2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/71452We rederive a relation between the genus-one GW-invariants of a quintic threefold in $\Pf$ and the genus-zero and genus-one GW-invariants of $\Pf$. In contrast to the more general derivation in a separate paper, the present derivation relies on a widely believed, but still unproven, statement concerning rigidity of holomorphic curves in Calabi-Yau threefolds. On the other hand, this paper's derivation is more direct and geometric. It requires a bit more effort, but relies on less outside work.55 pages, 5 figuresAlgebraic GeometrySymplectic Geometry14N35, 53D45text