Cremona transformations and diffeomorphisms of surfaces

TL;DR

This paper demonstrates that Cremona transformations can replicate the full complexity of surface diffeomorphisms, showing that algebraic automorphisms are dense in the smooth diffeomorphism groups for rational surfaces.

Contribution

It establishes the density of algebraic automorphisms in the diffeomorphism groups of rational surfaces, linking algebraic and smooth surface symmetries.

Findings

Cremona transformations exhibit full diffeomorphism complexity on real quadrics.

For rational surfaces, algebraic automorphisms are dense in the smooth diffeomorphism group.

The results connect algebraic and differential topology of surfaces.

Abstract

We show that the action of Cremona transformations on the real points of quadrics exhibits the full complexity of the diffeomorphisms of the sphere, the torus, and of all non-orientable surfaces. The main result says that if X is rational, then Aut(X), the group of algebraic automorphisms, is dense in Diff(X), the group of self-diffeomorphisms of X.