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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 $\sum_{I\subset\{1,2,3\}}(-1)^{|I|}\chi(O(\sum_{i\in I}C_i))=0$ which is somewhat similar to the theorem of the cube in the form $\bigotimes_{I\subset\{1,2,3\}}m_I^*\mathcal L^{(-1)^{|I|}}\cong O$. Is this just a coincidence, or is there any deeper reason for this similarity?

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  • $\begingroup$ Both are Moebíus-inversion-ish sums/products. $\endgroup$ Commented Mar 2, 2010 at 19:53

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The reason for both is that the third derivative of a (nonhomogeneous) quadratic function is zero.

$\chi(O(D))$ on a surface is a quadratic function of $D$, by Riemann-Roch.

Theorem of cube on an abelian variety is another face of the fact that the Riemann theta funciton $$\theta(z,\tau)=\sum_m \exp(2\pi i(m^2\tau/2+ mz))$$ written as a sum of eigenfunctions $\sum_m c_m q^m$, $q=\exp(2\pi i z)$, has coefficients $c_m$ which are multiplicatively quadratic in $m$.

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