2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/74018Type III unprojection plays a very important role in the birational geometry of Fano threefolds. It was first introduced by A. Corti on his calculations of Fano threefolds of genus 6 and 7. It seems that at present a general definition of the type III unprojection is still missing. After proving some general facts about residual ideals, we propose a definition for the generic Type III unprojection, and prove that it gives a Gorenstein ring.11 pages, to appear in Communications in AlgebraAlgebraic GeometryCommutative Algebratext