2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/79418We characterize contractible curves on proper normal algebraic surfaces in terms of complementary Weil divisors. Using this we generalize the classical criteria of Castelnuovo and Artin. As application we derive a finiteness result on homogeneous spectra defined by Weil divisors on surfaces.20 pages, submitted to J. Reine Angew. MathAlgebraic Geometry14C20, 14E10, 14J05, 14J17text