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From non-Kählerian surfaces to Cremona group of \(\mathbb{P}^2(\mathbb{C})\). (English) Zbl 1320.32022
Summary: 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 \(\mathcal{S} \to B\) 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 \(T \subset B\). 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.

MSC:
32J15 Compact complex surfaces
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