Blowings-up of \({\mathbb{P}}^ 2\) and their blowings-down (Q1065888)

For a smooth algebraic compact surface V defined over an algebraic closed field, and dominating \({\mathbb{P}}^ 2\), one defines an exceptional configuration as a \({\mathbb{Z}}\)-free basis of Pic V. Further, one defines a root system with simple roots, and a Weyl group W acting on the orthogonal of the canonical class. The author proves: if rk Pic \(V\geq 10\), then V has finitely many blowing-downs to \({\mathbb{P}}^ 2\) if and only if V has finitely many effective, irreducible W-conjugates of the simple roots, and they span a submodule of maximal rank. - Previous works in this direction were done by M. Nagata and E. Looijenga.