Algebraic characterizations of homeomorphisms between algebraic varieties

TL;DR

This paper characterizes when morphisms between algebraic varieties are homeomorphisms in both Zariski and a newly introduced strong topology, using algebraic properties like seminormalization and saturation.

Contribution

It provides algebraic criteria for homeomorphisms between algebraic varieties in different topologies, linking geometric and algebraic properties through seminormalization and rational functions.

Findings

Algebraic conditions equivalent to homeomorphisms in the Zariski topology.

Characterization of homeomorphisms in the strong topology via algebraic properties.

Interpretation of continuity in terms of rational functions on closed points.

Abstract

We address the question of finding algebraic properties that are respectively equivalent, for a morphism between algebraic varieties over an algebraically closed field of characteristic zero, to be an homeomorphism for the Zariski topology and for a strong topology that we introduce. Our answers involve a study of seminormalization and saturation for morphisms between algebraic varieties, together with an interpretation in terms of continuous rational functions on the closed points of an algebraic variety. The continuity refers to the strong topology which is the usual Euclidean topology in the complex case, whereas it comes from the theory of real closed fields otherwise.