Unobstructedness of filling secants and the Gruson-Peskine general projection theorem

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.