2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/60081The (small) quantum cohomology ring of a flag manifold F encodes enumerative geometry of rational curves on F. We give a proof of the presentation of the ring and of a quantum Giambelli formula, which is more direct and geometric than the previously known proof. Furthermore, this proof gives a relationship between quantum Schubert polynomials and universal Schubert polynomials, which arise in a degeneracy locus formula of Fulton.28 pages, LaTeXAlgebraic Geometrytext