From non-K\"ahlerian surfaces to Cremona group of P^2(C)

TL;DR

This paper investigates the parameter space of certain complex surfaces with global spherical shells, showing how these parameters relate to the Cremona group and birational structures on the surfaces.

Contribution

It demonstrates the effectiveness of blown-up points as parameters for surfaces with GSS and links these to the Cremona group via contracting germs.

Findings

Parameters given by blown-up points are generically effective.

Existence of a non-empty open set of surfaces with birational structures.

Surfaces with GSS can be described by contracting germs in the Cremona group.

Abstract

For any minimal compact complex surface S with n=b_2(S)>0 containing global spherical shells (GSS) we study the effectiveness of the 2n parameters given by the n blown up points. There exists a family of surfaces with GSS which contains as fibers S, some Inoue-Hirzebruch surface and non minimal surfaces, such that blown up points are generically effective parameters. These families are versal outside a non empty hypersurface T. We deduce that, for any configuration of rational curves, there is a non empty open set in the Oeljeklaus-Toma moduli space such that the corresponding surfaces are defined by a contracting germ in Cremona group, in particular admit a birational structure.