For $C$ a (possibly singular) curve on a nonsingular projective surface $X$, let's define $C^2=deg_C(\mathcal{O}_X(C))$.
Why is it not the same as $deg_C(N_{X|C})$ when $C$ is singular?
Why do lecture notes always assume $X$ is nonsingular projective? What if it's only normal, or non-projective?
Where can I find an accessible discussion on these stuffs?
Thank you very much.