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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23 views

About Runge's method

I have been reading about some Diophantine equations (like Runge's theorem and Cassel's theorem) and in the text says that these theorems are solved using Runge's method, but it doesn't say what ...
2
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0answers
27 views

Effective divisor vs curve on surface

Hartshorne in his book, with the term "a curve $C$ on a surface $S$" (over an algebraically closed field $k$) means that $C$ is an effective divisor on $S$. So, can I conclude that a "a curve $C$ on ...
2
votes
1answer
19 views

Locally free sheaf generated by global sections and vanishing cohomology on curves

Let $C$ be a smooth projective curve. Let $\mathcal{F}$ be a locally free sheaf on $C$ satisfying $H^1(\mathcal{F})=0$. Is it then true that $\mathcal{F}$ is generated by global sections?
2
votes
1answer
42 views

What is the symmetry group of the circle as a complex curve?

In a complex (projective) plane CP2 using homogeneous coordinates $(x, y, z)$ what is the group of (projective) transformations that leave the complex circle $$x^2 + y^2 = R^2 z^2 $$ invariant as a ...
2
votes
3answers
58 views

Existence of covers for given genus and degree.

Suppose we are given a genus $g$ and degree $n$; under which circumstances is there a curve $C$ of that genus admitting a map of degree $n$ to $\mathbb{P}^1$? For example: If $n = 2g-1$ there is no ...
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0answers
6 views

Tangent to Bernoulli lemniscate for node at infinity

The Bernoulli lemniscate in the complex plane CP2 using homogeneous coordinates has the equation: $F(x,y,z) = (x^2 + y^2)^2 -2f^2(x^2-y^2)z^2 = 0$. $f$ is the distance of the foci of the lemniscate ...
3
votes
0answers
45 views

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 ...
0
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2answers
90 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
votes
1answer
22 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 $\Phi|_{V \cap W}$ ...
2
votes
0answers
38 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 ...
0
votes
1answer
28 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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votes
1answer
27 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
votes
0answers
26 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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votes
0answers
47 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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0answers
42 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
0
votes
0answers
30 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 ...
1
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1answer
56 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 ...
0
votes
2answers
70 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 ...
0
votes
1answer
32 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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0answers
20 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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votes
1answer
44 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
votes
1answer
106 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 ...
0
votes
1answer
31 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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0answers
19 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 $T^{p}R_{f,g}(TY) = T^{q}R_{f,g}(Y)$?
1
vote
1answer
33 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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0answers
34 views

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 ...
7
votes
2answers
68 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 ...
1
vote
0answers
40 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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0answers
37 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 ...
1
vote
0answers
67 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 ...
1
vote
1answer
35 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 ...
3
votes
0answers
57 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 ...
0
votes
1answer
50 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
votes
0answers
25 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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votes
0answers
10 views

Bezout's Theorem-intersection multiplicity

Bezout's Theorem: $K$ algebraic closed Let $X=V(F),y=V(G)$ two projective curves of $\mathbb{P}^2(K)$ with degree $m$ and $n$ respectively that do not have a common component. Then: ...
1
vote
1answer
13 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 ...
0
votes
0answers
22 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 ...
3
votes
0answers
29 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 ...
0
votes
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 ...
0
votes
1answer
56 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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0answers
73 views

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 ...
6
votes
1answer
182 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 ...
1
vote
1answer
39 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
46 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} ...
2
votes
1answer
29 views

computing the full constant field of an algebraic function field

Let $K$ be a field such that char$(K) \neq 2$. Let $F=K(x,y)$ be an algebraic function field field of one variable $x$ where $$y^2 = f(x) \in K[x]$$. We want to compute the full constant field of $F$ ...
5
votes
0answers
54 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
votes
0answers
41 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
votes
0answers
44 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 ...
1
vote
1answer
33 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 ...