ag.algebraic geometry - Intersection numbers and theorem of the cube

On a smooth projective surface, the intersection number of two divisors $C,D$ may be defined as $chi(O)-chi(O(C))-chi(O(D))+chi(O(C+D))$ (Hartshorne, ex. V.1.1). Bilinearity of the intersection number translates to the formula $sum_{Isubset{1,2,3}}(-1)^{|I|}chi(O(sum_{iin I}C_i))=0$ which is somewhat similar to the theorem of the cube in the form $bigotimes_{Isubset{1,2,3}}m_I^*mathcal L^{(-1)^{|I|}}cong O$. Is this just a coincidence, or is there any deeper reason for this similarity?