Local cohomology and the connectedness dimension in algebraic varieties (Q1066217)

Let Z be a closed set of a noetherian scheme X. We define the connectedness dimension c(Z) of Z as the minimal value of dim(W), where W runs through all closed subsets of Z such that Z-W is disconnected. Using standard tools from local cohomology (Mayer-Vietoris sequences and vanishing theorems) we give bounds for c(Z) and connectedness criteria for Z. This gives a unified approach to study connectivity of intersections and of fibers. In particular one gets easy proofs for the connectedness theorem of Fulten-Hansen, and for (sharpened versions) of Zariski's connectedness theorem for projective morphisms and also for the connectedness statements of Bertini's theorems.