Monodromy and Connectedness

The purpose of this note is initially to present an elementarybut surprising connectedness principle pertaining to the intersectionof a fixed subvariety X of some ambient space Z with anothersubvariety Y which is ‘mobile’ (in the sense ofbeing movable, rather than actually moving). It is via thismobility that monodromy enters the picture, permitting the crucialpassage from ‘relative’ or total-space irreducibilityto ‘absolute’ or fibrewise connectedness (and sometimesirreducibility). A general form of this principle is given inTheorem 2 below. 1991 Mathematics Subject Classification 14C99,15N05.