I'm reading Fulton's algebraic curves and I have questions on page 46:
4
I know these fact are very basic, but I didn't understand why no elements of $\Gamma_h(V)$ can not be regarded as functions even when they are forms with the same degree.
See:
let $f,g$ be forms of the same degree, then
$\bar f=\bar g\Leftrightarrow f+I(V)=g+I(V)\Leftrightarrow f-g\in I(V)\Leftrightarrow (f-g)(x)=0\text{ for every $x$ in V}\Leftrightarrow f(x)=g(x)\text{ for every $x$ in V}\Leftrightarrow f=g$
So, there is an isomorphism between the polynomial functions from $V$ and $\Gamma (V)$ given by $f\mapsto f+I(V)$ in the affine case. Is there the same in the case I mentioned.
What am I missing?
Thanks in advance
$\def\Spec{\mathop{Spec}}$
I think you may be a little confused about what a regular function actually is. For an arbitrary scheme $S$, the set of regular functions can be obtained by covering $S$ by affines $U_{i} = \Spec R_{i}$ and choosing for each $i$, regular functions $\alpha_{i} \in R_{i}$ on the open subsets $U_{i}$ such that
$$\alpha_{i}|_{U_{i} \cap U_{j}} = \alpha_{j}|_{U_{i} \cap U_{j}}.$$
So let's see why the homogeneous coordinate ring of projective space does not give you functions except in the constant case. Take $\mathbb{P}^{1}_{k}$ with homogeneous coordinate ring $k[x, y]$. Then, we can cover it by affines $U_{0}, U_{1}$ where
$$U_{0} = \Spec k[y/x], U_{1} = \Spec k[x/y]$$
are the open affine subsets where the homogenous coordinate $x$ and $y$ are respectively nonzero. Now, choose some $\alpha \in k[y/x]$ and $\beta \in k[x/y]$. Then, this collection determines an element of the coordinate ring of $\mathbb{P}^{1}$ if and only if
$$\alpha|_{U_{0}\cap U_{1}} = \beta|_{U_{0}\cap U_{1}}$$
which just says that $\alpha = \beta$ in $k[y/x, x/y].$ This is only possible if $\alpha$ and $\beta$ are constant.
Here's another way to think about the problem. A regular function on $\mathbb{P}^{1}$ is just a function from $\mathbb{P}^{1}$ to $\mathbb{A}^{1}$. Let's assume we're over the complex field. Then this is a map that sends
$$[x : y] \mapsto f(x, y)$$
where $f$ is a polynomial in the $x$ and $y$. However, note that even if $f$ is homogeneous, for the map to be well-defined, rescaling $x$ and $y$ cannot do anything. This is only possible if $f$ is a scalar.
As a third remark, you will eventually see that the homogeneous polynomials are actually global sections of some geometric object on the projective variety. Instead of being global sections of the structure sheaf (i.e. a regular function), they are actually global sections of line bundles.