2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/114320We find a set of generators and relations for the system of extended tautological rings associated to the moduli spaces of stable maps in genus zero, admitting a simple geometrical interpretation. In particular, when the target is $\PP^n$, these give a complete presentation for the cohomology and Chow rings in the cases with/without marked points.some typos correctedAlgebraic Geometry14D20text