  We prove an unobstructedness result for deformations of subvarieties
constrained by intersections with another, fixed subvariety. We deduce
smoothness and expected-dimension results for multiple-point loci of generic
projections, mainly from a point or a line, or for fibres of embedding
dimension 2 or less. The current version is a substantial enhancement of the
previous one, including for the first time results on projection from a line.
