I have a question while reading Silverman's Arithmetic of Elliptic Curves. He defines the function field of an affine variety $V \subset \mathbb{A}^n(\overline{K})$ that is defined over $K$, to be the field of fractions of the coordinate ring of $V$. Then to define the function field of a projective variety $V \subset \mathbb{P}^n(\overline{K})$ defined over $K$, you intersect $V$ with one of the standard affine patches sitting inside $\mathbb{P}^n(\overline{K})$, identify that with a subset of $\mathbb{A}^n(\overline{K})$, and then take the function field of that.
What I don't understand is that he says that the function field of $\mathbb{P}^n$ may also be described as the subfield of $\overline{K}(X_0,…,X_n)$ consisting of rational functions $F(X)=f(X)/g(X)$ for which $f$ and $g$ are homogeneous polynomials of the same degree. For example, let's take $K=\mathbb{R}$. Then $\overline{K}=\mathbb{C}$. Let's take $V=\mathbb{P}^2(\mathbb{C})$. According to Silverman, to find the function field of $V$ one intersects $V$ with say, the copy $[x_0:x_1:1]$ of $\mathbb{A}^2(\mathbb{C})$. Then $V \cap \mathbb{A}^2$ is $\mathbb{A}^2(\mathbb{C})$, so the coordinate ring is $\mathbb{C}[X,Y]$, and the function field of $V$ is $\mathbb{C}(X,Y)$. Then apparently this can be identified with rational functions $f(X,Y,Z)/g(X,Y,Z)$ where $f$ and $g$ are homogeneous and have the same degree. I guess given a polynomial in $\mathbb{C}[X,Y]$ I can homogenize it to get a homogeneous polynomial in $X,Y,$ and $Z$, but when I take something in the field of fractions $\mathbb{C}(X,Y)$, how do I get it in the desired form?
Best Answer
Look at the map on the coordinates: $x$ is sent to $X/Z$, etc. So just map this way. E.g. $x^2/y\rightarrow(X/Z)^2/(Y/Z)=X^2/YZ$. The result is guaranteed to be homogeneous, as the individual terms $x,y$ are all sent to elements with degree $0$ (i.e. $X/Z, Y/Z$).