Structures du cube et fibres d'intersection

Abstract

We define the notion of a hypercube structure on a functor between two strictly commutative Picard categories which generalizes the notion of a cube structure on a $G_m$-torsor over an abelian scheme. We use this notion to define the intersection bundle of $n+1$ line bundles on a relative scheme $X/S$ of relative dimension $n$ and to construct an additive structure on the functor $I_{X/S}:PIC(X/S)^{n+1}\F PIC(S)$. Finally, we study a section of $I_{X/S}(L_1,...,L_{n+1})$ which generalizes the resultant of $n+1$ polynomials in $n$ variables and we interprete some classical formulas with this formalism.