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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Qing Liu's definition of an algebraic variety, a non-separated line

First, a little reminder. In Qing Liu's Book on algebraic Curves, algebraic varieties are defined as Let $k$ be a field. An affine variety over $k$ is the ...
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2answers
101 views

Relation between intersection and product of ideals

Let $C$ be a smooth projective (irreducible) curve in $\mathbb{P}^n$ for some $n$. Denote by $I_C$ the ideal of $C$. Let $g \in I_C\backslash I_{C}^2$, an irreducible element. Is it true that for any ...
3
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1answer
34 views

Union of holomorphic atlases is holomorphic atlas.

Let $S$ be a surface with open subsets $V$ and $W$ such that $s = V \cup W$. Suppose that $V$ and $W$ have holomorphic atlases $\Phi$ and $\Psi$ such that the holomorphic atlases $\Phi|_{V \cap W}$ ...
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62 views

Question about cusp cubic example in Hartshorne

In Hartshorne's Algebraic Geometry, in Chapter II.6 on Divisors he computes the Cartier class group (denoted $\operatorname{CaCl}$) of the cuspidal cubic cut out by $y^2z=x^3$ in $\mathbb{P}^2$. He ...
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1answer
29 views

why does $\varphi'(N)=0$ in this proof?

Fulton's book on page 105 defines $N$: Afterwards Fulton writes this solution for this lemma: I didn't understand why $\varphi'(N)=0$ Thanks
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1answer
32 views

(Family of) plane quartics with two double points

The wikipedia page on plane quartics (http://en.m.wikipedia.org/wiki/Quartic_plane_curve) mentions the possible number of singularities that such a curve can have, including some examples. I'd like to ...
2
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0answers
28 views

Curve minus a point on a surface

Let $S$ be a smooth complex projective surface and let $C\subseteq S$ a curve (maybe not integral). Suppose for example that $C$ is a fiber of a certain fibration of $S$ over $\mathbb P^1$. Now ...
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54 views

Can we classify plane cubics, What are they?

There are four qualitatively distinct pictures of the plane cubics. What are the polynomials corresponding to them? I know two of them have special names: nodal cubic and cuspidal cubic with ...
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1answer
31 views

Why is this derivation well-defined?

I'm reading Fulton's algebraic curves (page 105) and I'm trying to prove $\tilde D$ is well-defined: Let's define $\tilde D$ as $\tilde D(z)=y^{-1}(Dx-zDy)$, then if ...
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53 views

Bad notation in Fulton's algebraic curves book

I'm studying Fulton's algebraic curves book and I didn't understand this notation in the page 105: The problem is the author didn't define yet $F_{X_i}$ I'm a little confused here. thanks
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33 views

Position of singular points on a curve

Let $f\in\mathbb{R}[x,y]$ be of degree $d>3$. I am looking at the curve $\{(x,y)\in\mathbb{R}^2\,:\, f(x,y)=0\}$. Let $P:=\{p_1,\ldots,p_l\}$ be the set of singular points - where I assume this is ...
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1answer
60 views

What is $g^1_3$?

I'm trying to find the definition of $g^1_3$ in algebraic geometry Hartshorne's book, anyone who is used with this book could help me to find this definition? Thanks Remark: this extract is from ...
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2answers
119 views

Ray\curve mirror problem

I have an idea for a space station, but there is the following problem. I have a patch of grass on a space station. If the sun (yellow rays) shines from below it, what is the best shape of mirror ...
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1answer
38 views

Alebgraic curve and Riemann surfaces

How do we prove that any smooth complex algebraic curve $C\subset\mathbb{P}^2$ is a Riemann surface? Does there exist a complex version of the implicit function theorem?
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26 views

Genus in Falting's Theorem

Falting's Theorem states that algebraic curves of genus $g>1$ have only finitely many rational points on them. But how exactly is $g$ defined here? The notion of genus here obviously shouldn't ...
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1answer
45 views

Scheme almost everywhere reduced then reduced?

Let $K$ be a field and let $F\in K[x,y]$ be a polynomial such that $$ X_1:=\mathrm{spec}(K[x,y]/(F)) $$ is a reduced irreducible affine variety over $K$. You can also add smooth if you want to. Now ...
2
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1answer
115 views

Rational function and morphisms of quasiprojective varieties

Let be $k$ an algebraically closed field and let be $X$ a projective nonsingular curve. Notations We call $X_h : = X\setminus V(h)$ for any homogeneous polynomial $h$. A function ...
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1answer
38 views

Does a plane algebraic curve of degree d>1 with $\frac{1}{2}d(d-1)$ singularities exist?

In case it exists, it must be reducible, because the maximum number of singularities in an irreducible curve is $\frac{1}{2}(d-1)(d-2)<\frac{1}{2}d(d-1)$. Does it exist? Could we find an example?
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20 views

Algebraic Curves proof I am having a little trouble understanding (resultants)

$R_{f,g}$ is the resultant of polynomials $f$ and $g$.My question is, what purpose does $T$ play in this proof and how did they get the end result that $T^{p}R_{f,g}(TY) = T^{q}R_{f,g}(Y)$?
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1answer
38 views

(hyper) elliptic curve in characteristic two and the Jacobian criterion

Let $k$ be a field of characteristic two and let $E$ be a curve given by $$ y^2=x*(x+1)*(x^2+x+1)*(x^3+x+1)\quad\text{or}\quad y^2=f(x) $$ Now we have $dy^2/dy=2y=0$ and consider the Jacobian ...
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Exercise 2.19 algebraic curves by william fulton

Let $f$ be a rational function on a variety $V$. Let $U = \{P\in V; f \textrm{ is defined at }P\}$. Then $f$ defines a function from $U$ to $k$. Show that this function determines $f$ uniquely. So a ...
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71 views

If $f$ has a pole, does $f^2$ has a pole?

I don't understand something in the exercise 2.17 of Algebraic Curves of Fulton. Let $k = \overline{k}$ a field and $V$ be the variety defined by the zero of $ I = ( y^2 - x^2(x-1) ) \subset ...
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0answers
41 views

Where can I find these definitions in algebraic curves?

I'm trying to understand the definitions of trigonal curve, ramification points and linear systems. What is the best place to find it? I have as background just Fulton's algebraic curves. Thanks
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38 views

Singularity of $V(Y^2-X^3-X)\to\mathbb{P}^1$

In "The Arithmetic of Elliptic Curves, in example I.3.7, Silverman define $\Phi:V(Y^2Z-X^3-X^2Z)\to\mathbb{P}^1$ with $\Phi(X,Y,Z)=[Y,X]$. He says that $\Phi$ is not regular at $[0,0,1]$. How to prove ...
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76 views

Restriction of a very ample line bundle on complete intersection curves

Let $C$ be a smooth complete intersection curve in $\mathbb{P}^3_{\mathbb{C}}$, $f$ be a linear polynomial in $\mathbb{C}[X_0,...,X_3]$ which does not vanish identically on $C$. Denote by $U$ the open ...
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1answer
41 views

Pull-back of regular map and rational function field

I don't understand what I'm missing in this example. Let $X=V(X_1^2+X_2^2-X_0^2)$ the circle in $\mathbf{P}^2_k$, being $k$ an algebraically closed field. Let be also $f:X\longrightarrow ...
4
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63 views

What's an algebraic curve's polar line for?

I can take an algebraic curve, and I can draw its first polar. By this, I mean that I can take an arbitrary point not on the algebraic curve, and then I can identify all the points on that algebraic ...
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votes
1answer
66 views

Improvement over gamma correction for brightening images?

I'd like to brighten one of my own images for printing purposes, using a program I made. When I use the formula: pixelBrightness^0.6 to brighten an image (0.6 being an example, and where ...
0
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26 views

What is the order of the tangent of $C_F$ at the point $P$?

In my lecture notes there is the following about inflection points: Definition: A point $P=[x, y, z]$ of an algebraic curve $C_F=V(F)$ is called inflection point of $C_F$ if $P$ is not a singular ...
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1answer
16 views

Bezout-If two curves intersect at $m\cdot n$ points then the intersection multiplicity is $1$

In my lecture notes, after the Bezout theorem there is the following collary: If the plane projective curves, $x=V(F), y=V(G)$, intersect at exactly $m \cdot n$ discrete points, then the ...
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28 views

Projective algebraic curves-affine curves

At the projective algebraic curves there are similar identities to affine curves. Intersection points of projective algebraic curves. The meanings order of point of the curve $F$ intersection ...
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35 views

Calculate the singular points of affine curve

I want to calculate the singular points of the affine curve $$f(X,Y)=(1+X^2)^2-XY^2 \in \mathbb{C}[X,Y]$$ The point $P=(x,y)$ is singular $\Leftrightarrow$ If $x=0$ we find $y=0$ and then from the ...
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1answer
24 views

Find the lengths of the given curves

I have a problem where I need to find the length of a given curve using integration. I've probably put about $2$ whole hours into this question, but I'm completely stumped as to solving it. Here's the ...
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1answer
82 views

Exercise 2.17: Algebraic curves - William Fulton

Let $V=V(Y^2-X^2(X+1))\subset\mathbb{A}^2$, e $\overline{X}, \overline {Y}$ the residues of $X,Y$ in $A(V)$ its coordinate ring; let $z= \dfrac{\overline{Y}}{\overline{X}}\in K(V)$. Find the pole sets ...
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Embedding a $deg \geq 4$ curve in $\mathbb{P}^2$

Reading the paper by Kollar "The structure of algebraic threefolds", among some examples he does talking about intrinsic and extrinsic geometry over $\mathbb{C}$, he mentions that "an irreducible ...
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1answer
193 views

If six points of an elliptic curve are contained in a conic, then their sum is $O$.

Let $C$ be a projective cubic without singular points and $O\in C$ an inflexion point. We consider the addition in $C$ with $O$ as neutral element. If $R_{1},...,R_{6}\in C$ are different points such ...
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1answer
42 views

An example of a family of projective irreducible curves

I'd like to construct explicitely (namely with a parametric equation) the following example: A family of projective curves parameterized by $\mathbb P^1(\mathbb C)$ with 3 properties: All curves ...
2
votes
1answer
57 views

Weil divisors fail over singular varieties

Let be $k$ an algebraically closed field. We know that if $X$ is an irreducibile, normal variety, one can associate to every rational function $(f)\in k(X)^*$ a Weil principal divisor $$(f)=\sum_{Y} ...
5
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1answer
92 views

Conic by three points and two tangent lines

With the exception of degenerate situations, a conic is uniquely determined by five points lying on it. Likewise, five lines tangent to a conic uniquely define that conic. With four points and one ...
2
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0answers
48 views

Rational functions over curves define regular maps?

I've a rather simple question that is strucking me, perhaps because I'm a newbie in algebraic geometry. If we have a projective irreducible non-singular curve $C\subseteq \mathbf{P}^n_k$ where $k$ is ...
2
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0answers
47 views

Dessins d'Enfants and Real Algebraic Curves

I wrote a thesis on the Grothendieck theory of Dessins d'Enfants (after some articles by Leila Schneps). In Shafarevich, vol.2, there's a section on real algebraic curves. Is it possible to formulate ...
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1answer
37 views

Mordell's theorem-Finitely generated abelian group

In my lecture notes we have the following: Mordell proved the following: Let $C$ be a nonsingular cubic curve with rational coefficients. Then the abelian group of rational points on $C$ is ...
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Equation of a non-singular cubic curve

The equation of a non-singular cubic curve in affine coordinates is $$y^2+a_1 xy+a_3 y=x^3+a_2x^2+a_4x+a_6 .$$ If $\text{ch } K \neq 2, 3$ then it is written $$y^2=x^3+ax+b .$$ Why do we write it ...
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3answers
193 views

Program to find closest function to fit arbitrary data

I've wanted this for years, but have never come across anything; a program for Windows to find the closest function to fit arbitrary data. The data I feed it is simple: A table with two columns ...
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1answer
119 views

Curves that don't have lines as components

In my lecture notes we have the following: A point $P=\left [x, y, z\right ]$ of an algebraic curve $C_F=V(F)$ is called an inflection point of $C_F$ when $P$ is not a singular point of $C_F$. ...
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61 views

Principal divisor of rational functions over nonsingular curves and pullback

I'm studying the theory of divisor over algebraic varieties for a seminar and I came across a problem that I think I solved almost completely except for a point that I'm missing. Let be $k$ an ...
3
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1answer
78 views

Computation of the global sections of a normal sheaf

Let $Y\subset X=\mathbb{P}^r$ be the image of the Veronese embedding $\mathbb{P}^1\rightarrow\mathbb{P}^r$. I want to calculate $dim$ $H^{0}(C,\mathcal{N}_{Y|X})$, where $\mathcal{N}_{Y|X}$ is the ...
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20 views

Points at infinity correspond to asymptotic slopes

Let $ P^2\mathbb{C} = \{ [a, b, c] | a,b,c \in \mathbb{C}^* \} $ the complex projective plane. So $ [a,b,c] \sim [x,y,z] $ iff $ \exists \lambda \in \mathbb{C}^* \colon \lambda(a,b,c) = (x,y,z) $. In ...
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16 views

The Cusp $w^2 + p(z,w)=0$ is desingularizable in the origin $O \in \mathbb{C}$

I have just studied a method in projective geometry over complex numbers on how to desingularize a curve in a point but i'm a little bit confused. I don't know the name of this classical method in ...
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1answer
39 views

Property of the intersection multiplicity: $I(P, y \cap x)=1$

How can we show the following property of the intersection multiplicity? $$I(P, y \cap x)=1 , \text{ where the point } P \text{ is at } (x, y)$$ Edit: My try: $$I(P, f \cap g ) \geq m_P(f) ...