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

5
votes
2answers
92 views

Divisors of differentials.

Let $\mathbb{C}$ be the base field. Suppose $C \subset \mathbb{P}_2$ is a nonsingular projective curve of degree $d > 3$. Must it be the case that $D \in \text{Div}(C)$ is the divisor of a ...
1
vote
1answer
46 views

Examples of base points of linear systems

I'm reading Fulton's algebraic curves book and we have the following definitions: A divisor $D=n_1P_1+\ldots,n_kP_k$ ($n_i$'s are integers and $P_i$'s are points) over a curve. A linear system as ...
0
votes
0answers
22 views

if $D_1\sim D_1'$ and $D_2\sim D_2'$, then we have $D_1+D_2\sim D_1'+D_2'$

I'm studying Fulton's algebraic curves book and I'm trying to prove that if $D_1\sim D_1'$ and $D_2\sim D_2'$, then we have $D_1+D_2\sim D_1'+D_2'$. If the the first two equivalences work, then we ...
1
vote
1answer
29 views

Is a map from a smooth curve, bijective away from the singularities, already a normalization?

Over $\mathbb{C}$, I am considering a map between projective curves $f: C \rightarrow C'$, where $C$ is smooth. Suppose that $f$ is surjective as well as bijective away from the singularities of ...
0
votes
1answer
66 views

Why this linear system doesn't have base points?

I see somewhere that linear system of a non-negative degree divisor over a rational curve doesn't have base points, but I didn't understand why. I don't understand what the degree has to do with base ...
0
votes
0answers
41 views

Genus formula for a curve in a $2$-dimensional complex torus?

For a curve $C$ in a $2$-dimensional complex torus $T$, is there any formula to compute the genus of $C$? Say, in terms of self-intersection number? For a curve in $\mathbb{P}^2$ or a K3 surface, ...
1
vote
0answers
25 views

Linear system without base points

Let $D$ be a divisor and $V$ a vectorial subspace of $L(D)$. The set of the divisors $S=\{(f)+D\mid f\in V\}$ is called a linear system. A point $P\in C$ is a base point if for every divisor $D\in S$, ...
5
votes
1answer
58 views

Does every irreducible projective cubic curve have a nonsingular point of inflection?

Does every irreducible projective cubic curve necessarily have a nonsingular point of inflection? I've been trying to construct counterexamples, to no avail, which leads me to believe the ...
1
vote
1answer
24 views

How do we find the principal unit normal to this curve?

A curve is given in cylindrical coordinates: $r=r(t)$ $\theta=\theta(t)$ $z=z(t)$ The curve is unit-speed: $(\frac{dr}{dt})^2+r^2(\frac{d\theta}{dt})^2+(\frac{dz}{dt})^2=1$ How do we find the ...
9
votes
1answer
50 views

Irreducible projective cubic, exists continuous surjection?

Let $C$ be an irreducible projective cubic in $\mathbb{P}_2$ with a singular point $p$. So consider $f: \mathbb{P}_1 \to C$ defined as follows. Identify $\mathbb{P}_1$ with the set of lines in ...
5
votes
1answer
58 views

Necessarily a homeomorphism?

Let $D$ be the projective curve defined by $y^2z = x^3.$ Consider the map $f: \mathbb{P}_1 \to D$ defined by$$f[s, t] = [s^2t, s^3, t^3].$$Is it necessarily a homeomorphism? Any help would be greatly ...
2
votes
0answers
48 views

The degree-genus formula cannot be applied to singular curves in $\mathbb{P}_2$?

(The degree-genus formula) The Euler number $\chi$ and genus $g$ of a nonsingular projective curve of degree $d$ in $\mathbb{P}_2$ are given by$$\chi = d(3-d)$$and$$g = {1\over2}(d-1)(d-2).$$ My ...
9
votes
2answers
221 views

Why is it so difficult to find beginner books in Algebraic Geometry?

I don't understand why it is so difficult to find a really beginner book in Algebraic Geometry. Let's take for example Fulton's algebraic curves: An introduction to Algebraic Geometry. I've already ...
1
vote
0answers
29 views

What is a projective rational function in Fulton's book?

I'm reading Fulton's algebraic curves and I don't know if I understood correctly this definition on the page 46: What does the author mean by $f,g\in \Gamma_h(V)$ being of the same degree? Since ...
2
votes
1answer
39 views

Why this set is the pole set of $z$?

Suppose $V\subset \mathbb P$ a projective variety, $z$ a rational function on $V$ and let's define $J_z=\{F\in k[X_1,\ldots, K_{n+1}\mid \overline Fz\in \Gamma_h(V)\}$ and $S_z$ the polo set of $z$, ...
2
votes
1answer
48 views

I need help in this proof of this exercise from Fulton's book

I'm reading Fulton's algebraic curves book. I'm trying to understand this solution which I found online of the question 4.17 on page 97. What I didn't understand is why $V(J_z)$ are exactly those ...
4
votes
1answer
48 views

Proof in Fulton's *Algebraic Curves*

I'm reading Fulton's algebraic curves book on page 106 and I didn't understand this proof: I didn't understand why can we assume $F_Y\neq 0$? (what $F$ irreducible has to do with this?). ...
1
vote
1answer
38 views

Help to understand this proof in Fulton's book

I'm reading Fulton's algebraic curves book on page 105 and I didn't understand this proof: 1.Why if $R=k[X_1,\ldots,X_n]$, then $\Omega_k(R)$ is generated (as R-módule) by the differentials ...
1
vote
1answer
28 views

How do I find the orders of this rational function?

How can I find the orders of $z(x)=\frac{x}{1-x}$ over $k(\mathbb P^1)$ at the zero $x=0$ and the pole $x=1$? I saw in another question posted on MSE that the orders are both equal to $1$, but I ...
0
votes
1answer
46 views

Help me with this solution of the exercise 4.17 from Fulton's Algebraic Curves

I'm studying Fulton's algebraic curves book and in order to prove the well-definiteness of the divisor $div(z)$ on page 97 I'm trying to understand this solution which I found online. I didn't ...
1
vote
0answers
35 views

morphisms of curves and discrete valuation rings

Given a dominant morphism $\varphi\colon C\to C'$ of curves, a nonsingular point $Q\in C'$, such that $\varphi^{-1}(Q) = \{P_1,\ldots, P_m\}$ consists of nonsingular points only. Then it is clear to ...
1
vote
1answer
46 views

Example of a divisor of a function

I'm studying Fulton's algebraic curves book and on page 97 Fulton defines the divisor of the rational function $z\in k(C)$: I'm looking for an example of a divisor like this one. Thanks
4
votes
1answer
44 views

Book which covers these contents of the same level of Fulton's book.

My question is very specific. I'm studying Chapter 8 of Fulton's algebraic curves book and I would like to find another book (or online sources) which covers these contents: Divisors, the Vector ...
0
votes
0answers
24 views

Curves not contained in hypersurfaces

Consider a curve $C$ in $\mathbb{F}_q^m$, say. I am interested in the existence of curves not contained in any small degree hypersurface. For instance, a helix is not contained (or non-embeddable) in ...
2
votes
1answer
35 views

All polynomial parametric curves in $k^2$ are contained in affine algebraic varieties

I have started working through the textbook Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea and I am stuck on one part of an introductory question. The question begins by getting one to ...
4
votes
1answer
51 views

Link between Riemann surfaces and Galois theory

In my notes for a Geometry of Surfaces course that I'm studying, there is the following quote: (For those of you who like algebra and Galois theory) Studying compact connected Riemann surfaces is ...
0
votes
1answer
22 views

Integer solutions of a degree 3 curve

Suppose you have a square pyramid made out of rigid balls and all these balls are equal. Suppose now that you want to fill a square with the same number of balls that the pyramid is made. If $x$ ...
1
vote
1answer
27 views

Hartshorne's Example IV.3.3.5

In Hartshorne's book Algebraic Geometry he says in Example 3.3.5 in Chapter IV (p. 309): If $X$ is a plane curve of degree 4, then $D=X.H$ is a very ample divisor of degree 4. Here $H$ is a ...
9
votes
1answer
137 views

$p \in C - D$, inflection point for $C$ iff inflection point for $C \cup D$.

Show that if $C$ and $D$ are projective curves in $\mathbb{P}_2$ and $p \in C - D$ then $p$ is a point of inflection for the curve $C$ if and only if $p$ is a point of inflection for the curve $C \cup ...
0
votes
0answers
59 views

Finite solution of Power Diophantione Equation.

Given an equation $x^2+k=y^3$ where k is a constant and $y=f(x)$,$f(x)$ is differentiable and algebraic. for which- $$\frac{d}{dx}x^{2} \neq\frac{d}{dx} f(x)^3$$ 1. Can I infer that the ...
1
vote
1answer
41 views

Dimension of linear system of divisor of two points on curve of genus greater than 2

This should not be hard, but I am stuck on it nonetheless, so I would much appreciate a solution. Suppose $C$ is a projective non-singular curve of genus $g\geq 2$ and $P,Q$ are distinct points on ...
-2
votes
1answer
36 views

What is the degree of $(z-2)^5=0$ [closed]

Does anyone know what is the degree of the curve $(z-2)^5=0$ in $C$ (the complex numbers)? Thanks
0
votes
1answer
27 views

Which is the correct definition of degree of a curve

I found those definitions of what a degree is and they contradict each other so I don't know which is the correct. The first definition it was by the last year professor saying that the degree of an ...
3
votes
1answer
54 views

Algebraic Curves and Second Order Differential Equations

I am curious if there are any examples of functions that are solutions to second order differential equations, that also parametrize an algebraic curve. I am aware that the Weierstrass $\wp$ - ...
11
votes
1answer
143 views

Invariant point (flex) invariant under projective transformations.

Let $C$ be a projective curve in $\mathbb{P}_2$ defined by a homogeneous polynomial $P(x, y, z)$ and let $\alpha$ be a linear transformation of $\mathbb{C}^3$. Let $Q$ be the homogeneous polynomial $Q ...
3
votes
0answers
54 views

Calculating eigenvalues of the induced action on $H^0(2 K_C)$

Given a (smooth) curve $C$ and an automorphism $\phi$ of $C$. In the first part of their paper On the Kodaira dimension of the moduli space of curves Harris and Mumford calculate the eigenvalues of ...
0
votes
1answer
47 views

field of rational functions of a curve

Let $C$ be the algebraic curve defined by the modular polynomial $\phi_N$ of order $N>1$ over the rational numbers, i.e. \begin{equation}C:=\text{specm}(\mathbb{Q}[X,Y]/\phi_N(X,Y)). \end{equation} ...
1
vote
1answer
34 views

Benefit from local coordinates

I am reading Elliptic Curves by Anthony Knapp. Its the first time that I am dealing with local coordinates. In page 21 he introduces them as follows: Let $[x_0,y_0,w_0]\in \mathbb P_2(k)$ where $k$ ...
1
vote
1answer
39 views

Singular plane cubic curve birational to $\mathbb{P}^1$

Is it true that every singular plane cubic curve over an algebraically closed field is birationally equivalent to $\mathbb{P}^1$? I know that such a curve has to have only one singular point and that ...
3
votes
1answer
73 views

Divisor on curve of genus $2$

I suffer from lack of concrete examples in Algebraic Geometry, so I will appreciate it if somebody can help me in understanding a bit better this one: Let $\mathcal{C}$ be a genus $2$ curve ...
1
vote
0answers
40 views

Mapping a curve into projective space

Let $\mathcal{C}$ be a (smooth, complex, projective) genus 2 curve. Take two different points $p,q\in\mathcal{C}$ and let $K$ be the canonical divisor class. I know (by means of Riemann-Roch) that the ...
0
votes
0answers
28 views

Model for a Smooth Curve

If $K$ is a finite Galois extension such that $\mathbb{F}_q(x)$ such that $K\cap\bar{\mathbb{F}}_q = \mathbb{F}_q$ then there exists a smooth projective curve $C$ such that $\mathbb{F}_q(C) = K$. My ...
1
vote
1answer
49 views

Resolving a node singularity on a plane curve

So, I am trying to solve Exercise 1.5.6 b) on page 37 in Hartshorne's Algebraic Geometry. For completeness I will include the exercise in my post: If $P$ is a node on a plane curve $Y$, show that ...
1
vote
0answers
51 views

Genus of the product of two elliptic curves

In trying to understand the trichotomy of the genus of algebraic curves, I first consider the following two elliptic curves (over $\mathbb{Q}$), well-known to be of rank $2$, $ y^2 = x^3+17$ and $ ...
0
votes
1answer
43 views

Dual bundle of a stable vector bundle.

Given a vector bundle $E \to X$ over a complex curve $X$, let its rank and degree be $r$ and $d$ respectively. Then the slope of $E$ is defined to be $ \mu(E):= \frac{d}{r}. $ $E$ is defined to be ...
3
votes
0answers
41 views

Use of the Bezout's theorem in Abstract Algebra

The Bezout's theorem: Let $C$ and $D$ be two plane curves described by equations $f(X,Y) = 0$ and $g(X,Y) = 0$, where $f$ and $g$ are nonzero polynomials of degree $m$ and $n$, respectively. ...
0
votes
0answers
22 views

Consider the Plane Curve?

Consider the plane curve $$\gamma(t) = \left( \cosh(t) \cos(t), \cosh(t) \sin(t) \right), \;\; t \in \mathbb R.$$ Is $\gamma$ regular? If $\gamma$ is not regular, can you restrict the parameter ...
0
votes
1answer
25 views

Rate of Change of a Multivariable Function

The problem says, Find the rate of change of $$(x,y,z) = x/z + y/z$$ with respect to t along the curve $$r(t) = \sin^2{t}[ i] + \cos^2{t}[j] + 1/(2t)[k]$$ The answer is apparently ...
6
votes
1answer
95 views

‘Integral’ of a Weierstrass $ \wp $-function.

I'm revising for my finals and I've seen a question which asks: Is there a meromorphic function $f: \mathbb{C}/\Lambda \to \mathbb{P}^1$ such that $f' = \wp$? There is a hint which says consider the ...
12
votes
0answers
170 views

Maximum number of intersection points of two different Bernoulli lemniscates

What is the maximum number of intersection points of two different Bernoulli lemniscates in the real plane? (Of course two identical lemniscates share an infinite number of points.) Here are ...