A question to the exercise I.5.4(c) Hartshorne Much before Bezout's theorem the following exercise is given:
If $Y$ is a curve of degree $d$ in ${\mathbb{P}}^2$ and if $L$ is a line in ${\mathbb{P}}^2$, $L \neq Y$, show that $(L \cdot Y) = d$. The definition of $(L \cdot Y)$ is given by $(L \cdot Y) = \sum (L \cdot Y)_P$ taken over all points $P \in L \cap Y$, where $(L \cdot Y)_P$ is defined using a suitable affine cover of ${\mathbb{P}}^2$.
Of course the definition $(L \cdot Y)_P$ in an affine space is simply $\ell_{\mathcal{O}_P}({\mathcal{O}_P}/{\langle f, g \rangle})$, where $L = Z(f), Y = Z(g)$. And even it is easy to verify $\ell_{\mathcal{O}_P}({\mathcal{O}_P}/{\langle f, g \rangle}) = {\mathrm{dim}}_k~{\mathcal{O}_P}/{\langle f, g \rangle}$.
The solution can be found here : Problem I.5.4(c) in Hartshorne and to the sources quoted as well.
My question: is the term $(L \cdot Y)_P$ depends on the choice of the affine cover we take?
My perception is "yes" as the equation need not be symmetric towards the dehomogenization taken for the corresponding cover. However, I can't see get a precise counterexample.
Here is another reason why I am asking this: the answer to the exercise begin with the assumption that $(L \cdot Y)_P$ is invariant under global linear change and hence we assume $L = Z(y)$. However, does dehomogenization respect this global linear change?
 A: No, the quantity $(L\cdot Y)_P$ does not depend on the open neighborhood of $P$ one uses to calculate this. Recall Hartshorne's definition of $\mathcal{O}_{P,X}$: this is the ring of equivalence classes $(U,f)$ where $U\subset X$ is an open neighborhood of $P$ and $f$ is a regular function defined on $U$ where we say that $(U,f)\sim (V,g)$ if $f-g=0$ on $U\cap V$. It is trivial to see that if we instead calculate this from some open $X'\subset X$ which contains $P$, we still get the same ring: just pick representatives of each equivalence class which are entirely contained inside $X'$.
It is also not hard to see that an automorphism $\varphi:X\to X$ induces an isomorphism of $\mathcal{O}_{P,X}$ with $\mathcal{O}_{\varphi(P),X}$: this is Hartshorne exercise I.3.3(b), for instance. So since a global linear change of coordinates is an automorphism, we get that the quantity described is invariant under such actions (because it's an isomorphism invariant). In short: everything is alright and all the statements you're worried about are okay.
