Generic residual intersections and intersection numbers of movable components (Q2436714)

Let \(K\) be a field. Based on Stückrad and Vogel's approach to an algebraic theory of intersections in \(\mathbb{P}^n_K\), given two equidimensional closed subschemes \(X,Y \subseteq \mathbb{P}^n_K\) without embedded components, one can define the intersection cycle \[ z(X,Y) = z_0 + \ldots + z_{n+1} = \displaystyle \sum_{C \in \mathcal{C}(X,Y)} j(X,Y;C) [C] \text{ on } X_L \cap Y_L, \] where \(L = K(u_{ij})\) is a purely transcendental field extension of \(K\). Subvarieties \(C \in \mathcal{C}(X,Y)\) which are not defined over \(K\) (but over \(L\)) are called movable components. The authors study the Stückrad-Vogel cycle, and give a precise description of the movable parts. To do this they work in a more general setting: let \(R\) be an equidimensional commutative noetherian ring with unit, and let \(x = x_1,\ldots,x_r\) be a sequence of elements in \(R\). For a finitely generated \(R\)-module \(M\) they define abstract cycles \(z_i^R(x,M)\) which for \(i\geq 1\) are elements of the cycle group of the extension ring \(R[u_{1,1},\ldots,u_{1,r},u_{i-1,1}\ldots,u_{i-1,r}]\). When \(R=J(X,Y)\) is the ruled join of subschemes \(X\) and \(Y\) as above, then they recover the intersection cycle \(z(X,Y)\). They define \(R\)-rational primes and primes that correspond to the movable components, and in the main theorem they describe them and they get expressions for the intersection numbers \(j(x,\mathfrak{p},M)\) in both cases. As an application of the Main Theorem, they formulate and prove a more general version of a Theorem of Bézout.