2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/76308We classify codimension two analytic submanifolds X of projective space having the property that any line through a general point p having contact to order two with X at p automatically has contact to order three. We give applications to the study of the Debarre--de Jong conjecture, and of n-dimensional varieties whose Fano variety of lines has dimension 2n-4.v.1, 11 pagesAlgebraic GeometryDifferential Geometry14N15; 51N15; 53A20text