Torus actions on small blow ups of CP²

A manifold obtained by k simultaneous symplectic blow-ups of CP^2 of equal sizes epsilon (where the size of CP^1 in CP^2 is one) admits an effective two-dimensional torus action if k <= 3. We show that it does not admit such an action if k >=4 and epsilon <= 1/(3k 2^{2k}). For the proof, we correspond between the geometry of a symplectic toric four-manifold and the combinatorics of its moment map image. We also use techniques from the theory of J-holomorphic curves.