When Polynomial Multiplication Counts Geometry: Schubert Polynomials and Monk's Rule

Abstract

We give a geometric introduction to Schubert polynomial multiplication through point-line flags in the complex projective plane. Moving a prescribed point onto a prescribed line gives a family with local equation $au=λ$, whose limiting curve has two components. The incidence equations and their local multiplicities give a two-term divisor product, using the standard intersection product on classes of algebraic cycles. Permutation labels and Schubert polynomials express the same decomposition as an instance of Monk's rule. For three incidence conditions, each allowing either a point on a prescribed line or a line through a prescribed point, the product has degree six. Under explicit general-position assumptions, direct construction and local equations give six distinct solutions of multiplicity one.