Polynomial Mappings with Small Degree

Abstract Let Xn be an affine variety of dimension n and Yn be a quasi-projective variety of the same dimension. We prove that for a quasi-finite polynomial mapping ƒ : Xn → Yn, every non-empty component of the set Yn\ ƒ (Xn) is closed and it has dimension greater or equal to n μ(ƒ), where μ(ƒ) is a geometric degree of ƒ. Moreover, we prove that generally, if ƒ : Xn → Yn is any polynomial mapping, then either every non-empty component of the set is of dimension ≥ n μ(ƒ) or ƒ contracts a subvariety of dimension ≥ n μ(ƒ) + 1.