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We prove an unobstructedness result for deformations of subvarietiesconstrained by intersections with another, fixed subvariety. We deducesmoothness and expected-dimension results for multiple-point loci of genericprojections, mainly from a point or a line, or for fibres of embeddingdimension 2 or less. The current version is a substantial enhancement of theprevious one, including for the first time results on projection from a line.

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