A characterization of proper regular mappings (Q2724126)

Let \(X\), \(Y\) be complex affine varieties and \(f: X\to Y\) a regular morphism. It follows from the constructibility theorem of Chevalley that if \(f\) is proper in the classical topology then \(f\) is closed in the Zariski topology. The aim of the present paper is to prove the converse:NEWLINENEWLINE If \(\dim X\geq 2\) and \(f\) is a non-constant closed morphism, then \(f\) is proper in the classical topology.NEWLINENEWLINE The crucial role in the proof plays the Łojasiewicz exponent at infinity of regular mappings on algebraic sets and a theorem on selection of an algebraic curve on which the Łojasiewicz exponent at infinity is attained.