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 ...
16
votes
4answers
1k 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 ...
0
votes
0answers
68 views
A problem of irreducibility
Consider the cone in $\mathbb{C}^3$ over the curve $\{y^2-x^2-x^3=0\}\subset\mathbb{C}^2$. Show that $g=zy^2-zx^2-x^3$ with $g = 0$ giving the cone, is irreducible in $\mathcal{O}_{\mathbb{C}^3,~0}$, ...
15
votes
3answers
606 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 ...
9
votes
2answers
220 views
Can there be a point on a Riemann surface such that every rational function is ramified at this point?
Let $X$ be a compact connected Riemann surface, and let $S\subset X$ be a finite subset.
Does there exist a morphism $f:X\to \mathbf{P}^1(\mathbf{C})$ which is unramified at the points of $S$?
I'm ...
6
votes
5answers
222 views
For $(x+\sqrt{x^2+3})(y+\sqrt{y^2+3})=3$, compute $x+y$ .
If $(x+\sqrt{x^2+3})(y+\sqrt{y^2+3})=3$, compute $x+y$.
5
votes
1answer
236 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 $f \colon M \to \mathbb{P}^1_{/\mathbb{C}}$, ...
0
votes
2answers
834 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 ...
5
votes
1answer
170 views
Basis for the Riemann-Roch space $L(kP)$ on a curve
Let $C$ be a smooth algebraic plane complex curve of degree $d$ defined as the zero locus of a homogeneous polynomial $F$.
Let $P\in C$. For a positive integer $k$ consider the divisor
$$D=kP$$
...
6
votes
2answers
138 views
Irreducible components of the variety $V(X^2+Y^2-1,X^2-Z^2-1)\subset \mathbb{C}^3.$
I want to find the irreducible components of the variety $V(X^2+Y^2-1, \ X^2-Z^2-1)\subset \mathbb{C}^3$ but I am completely stuck on how to do this. I have some useful results that can help me ...
3
votes
1answer
264 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.
2
votes
1answer
101 views
writing down the minimal discriminant of an elliptic curve
Let $j$ be an integer.
Does there exist an elliptic curve $E_j$ over $\mathbf{Q}$ with $j$-invariant equal to $j$ whose minimal discriminant we can write down in a practical way?
For example, can ...
2
votes
1answer
73 views
Homogenous polynomials
In section 3.1 (3rd paragraph on page 4) in this paper, I cannot understand why $Q$ and $R$ are homogeneous:
(Given $A$, $B$ are homogenous. Capital letters denote homogenous polynmials.)
2
votes
1answer
73 views
Is the intersection of the diagonal with a graph always transverse in characteristic zero
Let X be a projective smooth connected curve over $\mathbf{C}$. Let $f:X\to X$ be a non-constant morphism.
Is the intersection of the diagonal $\Delta_X$ and the graph $\Gamma_f$ on $X\times X$ ...
1
vote
1answer
53 views
Transforming a Continuous Function
My math is quite limited so please bear with me. I will get to the point:
Is there a way to transform a continuous function into a bounded one?
In essence I have a normalized Gaussian distribution ...
0
votes
1answer
57 views
Representing a curve as a plane curve in different ways
Let $X$ be a (smooth projective geometrically connected) curve over $k$, where $k$ is a field of characteristic zero. We assume $X$ has genus at least $2$.
I know that $X$ has a plane model. More ...
