Let $k$ be an algebraically closed field with characteristic distinct from $2$.
I want to compute the multiplicity of the intersection of the points which lie in $V_{1} \cap V_{2}$ where:
$V_{1}=V(x^{2}+y^{2}-z^{2}) \subseteq \mathbb{P}^{2}$ and $V_{2}=V(x^{2}+y^{2}-2z^{2}) \subseteq \mathbb{P}^{2}$.
Well doing the algebra shows they intersect at two points $[1:i:0]$ and $[1:-i:0]$. By Bezout's theorem we know that the sum of the multiplicities is equal to $4$ right?
Now, as I understand, to compute the multiplicity at $[1:i:0]$ we need to take a chart containing this point yes? so we can take say $x=1$ then we need to compute the dimension of the following vector space:
$k[y,z]/(1+y^{2}-z^{2},1+y^{2}-2z^{2})$
Macaulay says that the dimension is equal to $4$ but isn't this impossible? it would force that that the intersection multiplicity of the other point is zero but this is impossible because the point lies in the intersection.
What am I doing wrong?
EDIT: is the mistake that we are taking a chart which also contains the point $[1:-i:0]$?, that is, do we need to take a chart that contains $[1:i:0]$ but not $[1:-i:0]$? can we take then $x=1$ and $z=1$?