Rational Maps $\mathbb{P}^n \rightarrow \mathbb{P}^m$

Question

In general, any rational map $\mathbb{P}^n \rightarrow \mathbb{P}^m$ can be given by an $m+1$-tuple of polynomials of the same degree, with no common factor.

Can anyone give me a reference or a proof of this fact?

Answer 1

This follows from the following four facts.

Fact 1: If $X$ is a normal variety (e.g. $\mathbb{P}^n$), then a rational map $X\to\mathbb{P}^m$ can be extended over codimension 1, so that it is defined away from a closed subvariety of codimension at least 2 in the target.

Fact 2: A map $\pi:X\to\mathbb{P}^n$ is given by $[s_0:\ldots:s_n]$, where the $s_i$ are global sections of a line bundle $\mathcal{L}\cong\pi^*\mathcal{O}_{\mathbb{P}^n}(1)$ on $X$, and where the $s_i$ do not simultaneously vanish at any point of $X$.

Fact 3: If $X$ is integral and locally factorial (e.g. $X=\mathbb{P}^n$) and $U\subset X$ is an open subvariety whose complement has codimension at least 2, then the restriction map $Pic(X)\to Pic(U)$ is an isomorphism. Moreover, a global section of a line bundle on $U$ may be extended uniquely to a global section of the corresponding line bundle on $X$.

Fact 4: The line bundles on $\mathbb{P}^n$ are those of the form $\mathcal{O}(d)$, whose global sections are homogeneous polynomials of degree $d$ in $n+1$ variables.

Now, suppose we have a rational map $\mathbb{P}^n\rightarrow\mathbb{P}^m$. We may then assume it is given by an honest morphism $U\to\mathbb{P}^m$, where $U$ is an open subvariety of $\mathbb{P}^n$, and moreover assume the complement of $U$ has codimension at least 2, by fact 1. By fact 2, this map takes the form $[s_0:\cdots:s_n]$, where the $s_i$ are global sections of some line bundle on $U$. By fact 3, these global sections uniquely extend to global sections of a corresponding line bundle on $\mathbb{P}^n$, and thus by fact 4 the $s_i$ may be taken to be homogeneous polynomials of some degree $d\ge0$. It is not hard to see that we can assume these don't have a non-constant common factor.