Homeomorphism versus isomorphism for varieties

Abstract

It is well known that a morphism onto a weakly normal algebraic variety that is both birational and a universal homemorphism is an isomorphism of varieties. in this note, the author shows that a birational morphism onto a weakly normal algebraic variety X with a continuous inverse is necessarily an isomorphism of varieties if and only if X had no one-dimensional components. Thus the characterization of the weak normalization of a variety X as the largest variety that is birationally homeomorphic to X is only valid in the absence of one-dimensional components. All varieties are over a fixed algebraically closed field.