1
vote
0answers
31 views

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 ...
1
vote
0answers
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 $ ...
0
votes
0answers
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 - ...
0
votes
1answer
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 ...
0
votes
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 ...
2
votes
0answers
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 ...
2
votes
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 ...
1
vote
0answers
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
2
votes
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 ...
2
votes
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) := ...
0
votes
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 ...
1
vote
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 $\Gamma(V)=k[X_1,\ldots,X_n]/I(V)$. ...
0
votes
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 ...
1
vote
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 ...
1
vote
0answers
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 $$D=\sum_in_ix_i$$ ...
0
votes
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 ...
2
votes
0answers
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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
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 + ...
0
votes
0answers
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
0
votes
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 ...
0
votes
0answers
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: ...
0
votes
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.
2
votes
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 ...
0
votes
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 ...
0
votes
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 ...
3
votes
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 ...
0
votes
0answers
20 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 ...
-1
votes
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 $a\pm b,ab$ ...
0
votes
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 ...
1
vote
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
votes
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 ...
2
votes
0answers
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 ...
1
vote
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 ...
1
vote
0answers
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 ...
3
votes
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? ...
0
votes
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?
0
votes
0answers
53 views

How can we compute the order of 1-form on Riemann surfaces

Let X be a hyperellictic curve defined by $y^2=h(x)$. Let $\pi:X\rightarrow\mathbb{P}^1$ be the double covering map seding $(x,y)$ to $x$. Let $\omega=\pi^*(dx/h(x))$. How can we compute the orders of ...
1
vote
0answers
44 views

On existence of a tangent line passing through a given point

Question Suppose $k$ is an algebraically closed field of characteristic $0$, and $C\subseteq\mathbb P^2(k)$ is an irreducible projective plane curve of degree $n>1$, and $P$ is a point on $\mathbb ...
2
votes
0answers
52 views

Irreducible components of a lifted curve

I am looking at Terry Tao's blog where he reviews Bombieri's proof of the Hasse Weil bound. At some point he argues as follows. Let $C$ be a curve defined over $\mathbb{F_q}$ and let $\pi : C \to ...
1
vote
1answer
63 views

Birrational curves and singularities [closed]

If $C$ and $D$ are two birrational plane curves. Is there some relation between their singular points?
2
votes
1answer
67 views

Holomorphic Differentials on a non-singular curve.

So I've been working on this for an exam I have coming up and I'm not sure I really understand. If I have a curve defined by some homogenous polynomial P, I can show that the canonical divisor class ...
2
votes
0answers
41 views

Finiteness of morphism of curves with fixed image

This question comes from the proof of "bend and break" lemma in "Higer-dimensional algebraic geometry" (p.59-60). I use the notations in compatible with the notation given there for convenience. Let ...
3
votes
0answers
58 views

Are $k$ points on a smooth algebraic plane curve ever in general position?

Let $C$ be a smooth plane curve of degree $d$ and genus $g=\frac{(d-1)(d-2)}{2}$. Let us choose $k\leq g+3d-1$ points on $C$. Is it true that the dimension of the space of plane curves of degree $d$ ...
1
vote
1answer
64 views

An question on effective divisor (Clifford 'S theorem)

For an effective divisor $D\ge 0$ on a curve $Y$, define $$\lvert D\rvert =\{ D' \in \mathrm{Div}(Y) \mid D'\ge 0 \;\text{ and }\; D' \sim D \}$$ where $D\sim D$ means $\exists$ a rational ...
1
vote
1answer
41 views

A question on the morphism of projective varieties

The continuation of this, my question I want to show that $X$ and $Y$ are smooth and irreducible curves then $f(X)$ is either $Y$ or a point. Note that I know the proof of this ...
2
votes
1answer
69 views

the diagonal $\Delta (Y) =\{(y,y)\in Y \times Y\}$ is closed in $\Bbb P^m \times \Bbb P^m $

Asumme that $\Pi : \Bbb P^n \times \Bbb P^m \rightarrow \Bbb P^m $is a closed map. $X\subset \Bbb P^n$ And $Y\subset P^m$ $f: X\rightarrow Y$ be a morphism of projective varieties. ...
2
votes
0answers
42 views

Can a “negative degree” line bundle on a reducible curve have global section?

Suppose $A,B$ are curves on smooth projective surface, having no common components and intersect, so $(A.B)>0$,do we have $H^0(O_A(-B|_A))=0?$ (here $A,B$ are effective divisors, may be irreduced ...
2
votes
1answer
48 views

intersection multiplicity from Shafarevich

In Basic Algebraic Geometry, Shafaravich proves the following theorem: Theorem. If $X$ is an irreducible affine curve, and $P \in X$ is a nonsingular point, then there is a function $t$, regular at ...