Cube structures and intersection bundles

We define the notion of a hypercube structure on a functor between two commutative Picard categories which generalizes the notion of a cube structure on a Gm-torsor over an abelian scheme. We prove that the determinant functor of a relative scheme X/S of relative dimension n is canonically endowed with a (n+2)-cube structure. We use this result to define the intersection bundle IX/S(L1,…,Ln+1) of n+1 line bundles on X/S and to construct an additive structure on the functor IX/S:PIC(X/S)n+1→PIC(S). Then, we construct the resultant of n+1 sections of n+1 line bundles on X, and the discriminant of a section of a line bundle on X. Finally we study the relationship between the cube structures on the determinant functor and on the discriminant functor, and we use it to prove a polarization formula for the discriminant functor.