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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456 views

Good books/expository papers in moduli theory

I have been studying mathematics for 4 years and I know schemes (I studied chapters II, III and IV of Hartshorne). I would like to learn some moduli theory, especially moduli of curves. I began ...
4
votes
1answer
297 views

A question on an exercise in Fulton's book Algebraic curves

Is there a neighborhood of $(0,0,0)$ on $V(x^2-y^3, y^2-z^3)$ that is isomorphic to an open subvariety of a plane curve?
4
votes
2answers
198 views

isomorphisms of algebraic closures

let $K$ be an algebraically closed field. Consider the algebraic closure $\overline{K(X)}$ of $K(X)$, with $X$ trascendent over $K$. Are there cases in which $\overline{K(X)}\cong K$? where $\cong$ is ...
2
votes
0answers
111 views

Singularity type and number of irreducible local analytic curve components

Let $V$ be an irreducible complex plane algebraic curve, $V=V(f)$, and let $\mathcal{O}_p$ be the local ring of holomorphic functions defined in some neighborhood of $p$. If $p=(0,0)$ is a smooth ...
7
votes
1answer
436 views

Abstract Nonsingular Curves

In section I.6 of Algebraic Geometry, Hartshorne establishes a that every curve is birationally equivalent to a nonsingular projective curve. To do this, he defines for any given curve $C$ with ...
4
votes
1answer
430 views

Set that is not algebraic

I'd like some hints for the problem: Show that the following set is not algebraic: $ \{ (\cos(t),\sin(t),t) \in \mathbb{A}^3 : t \in \mathbb{R} \} $ thanks.
6
votes
2answers
211 views

endomorphisms of the jacobian of a general hyperelliptic curve

Let $C$ be a curve of genus $g$. If $C$ is very general, we know that the Jacobian $JC$ of $C$ is simple and thus $End(JC)=\mathbb{Z}$. Do we know something about $End(JC)$ if $C$ is a very general ...
1
vote
2answers
193 views

Every curve has a finite number of multiple points?

I've encountered this assertion and I'm wondering how it is proved. (Here, a multiple point is defined as a point whose local ring is not a DVR, [EDIT] and a curve is a variety whose function field ...
9
votes
3answers
809 views

rational points of an algebraic variety

In http://en.wikipedia.org/wiki/Rational_point we read : a $K$-rational point is a point on an algebraic variety where each coordinate of the point >belongs to the field $K$. This means that, if ...
8
votes
1answer
254 views

How to compute the order $\text{ord}_P (f)$ for $f \in K(C)$

First lets fix some notation. Let $C$ be a projective curve (i.e. projective variety of dimension 1) defined over a field $K$. Suppose that $P \in C$ and that $P$ is a smooth point. It is known that ...
0
votes
2answers
173 views

some notions on algebraic curve

1) I want to learn about algebraic curves and i'm confused, please correct me if i'm wrong : when we say an Affine algebraic curve over the field $F$ : here affine to distinguish it from projective ...
6
votes
2answers
359 views

The genus of an algebraic curve is invariant under isomorphisms

I would like to know how to prove (or even better to see a full proof) of the following "fact". Let $C_1$ and $C_2$ be two smooth curves and let $\phi : C_1 \rightarrow C_2$ be an ...
14
votes
1answer
961 views

Proving that the genus of a nonsingular plane curve is $\frac{(d-1)(d-2)}{2}$

I'm studying from Joseph Silverman's book The Arithmetic Of Elliptic Curves and I'm trying to do as many exercises as I can. Right now I'm trying to do Exercise 2.7 from chapter II which reads as ...
15
votes
2answers
543 views

What is Riemann-Roch in arithmetic all about?

I learn number theory recently and I could not understand what Riemann-Roch was all about in arithmetic; could someone give me a bit hint? What is the advantage of viewing all this stuff geometrically ...
6
votes
1answer
414 views

Computing the monodromy for a cover of the Riemann sphere (and Puiseux expansions)

Given an irreducible polynomial $P(X,Y) \in \mathbb{C}[X,Y]$, one obtains by analytic continuation a Riemann surface $M$ with a branched covering $f \colon M \to \mathbb{P}^1_{/\mathbb{C}}$, ...
3
votes
2answers
256 views

Chinese remainder type theorem in Fulton's Algebraic Curves

The book "Algebraic Curves" by Fulton is available free for download on his website. On page 27, Fulton constructs an isomorphism which is used several times throughout the book. His construction is ...
7
votes
2answers
345 views

Hartshorne exercise about sheaves on $\mathbb{P}^1$

I've been stuck on Exercise II.1.21(e) from Hartshorne's book for quite a while. It concerns the projective line $\mathbb{P}^1$ over an algebraically closed field $k$: write $\mathscr{H}$ for the ...
2
votes
2answers
1k views

What is the equation for plotting points on a curve with fixed end points?

What is the equation for plotting points on an exponential curve with fixed end points? For example, if I want to plot 10 point along a curve that starts with 10,000 (x=1, y=10000) and ends with ...
3
votes
1answer
235 views

every divisor of degree $0$ on a smooth cubic curve $\mathcal{C}\subseteq\mathbb{P}^2$ is equivalent to $A-A_0$ for a fixed $A_0\in\!\mathcal{C}$

NOTATION: Let $\mathcal{C}=\mathcal{V}(F)\subseteq\mathbb{P}^2$ be a curve of degree $3\!=\!deg(F)$ with no singularities and let $A_0\!\!\in\!\mathcal{C}$ be fixed. Let $Div(\mathcal{C})$ denote the ...
4
votes
0answers
231 views

intersection multiplicity and partial derivatives of algebraic curves

this will probably be an easy-to-answer and a not-well-posed question, since I'm a total beginner in the field, but here goes: Let $V(F)$ and $V(G)$ be two projective curves in $\mathbb{P}^2$ ...
21
votes
3answers
1k views

What is a local parameter in algebraic geometry?

Shafarevich offers the following theorem-definition: "At any nonsingular point $P$ of an irreducible algebraic curve, there exists a regular function $t$ that vanishes at $P$ and such that every ...
5
votes
1answer
107 views

How to prove that this kind of differential form exists on an algebraic curve?

The following is a problem in Miranda's Algebraic Curves and Riemann Surfaces. Given any algebraic curve $X$ and a point $p \in X$, show that there is a meromorphic $1$-form $\omega$ on $X$ whose ...
2
votes
0answers
106 views

Deducing characteristics of a map induced by a divisor

Given a divisor $D$ on an algebraic curve $X$, there is a corresponding map $\phi_D$ from $X$ to the projective space (of dimension $\dim L(D)-1$). In particular, we know that if $D$ is a very ample ...
5
votes
0answers
331 views

The Milnor Conjecture on the Unknotting Number of a Torus Knot

Let $f \colon (\mathbb{C}^{n},\mathbf{0}) \to (\mathbb{C},0)$ be a complex analytic function with isolated critical point at the origin. Define the singular hypersurface $V_{f, \kappa} = ...
0
votes
3answers
1k views

Intersection of Cubic curves

This is the question which i am attempting to solve, and it seems to difficult to get rid of the exponents. Show that a the two cubic curves $Y^3 = X^2 + X^3$ and $X^3 = Y^2 + Y^3$ intersect in ...
0
votes
2answers
2k views

equation of a curve given 3 points and additional (constant) requirements

Given 3 pairs of coordinates, $x_1, y_1, x_2, y_2, x_3, y_3$, I need a function $y(x)$ that will return the $y$ coordinate of any $x$ coordinate between $x_1$ and $x_3$ (it can be assumed that $x_1 ...
19
votes
4answers
2k views

intuitive explantions for the concepts of divisor and genus

when trying to explain AG-codes to computer scientists, the major points of contention i am faced with are the concepts of divisors, Riemann-Roch space and the genus of a function field. are there any ...