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 ...

learn more… | top users | synonyms

0
votes
0answers
22 views

Common zeros and GCD of polynomials

Facing another algebraic geometry problem: Let $p,q \in T[x,y]$. Prove the set $V(p,q)$ is finite if and only if set $V(GCD(p,q))$ is finite. ($V(p)$ of course meaning the subset of $A^2(T)$ where p ...
3
votes
3answers
173 views

Proof that two simultaneous line equations do not intersect?

Apologies if this isn't at the level of questions expected here! I've got two simultaneous equations to solve. (Equation 1): $ x y = 4 $ (Equation 2): $ x + y = 2 $ They produce the following ...
2
votes
0answers
39 views

Where do divisors on curves come from?

I've been learning about Riemann surfaces and the Riemann-Roch theorem, and I've been able to experience some of its great power. However, I'm curious as to what the history behind divisors on curves ...
0
votes
1answer
21 views

Decomposition into irreducible algebraic sets

I am facing following problem and would really appreciate anyone's help: I am to find decomposition of $V(x^2 - y^4, x^3 - xy^2 + x^2y^2 -y^4)$ into irreducible algebraic sets in $A^2( \mathbb{C})$. ...
1
vote
0answers
12 views

Singular Points on Irreducible Cubic Curves Defined over Not Necessarily Algebraically Closed Fields

Let $C$ be a cubic curve defined over a field $k$. Take, for example, an affine curve: $$ C = \{(x,y) \in k\times k : a x^{3} + b x^{2} y + c x y^{2} + d y^{3} + e x^{2} + f x y + g y^{2} + h x + i y ...
0
votes
2answers
70 views

Algebraic variety in $\mathbb{C^{2}}$

would somebody algebraic-geometry-savvy please help me with this problem, as I am pretty new to algebraic geometry: I have to find smallest algebraic variety (irreducible algebraic set) in ...
0
votes
0answers
24 views

Determine the number of real roots of the system.

Determine the number of real roots of the system,$1.$$x^3y - y^4 =a^2$ $2.$$x^2y+2xy^2+y^3=b^2$ where $a$ and $b$ are real parameters.
0
votes
0answers
23 views

What is the definition of Riemann surface of an algebraic function?

What does it mean by the Riemann Surface of a function $y=\sqrt{x^3}$? I saw how to use the cut and glue method to obtain a sphere where $y=\sqrt{x}$ can be defined. But I was not clear in what sense ...
2
votes
1answer
41 views

Solutions to a system of equations

I know that Bezout's theorem says that if you take two plane curves, then their maximal number of intersection points is the product of their degrees. However, assume that I have two irreducible ...
1
vote
0answers
29 views

A question about hyperelliptic curve

This question is from the Qing Liu's book Algebraic Geometry and Arithmetic Curves 7.4.10 Let P(t) $\in$ k[t] be a seperated polynomial of even degree $\geq$ 2 over an algebraically closed field ...
3
votes
2answers
60 views

$\dim (D-P)=\dim (D)-1$

I'm trying to prove this question: Let $D$ be a divisor in $F|K$ such that $\dim (D)\gt 0$ and $0 \neq f\in \mathscr L(D)$. Thus $f\notin \mathscr L(D-P)$ for almost all $P$. Then show that ...
0
votes
0answers
16 views

Algebraic curves, problem of cubics, with intersection, parabolic branch…

Find a cubic that intersect V(X^2-XY+Y^3)with at least multiplicity 6 in the origin, and besides it has an horizontal parabolic branch and the point (0,1) belongs to that cubic. First of all, to get ...
3
votes
0answers
34 views

When are the coordinates of the intersection points of plane curves actually algebraic conjugates

Suppose $f(x,y)\in\mathbb{Z}[x,y]$ is an irreducible polynomial defining a plane curve. Say I want to find the intersection of this plane curve with the line defined by g(x,y)=y-ax+b. One way to do ...
0
votes
0answers
44 views

An Estimation for Multiple Points in Fulton's Curve Book

I am learning basic algebraic geometry by following Fulton's "Algebraic Curves". In 5.4, the auther stated the following theorem: Theorem If $F$ is an irreducible projective plane curve ...
1
vote
0answers
39 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
65 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
41 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
51 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
53 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
99 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 ...
2
votes
0answers
44 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
32 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
119 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
55 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
49 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
92 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
27 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 ...
1
vote
0answers
73 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
44 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
43 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
32 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
33 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 + ...
1
vote
0answers
19 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 ...
0
votes
0answers
20 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
0answers
28 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 ...
0
votes
1answer
53 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
77 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
44 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
38 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
27 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
47 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
0answers
36 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 ...
2
votes
1answer
57 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 ...
0
votes
2answers
75 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 ...