Define the Euler characteristic of a scheme to be the Euler characteristic of its structure sheaf. I remember being told that for curves, this invariant satisfies inclusion-exclusion. That is, if $C_1, C_2$ are curves , then
$$\chi(C_1 \cup C_2) + \chi(C_1\cap C_2) = \chi(C_1) + \chi(C_2)$$
The intersection is scheme theoretic.Does any one knows a proof or a reference for this ?