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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Isomorphism between $Pic_X[2]$ and $(\mathbb{Z}/2\mathbb{Z})^{2g}$

Let $X$ a Riemann surface of genus $g$. How could I prove that there is an isomorphism between $Pic_X[2]$ (i.e the line bundles $L$ such as $L^2 \simeq \mathcal{O}_X $) and ...
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61 views

Embedding an affine curve in a proper curve.

I am trying to figure out the following problem from Q.Liu's algebraic geometry in chapter 4. Let $U$ be an integral affine algebraic curve over $k$. (a) show that there exists a proper curve $ ...
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34 views

Is it important to study plane algebraic curves before read Fulton's book

I'm studying Fulton's algebraic curves book and I would like to know how important study plane algebraic geometry before read Fulton's book. Example of books on this subject: Algebraic Curves - ...
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31 views

What does linearly equivalent mean in this context

I'm trying to understand this proof of Fulton's algebraic curves book page 107: I didn't understand what does linearly equivalent mean in this context and why this implies it suffices to show that ...
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1answer
48 views

Why this order is well-defined?

I'm trying to understand why this order defined in Fulton's algebraic curves book is indeed well-defined: I have two questions: Why $u\in \mathcal{O_P}(X)$ (he is implicitly assuming this fact so ...
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52 views

What do I need to understand this article

I've just finished Fulton's algebraic curves book and I would like to know what do I need to know to understand this article: Weierstrass semigroups and the canonical ideal of non-trigonal curves. I ...
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1answer
98 views

Help in this notation in Fulton's Algebraic Curves book

I'm reading Fulton's Algebraic Curves book, I'm stuck in the following proposition (page 105): In fact, what I didn't understand is the following notation in the proof of this proposition: Why ...
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42 views

What does “$\overline{G}_*$ is the residue of $G_*$ in $\mathscr{O}_P(F)$” mean in Fulton's book on algebraic curves?

I'm trying to understand this phrase in Fulton's algebraic curves book page 53: Anyone could help me? Thanks
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1answer
30 views

Help in this characterization of the gaps of the symmetric numerical semigroups

Before my question, some background: Definition 1: A numerical semigroup is a subsemigroup $N$ of the additive semigroup $\mathbb N$ of the non-negative integers such that $\mathbb N-N$ is ...
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1answer
117 views

Meaning of notation $\operatorname{ord}_Q(g)$ in “Algebraic Curves” by Fulton

I didn't understand this notation in the chapter 7 page 93 of Fulton's algebraic curves book: What the author means by $\text{ord}_Q(g)$? Maybe he would like to say $\text{ord}_Q(G) := ...
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1answer
54 views

Help in this equivalence in Fulton's book

I'm studying Fulton's algebraic curves book and some of its notation look like confusing. I need help to understand this equivalence in the page 35. The preceding paragraph he mentioned I put ...
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1answer
48 views

The intuition behind the coordinate ring $\Gamma(F)$

I'm studying Fulton's algebraic curves book. He gives the following definitions: We can define the coordinate ring of a nonempty variety $V\subset \mathbb A^n$ as $\Gamma(V)=k[X_1,\ldots,X_n]/I(V)$. ...
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1answer
21 views

Why this $F_*=F(X,Y,1)$

I'm studying Fulton's algebraic curves book. Someone could help me to prove this phrase highlighted: I didn't understand why the $F_*$ he defined is the same of the known $F_*=F(X,Y,1)$. Thanks ...
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3answers
86 views

The topology on $\mathbb A^2$ is not the product topology [duplicate]

I'm trying to prove the Zariski topology on $\mathbb A^2$ is not the product topology on $\mathbb A^1\times \mathbb A^1$. I'm looking for a counter-example based on the fact the closed subsets in ...
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0answers
22 views

Curve and Constant Curvature

I have initial position vector $p_0$, given curve-linear length $1$. It can be parameterized by $s\in[0,1]$. Assume we have the equation to generate the curve from given starting point and constant ...
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65 views

Divisor question on the normal projective curve

Let $X$ a normal projective curve over an infinite field $k$, let $x_1,\dots,x_n$ be pairwise distinct closed points in $X$ and let $n_1,\dots,n_r\in\mathbb Z$. Let $$D=\sum_in_ix_i$$ ...
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1answer
41 views

I need help to understand blowups of points in curves in $\mathbb A^2$

I'm trying to understand how to blowup curves which I'm finding very difficult. Example $V=V\bigg(y^2-x^2(x+1)\bigg)$ Blowup map $\pi$: $$B=\{(x,l)\in \mathbb A^2\times \mathbb P^1|x\in l\}\to ...
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43 views

Theta-divisor and special divisors

I'm studying the Jacobi inversion problem. Could you help me with the next question? Let us consider an algebraic curve $C$ and it's Jacobian $J(C)$. For a initial point $P_0$ we can construct the ...
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0answers
42 views

Why this equality?

I'm trying to understand this proof in Fulton's algebraic book: I understood why we can assume $C$ a closed subvariety of $\mathbb P^n$ such that $C\cap U_i\neq \emptyset$, $i=1,\ldots,n+1$ . My ...
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1answer
31 views

Doubt in the definition of closed subvarieties

I'm trying to understand this definition in Fulton's algebraic curves: In order to be $Y$ a variety, $\overline Y$ has to be an irreducible algebraic set of $\mathbb ...
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0answers
32 views

Taylor expansions in two variables

I need help in this proof Can I use Taylor expansion in Algebra? someone could give more detail of this Taylor expansion? Thanks in advance EDIT The main question is how the author get this "Y + ...
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0answers
18 views

About the equivalence between smooth projective curves and compact Riemann surfaces

We know that there exists an equivalence of categories between compact Riemann surfaces and smooth projective curves (over $\mathbb C$). But the Riemann surfaces are connected by definition, so I ...
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19 views

$P$ is a simple point of $F$ $\Leftrightarrow O_P(F)$ is a DVR

I'm trying to find some sources with another proof of this theorem in Fulton's book: Does someone know other proofs of this theorem? maybe more algebraic? Thanks in advance
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22 views

How to find the tangent vector to a curve at a special point

The two points $A=(x_{0},y_{0},z_{0})$ and $B=(x_{1},y_{1},z_{1})$ are given. The line segments $AC$ and $BC$ make equal angle $\alpha$ with the horizontal plane through $C$. The angle ...
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1answer
50 views

How to prove the uniqueness

I'm trying to solve this question from Fulton's algebraic curves: I've already easily solved (a) and the existence part of (b). I'm having problems to prove the uniqueness of part (b). I need ...
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51 views

Why is this theorem important in fulton's book?

In Fulton algebraic curves book, we have the following proprieties which help us to find the intersection number of a pair of function at a given point. afterwards he states this theorem: ...
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1answer
75 views

How to prove this comment of Fulton

I'm trying to understand why this is true in Fulton's Algebraic Curves: Why we add this point $(0,\ldots, 0)$? Why this equality is true? I really need help. Thanks in advance.
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1answer
24 views

Sources about transcendence degree

I asked this question: Characterization of the transcendentals over a field I realized I need some knowledge about transcendence degree to prove some facts in the book I'm reading. I would like to ...
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1answer
43 views

The ring of fractions $K(x)$ is the field generated by $K$ and $x$.

I would like to show that the ring of fractions $K(x)$ of $K[x]$ in an extension $L$, where $K\subset L$ fields, is the field generated by $K$ and $x$ (let's call it by $\tilde{K(x)}$). I know just ...
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2answers
19 views

Rational functions are decomposed in polynomial products

I'm trying to understand why this is true: Since $K(x)$ is a field, $K(x)$ is an UFD, then $K(x)$ can be written uniquely as products of irreducible elements of $K(x)$. I didn't understand why ...
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1answer
36 views

Characterization of the transcendentals over a field

I'm studying Algebraic Function Fields and Codes book from Henning Stichtenoth and I didn't understand this remark in the first page: I couldn't solve any part of the equivalence, I think maybe ...
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19 views

projective change of coordinates and tangent line

Let $C/k$ be a projective algebraic curve given by a polynomial $F \in k[X,Y,Z]$. If $L$ is tangent to $C$ at a point $P$ does $L$ remain tangent to $C$ at a point $P'$ after a projective change of ...
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1answer
45 views

$a,b$ integral $\implies$ $a+b$ integral

I'm sure it's just silly thing. I'm reading Fulton's algebraic curves book and I don't understand this phrase of this proof: I didn't understand why according to the proposition we have $a\pm b,ab$ ...
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34 views

Curve avoiding semi-rational points

A rational point is a point in $\mathbb{R}^d$ all of whose $d$ coordinates are rational. Let me define a semi-rational point as one that has at least one rational coordinate (but whose other ...
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1answer
51 views

Prove that $H^1(\mathcal{M}^*)=0$.

Let $X$ be a compact Riemann surface. For an open set $U$, let $\mathcal{M}^*(U)$ be the multiplicative group of nonzero meromorphic functions on $U$ ("nonzero" meaning "not identically zero"). This ...
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2answers
74 views

Dimension of $\dim_{\mathbb C}\mathbb C[X,Y]/I(Y^2-X^2,Y^2+X^2)$

I'm trying to solve the question 1.36 from Fulton's algebraic curves book: Let $I=(Y^2-X^2,Y^2+X^2)\subset\mathbb C[X,Y]$. Find $V(I)$ and $\dim_{\mathbb C}\mathbb C[X,Y]/I$. Obviously ...
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1answer
65 views

Decomposition of an algebraic variety into irreducible components

I'm studying the Fulton's algebraic curves book and I have the following doubts in the end of the page 9: I didn't understand why the following equations hold: $$I\left(\bigcup_i ...
2
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0answers
81 views

When do two integral superellipses have 'nice' intersections?

A recent question posed the nonlinear system \begin{cases} 3x^3+4y^3=7\\ 4x^4+3y^4=16 \end{cases} for real $(x,y)$ and asked for the sum $x+y$. As noted by commentary in the question, this regrettably ...
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1answer
27 views

integral point on conics

Suppose we have a conic $ax^2 + bxy + cy^2 + dx + ey + f = 0$ where $a,b,c,d,e,f \in \mathbb{Q}$. Is there a way of computing the integer points on this curve. Since it is affine an not projective we ...
2
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32 views

Mobius transformation of Algebraic curve

I am working on the uniformization of algebraic curve problem. Currently, my adviser gave me a question about build a Mobius transformation between algebraic curves, and then lift it to the Rimeann ...
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1answer
79 views

Finding the area enclosed by $px^4+qxy+ry^2+sy+t=0$

When $px^4+qxy+ry^2+sy+t=0\ (p,q,r,s,t\in\mathbb R)$ represents a simple closed curve on the $xy$ plane, can we represent the area enclosed by this curve by $p,q,r,s,t$? If yes, then how? Example 1 ...
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1answer
62 views

Merge two or more cubic Bézier curves for optimization

I am looking for an algorithm which can merge several cubic Bezier curves. For instance, I have a lot of cubic Bezier that are joined to form a poly-Bezier curve. The idea is to merge dynamically some ...
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11 views

is it possible to express the moduli of ppav's using torelli loci?

This is a probably vague question from an outsider: It is well known that it is possible to embed "the" moduli space of curves $M_g$ with fixed genus g into the moduli space of principally polarized ...
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1answer
25 views

Covering of Projective line

This is an exercise given during the course in Riemann Surfaces that I attended this year. Let $X$ be a compact Riemann Surface that is a degree $3$ cover of $\mathbb{P}^1(\mathbb{C})$ given by ...
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43 views

Torsion points on elliptic curves, $E^1$

Let $E$ be an elliptic curve over $\mathbb{Q}_p$ and let $E(\mathbb{Q}_p) \supset E^0(\mathbb{Q}_p) \supset E^1(\mathbb{Q}_p) \supset \cdots$ be its $p$-adic filtration, where $E^n(\mathbb{Q}_p) = \{P ...
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32 views

Parabola tangent to four lines

Suppose that in the affine plane R^2 four lines are given, with the property that no two are parallel and no three are concurrent. Show that there exists a unique parabola tangent to each of the four ...
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1answer
80 views

Why is it called “elliptic” curve?

One of my favourite and most studied algebraic curve is the elliptic curve. But something that I have never asked myself is: Why do they call this nonsingular cubic curve an "elliptic" curve? ...
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1answer
54 views

Why a cubic plane curve meets a line three times?

Can someone explain to me why a cubic curve in a projective plane always meets a line three times?
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1answer
51 views

$p$-adic numbers and projective coordinates

Let $E/\mathbb{Q}_p$ be an elliptic curve and let $E^0(\mathbb{Q}_p)$ denote its nonsingular points. We accept that $E^0(\mathbb{Q}_p)$ is a subgroup of $E(\mathbb{Q}_p)$. Then let ...
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1answer
57 views

elliptic curves over $\mathbb{Q}_p$

Let $E: Y^2 = X^3 + AX + B$ be an elliptic curve over $\mathbb{Q}_p$, i.e. $A,B \in \mathbb{Q}_p$ and $4A^3 + 27B^2 \neq 0$. Then, according to page 47 of Cassels' Lectures on Elliptic Curves, if ...