Intersections of projective varieties and generic projections
Let \(X, Y \subset {\mathbb{P}}^N\) be closed subvarieties of dimensions \(n\) and \(m\), respectively. \textit{J. Stückrad} and \textit{W. Vogel} [in: The curves seminar at Queen's, Vol. 2, Kingston 1981-82, Queen's Pap. Pure Appl. Math. 61, Exposé A (1982; Zbl 0599.14003)] introduced cycles \(v_k = v_k (X,Y)\) of dimension \(k\) on \(X \cap Y\) and \(\beta_k\) on the ruled join variety \(J(X,Y)\) of \(X\) and \(Y\) which are obtained by a simple algorithm. These cycles were important in their proof of a Bezout theorem for improper intersections. In this paper the authors give an interpretation of these cycles in terms of generic projections \(p_k:{\mathbb{P}}^N \to{\mathbb{P}}^{n+m-k-1}\). To do this they introduce a relative ramification locus \(R(p_k, X, Y)\) of \(p_k\) which is of dimension at most \(k\) and generalizes the usual ramification cycle in the case \(X=Y\). The main result of the paper is that for \(0 \leq k \leq \dim X \cap Y -1\), this cycle is just \(v_k\) on the smooth locus of the pair \((X,Y)\) (i.e.\ the points of \(X \cap Y\) where \(X\), \(Y\) and \(X \cap Y\) are smooth). Moreover, the cycles \(\beta_{k+1}\) (for \(-1 \leq k \leq \dim X \cap Y -1\)) may be interpreted geometrically as the cycle of double points of \(p_k\) associated to the closure of the set of all \((x:y)\) in the ruled join such that \((p_k (x):p_k (y))\) is in the diagonal \(\Delta_{{\mathbb{P}}^{n+m-k-1}}\) of \(J({\mathbb{P}}^{n+m-k-1}, {\mathbb{P}}^{n+m-k-1})\). Along the way, the authors prove some results about generic residual intersections.
scientific article; zbMATH DE number 1002954