Configurations of \(-2\) rational curves on sectional surfaces of \(n\)-folds
Let X be an n-dimensional projective manifold. Let L be a line bundle on X which is ample and spanned by global sections. Assume that \(K_ X\otimes L^{n-1}\) is nef and big and that \(\kappa (X,K_ X\otimes L^{n-2})=n.\) It is well-known that there exists a reduction (X',L') of (X,L) such that \(K_{X'}\otimes L'{}^{n-2}\) is semiample. We classify the positive dimensional fibres of \(\phi_ n\), the morphism associated to the linear system \(| (K_{X'}\otimes L'{}^{n-2})^ N|\) for \(N\gg 0\). Such classification gives us information about the maximal connected reduced curves C on S' satisfying \(K_{S'}\cdot C=0\), where S' is the intersection of n-2 general members of \(| L'|\). We see that if dim \(X\geq 4\) then the only possible Dynkin diagram associated to C is \(A_ n\) with \(n=1, 2\).
scientific article; zbMATH DE number 3933211