2
votes
0answers
35 views

Rational function on a noetherian scheme

Let $X$ be a noetherian scheme and $f$ a rational function on $X$, so by definition the domain of $X$ includes all associated points of $X$. I think the following is true: $f$ is regular on $X$ if and ...
0
votes
1answer
33 views

Rational parametrization of points on sphere of irrational radius

I am trying to figure out if a closed form parametrization can exist for finding all of the rational points on a sphere of radius $\sqrt{2}$. ie, x=f(u,v), y=g(u,v), and z=h(u,v) where f,g,and h are ...
1
vote
1answer
30 views

Ring of rational functions ideal generators

There is an affine variety $X\subset \mathbb{A}^n$ with its ring of rational functions which is the quotient ring of $\mathbb{k}[X]$ (each $f\in \mathbb{k}(X)$ has a form $\frac{p}{q}$ where $q$ does ...
0
votes
1answer
34 views

Rational locus of a function defined on $x^2+x^3=y^2$

We have a curve $X$ on $\mathbb{A}^2$ given by $y^2=x^2+x^3$. Consider the rational function $f$ on $X$ which maps $(x,y)\in X$ to $\frac{y}{x}$. There is a nice geometric interpretation of $f$: if we ...
1
vote
1answer
55 views

Equality of rational functions as functions and it as rational functions.

Let $k$ be an algebraic closed field, $\mathbb{A}^n$ be an affine variety over $k$, $U$ be open set in $\mathbb{A}^n$ and $f,g\in k(x_1,...,x_n)$ and their denominators is not zero over $U$.If ...
3
votes
1answer
56 views

Is this parabola the same as this circle?

Let $K$ be a field of characteristic not 2. Is there an invertible rational function between $\mathbb{P}^1(K) = \{ [s:t] : s,t \in K, (s,t) \neq (0,0) \}$ and $V= \{ [x:y:z] : 2xy+z^2 = 0 \}$? Here ...
4
votes
3answers
105 views

When are the sections of the structure sheaf just morphisms to affine space?

Let $X$ be a scheme over a field $K$ and $f\in\mathscr O_X(U)$ for some (say, affine) open $U\subseteq X$. For a $K$-rational point $P$, I can denote by $f(P)$ the image of $f$ under the map ...
2
votes
3answers
270 views

Zeros and poles of rational functions on locally Noetherian schemes

Let $X$ be a locally Noetherian scheme and let $f$ be a rational function on $X$ (i.e. the equivalence class of a pair $(U,f)$, where $f \in \mathcal{O}_X(U)$ and $U$ contains the associated points of ...
4
votes
2answers
126 views

Connecting the intuitive way to compute divisors of rational functions to the rigorous definition.

Let $X$ be the curve $xy-z^2 \subset \mathbb{P}^2$, and let $f$ be the rational function $x/y$ (Edit: I'm trying to simplify this as much as possible, but of course $x$ itself isn't a rational ...
2
votes
2answers
67 views

Factorizing rational functions of curves

Let $f:X\to \mathbf{P}^1$ be a rational function of degree $d\geq 2$ on a curve $X$. Let $n\geq 2$ be a divisor of $d$. Does there exist a curve $Y$ with a rational function $g:Y\to \mathbf{P}^1$ of ...
6
votes
1answer
185 views

Does every curve over a number field have infinitely many rational functions of fixed degree

Let $X$ be a curve over a number field $K$ of genus $g\geq 2$. Does there exist an integer $d$ such that $X$ has infinitely many rational functions (i.e., finite morphisms $f:X\to \mathbf{P}^1_K$) of ...
10
votes
2answers
1k views

Luroth's Theorem

I have just begun to read Shafarevich's Basic Algebraic Geometry. In the first section of the first chapter, he quotes Luroth's theorem, which states that any subfield of $k(x)$ that is not just $k$ ...