambda Z), \qquad \lambda \ne 0,1$$ where $F_\lambda$ and $F_\mu$ are isomorphic if they have the same modulus $$J(\lambda) = \frac {27} 4 \frac {(1 -…
isekaijin
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Why is this partial derivative zero? (Algebraic functions)

Why is $F'(a,z_i) \ne 0$? An algebraic function $y=f(x)$ is defined by the algebraic equation $$ F(x,y) := g_n(x)y^n + g_{n-1}y^{n-1} + \cdots + g_0(x) = 0 $$ where $g_j$ are polynomials. In what follows for simplicity we will assume that…
oliver maclean
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Exercise 8.3 Fulton

Let $C=X$ be a nonsingular cubic. a) Let P,Q $\in{C}$. Show that $P \equiv Q$ if and only if $P=Q$. (Hint: Lines are adjoints of degree 1) Where $P \equiv Q$ if and only if $P=Q+div(z)$ Please give me an idea, I was thinking about it a lot of time.…
elado
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Is the curve smooth?

Are the plane curves $C: X^5+Y^5+Z^5+(X^2+Y^2)Z^3 $ and $D: X^5+Y^5+(X^2+Y^2)Z^3$ smooth or not? It seems $D$ has only a node at $(0,0,1)$.
Rahul
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Can we say that, every algebraic curve is a piecewise ${C^\infty }$ curve?

Let $L$ be algebraic curve. Can we say that, $L$ is a piecewise ${C^\infty }$ curve?
Under sky
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An application of the Max Noether's theorem

I'm reading chapter IV of Robert J Walker's book 'algebraic curves'. The last section of this chapter is about Max noether's AG+BF theorem. I am stuck on an exercise in this section. The exercise states : the tangents at the six intersections of a…
user282089
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Relation between curvature of curve and dual curve?

For a plane algebraic curve, does there exist a relationship between the curvature of the curve and the curvature of its dual curve?
Gerard
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morphisms between algebraic curves

I have a cubic $(E)$: $y^2=ax^3+bx^2+cx+d$ over rational field. Setting: $d=0, \: \: a'=-2a,\:\: b'=a^2-4b$, we find that $(E)$ is isomorphic to her "little sister" $(E'): \: y^2=x(x^2+a'x+b')$. For example, with $a=8, \: \:…
beginarray
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Asymptotes' intersection with curve at infinity

I read in my course book that asymptotes have at least two intersection with algebraic curve at infinity. How can I take this fact in my head in a visualized way? And what does that at least means? What are these multiple intersections on infinity ?
Vicrobot
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$a + b + c = ab$ and $a$, $b$, and $c$ are numbers different from zero

Suppose $a+b+c = ab$, and $a,b,c$ are nonzero. Then $$\frac{ab + ac + bc + c^2}{abc}$$ is equal to what? I've found an answer: $\frac ba$.
Physicer
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