2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/123335Zero-schemes on smooth complex projective varieties, forcing all elements of ample and free linear systems to be reducible are studied. Relationships among the minimal length of such zero-schemes, the positivity of the line bundle associated with the linear system, and the dimension of the variety are established. A generalization to higher dimension subschemes is studied in the last section.23 pages. Refereed version, to appear in JPAAAlgebraic Geometrytext