# Tagged Questions

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 ...

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### How to linearlize level curves at a saddle point

Let $f(x,y)$ be a real-valued function on a domain $D$ in $\mathbb{R}^2$, and let $(x_s, y_s)$ be a saddle point of $f(x,y)$ in $D$. That is to say, \begin{align} \frac{\partial f}{\partial x}(x_s, ...
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### Action of $\mathbb{Z}/3\mathbb{Z}$ on $P^{1}$

I am reading from the book Topics in Galois theory by Serre. I have the following question , take $G=\mathbb{Z}/3\mathbb{Z}$. The group $G$ acts on $P^1$ by $$\sigma x\;=\;1/(1-x)$$ where $\sigma$ ...
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### What is the difference between surface and algebraic curve in general?

The question may seem dumb at first glance. But I couldn't figure out a satisfying answer after some research. A friend of mine told me that in an interview, she was asked to explain the sliding mode ...
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### branched cover of projective line over rationals

I was reading something when I came across the following phrase "branched cover of projective line over rational " . To understand what does author mean , I started reading , Now I know about ...
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### Noether normalization in algebraically closed field

The Noether normalization lemma states that if $k$ is a field, and $A$ a finitely generated $k$-algebra, then there exist elements $y_1,...,y_m\in A$ such that $y_1,...,y_m$ are algebraically ...
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### Definition of singular points on an algebraic curve

From what I understood, given a point $p$ on a scheme $X$ over a field $k$, we have \dim \mathcal{O}_{X,p} \leq \dim_{\mathcal{O}_{X,p}/\mathfrak{m} }\mathfrak{m}/\mathfrak{m}^2 \end{...
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### Is this true that, any algebraic curve has finitely many singularities?

Can we say that any algebraic curve has finitely many singularities?
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### Exterior Algebra VS Torsion

Let $C$ be an irreducible and reduced rational curve, and $f: \mathbb P^1\rightarrow C$ be the normalization. If $\mathcal F$ is a coherent sheaf of rank $r$ over $C$, then I was wondering if we can ...
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### intersection multiplicity from kirwan’s book

I'm working on F. Kirwan's Complex Algebraic Curves. To define intersection multiplicity, Kirwan choose some special projective coordinates and calculate the resultant. She claims before the ...
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### Vanishing polynomial in complex projective space

Assume we are working in $n$-dimensional complex projective space. Why does a (homogeneous) polynomial of degree less than or equal to $d - 1$ which equals $0$ on $d$ points on a line $L$ in ...
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### Help with corollary 4.6 in Griffiths

Corollary 4.6 (P.72) in Griffith's 'Introduction to Algebraic Curves' proves that $\mathcal{O}=\mathbb{C}\{x,y\}=$set of all holomorphic functions in $x,y$ is a UFD, using the Weierstrass preparation ...
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### Riemann-Roch Space for Quotient Curve

Let $C$ be a curve defined over a finite field $\mathbb{F}_q$. Let $\{f_1,..f_m\}$ be a basis for the riemann-roch space of functions, L(D), for the divisor $D= t\infty$. Suppose you have a subgroup ...
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### Bounding the number of points at infinity of a curve of degree $n$.

I'm trying to prove the following statement. Let $C$ be a curve of degree $n$. Give a bound for the number of points at infinity. I tried it for $C$ defined by a polynomial in two variables only. ...
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### How do I show the tangent to an elliptic curve over the complex numbers meets the elliptic curve at another point?

If $E(\mathbb{C})$ is an elliptic curve given by $y^2=ax^3+bx+c$ for $a,b,c\in \mathbb{C}$, and $\ell$ is a line tangent to $E(\mathbb{C})$ at some point $p$, then why does $\ell$ meet $E(\mathbb{C})$ ...
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### Finding Primitive Elements of Separable Function Field Extensions

Suppose you have a a curve $C$ defined by an equation in $x$ and $y$. There is a map from $C$ to $\mathbb{P}_1$ by projection onto $x$. This corresponds to a separable extension of function fields ...
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### Question about the rational normal curve and different representations of it.

I know the rational normal curve as the image of a polynomial map \begin{gather} \phi:K\rightarrow K^n\\ \phi(t)=(t,t^2,\dots,t^n) \end{gather} My question is proving the variety defined by the set ...
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### History and future of algebraic curves and the like?

Now that Fermat's last theorem has been proven, and also elliptic curves see widespread use in simple everyday applications, I would love to learn how the related theories came into beeing, how they ...
### Let $f = c{f^{r_1}}_1 … {f^{r_s}}_s$ be the unique factorization of the polynomial
$\ \ \space$ We are in the field of the theory of algebraic curves. Here $F$ is a projective curve. Choosing a plan affine $L$ whose equation is given by $ax + by + cz = d$ with coefficients of non-...