2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/72896We study relative Fourier-Mukai transforms on genus one fibrations with section, allowing explicitly the total space of the fibration to be singular and non-projective. Grothendieck duality is used to prove a skew-commutativity relation between this equivalence of categories and certain duality functors. We use our results to explicitly construct examples of semi-stable sheaves on degenerating families of elliptic curves.Major revision, clearer notation, improved presentation, simplified proofs and new examples included; 27 pagesAlgebraic Geometry18E30, 14H60, 14H20, 14J27, 14H10text