2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/97534The Castelnuovo-Mumford regularity of varieties of degree r and dimension n in the r-dimensional projective space that have an extremal secant line, is at least d-r+n+1. We classify these varieties and show that their regularity is exactly d-r+n+1, as predicted by the regularity conjecture.12 pages, part of it contains results from my Ph.D. thesis (Columbia University NY)Algebraic Geometry14N05;14F05text