Let $Z(f)$ be the zero set of a degree $d$ homogeneous polynomial $f\in\mathbb{C}[x_0,x_1,x_2]$, then it defines a submanifold in $\mathbb{CP}^2$. How can we compute its intersection index with the curve $\mathbb{CP}^2$?
My current definition of local intersection index is based on local orientation (e.g Let $u_i$ $v_j$ be the local basis of curves $U$, $V$ at a point, check if $u_1,...,v_1,...$ differs from the local basis of $\mathbb{CP}^2$ by an even or odd permutation, then the local index is 1 or -1 correspondingly). Please avoid the dimension of quotient definition in algebraic geometry if possible.
My guess is that they would intersect $d$ times (after some local operation, like translating, that maximizes intersection points) and local index $1$ at every intersection point, but I have no idea how to start the proof.