Constant scalar curvature Kähler metrics on rational surfaces

We consider projective rational strong Calabi dream surfaces: projective smooth rational surfaces which admit a constant scalar curvature Kähler metric for every Kähler class. We show that there are only two such rational surfaces, namely the projective plane and the quadric surface. In particular, we show that all rational surfaces other than those two admit a destabilising slope test configuration for some polarisation, as introduced by Ross and Thomas. We further show that all Hirzebruch surfaces other than the quadric surface and all rational surfaces with Picard rank 3 do not admit a constant scalar curvature Kähler metric in any Kähler class.